The local limit theorem on nilpotent Lie groups
Timothée BénardThe first author has received funding from the European Research
Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No. 803711).  and Emmanuel Breuillard

We establish the (non-lattice) local limit theorem for products of i.i.d. random variables on an arbitrary simply connected nilpotent Lie group $G$, where the variables are allowed to be non-centered. Our result also improves on the known centered case by proving uniformity for two-sided moderate deviations and allowing measures with a moment of order $2(\dim G)^2$ without further regularity assumptions. As applications we establish a Ratner-type equidistribution theorem for unipotent walks on homogeneous spaces and obtain a new proof of the Choquet-Deny property in our setting.

§ INTRODUCTION

Given a Lie group $G$ and a probability measure $\mu$ on $G$, we may ask whether there are sequences $a_n>0$ and $g_n \in G$ such that the measures
\begin{equation}\label{llconv} a_n \,\mu^{*n} * \delta_{g_n}\end{equation}
converge weakly towards some non-trivial Radon measure on $G$. The measure $\mu^{*n} * \delta_{g_n}$ is the law of the random product $X_{1}\ldots X_{n} g_n$, where $X_1,\ldots,X_n$ are independent random variables with law $\mu$. An estimate of this kind is called a local (or local central) limit theorem (LLT for short) on $G$. Such results are known for various kinds of Lie groups (see the historical remarks at the end of the introduction in <Ref>). In this paper we shall answer this question affirmatively for the class of simply connected nilpotent Lie groups. These groups are defined algebraically as those simply connected real Lie groups $G$ such that $G^{[s+1]}=\{1\}$ for some finite integer $s$, where $G^{[i+1]}:=[G,G^{[i]}]$, $G^{[1]}=G$, is the central descending series.  They can also be characterized precisely as the connected subgroups of unipotent upper triangular matrices (see <cit.>).

In the abelian case $G=\R^d$, the local limit theorem is a classical result [38, 18]. Given a sum of $\R^d$-valued independent random variables $S_n=\sum_{k=1}^nX_{k}$ with common law $\mu$, drift $\chi=\E(X_1)$ and covariance the identity, we have as $n\to+\infty$,

\begin{equation}\label{classicalLLT}
(2\pi n)^{d/2} \,\mathbb{P}(S_{n}-n\chi \in B) =|B|+o_{\mu,B}(1),
\end{equation}

where $B$ is any box  $B=\prod_{i=1}^d[s_i,t_i]$ of Euclidean volume $|B|$,
provided $\mu$ is non-lattice (i.e. the support of $\mu$ is not contained in a coset of a proper closed subgroup of $\R^d$). A uniform version of the local limit theorem is also available [84]:

\begin{equation}\label{classicalLLT-uniform}
\, \sup_{x \in \R^d} |\mathbb{P}(S_{n}-n\chi \in B+x) - (\sqrt{n}_{\star}\mathscr{N})(B+x)|  = o_{\mu,B}(n^{-d/2}),
\end{equation}

where $\sqrt{n}_{\star}\mathscr{N}$ is the push-forward by the homothety $x\mapsto \sqrt{n} \,x$ of  the standard  Gaussian distribution $\mathscr{N}:= (2\pi )^{-d/2}e^{-\frac{\|x\|^2}{2}}dx$ on $\R^d$.

In the context of non-commutative nilpotent Lie groups, the question of a local limit theorem has been investigated by the second-named author [21] and by Diaconis and  Hough [29, 49], under the assumption that the measure $\mu$ is centered in the abelianization $G/[G,G]$. Both [21] and [29] deal with the case where the ambient group is the Heisenberg group, that is the simplest instance of a non-commutative nilpotent Lie group, while [49] contains a generalization to arbitrary simply connected nilpotent Lie groups. As in [21, 49], we will focus on the non-lattice (equivalently aperiodic) case: we work under the assumption (see the remarks in <Ref>) that the projection $\mu_{ab}$ of $\mu$ under the abelianization map $G \to G/[G,G]$ is not supported on a coset of a proper closed subgroup of $G/[G,G] \simeq \R^d$.

[Heisenberg group]  Let $G$ be the $3$-dimensional Heisenberg group, i.e.  $\R^3$ endowed with the group structure $x*y=(x_{1}+y_{1}, x_{2}+ y_{2}, x_{3}+y_{3}+\frac{1}{2}(x_{1}y_{2}-x_{2}y_{1})$. Let  $\mu$ be a probability measure on $G$ whose support is compact  and has  non-lattice projection in $\R^2$. Assume that $\mu$ is centered in the sense that $\E_{\mu}(x_{1})=\E_{\mu}(x_{2})=0$.  It was shown in [21] that for every  compactly supported  continuous function $f\in C_{c}(G)$, as $N\to+\infty$,

\begin{equation} \label{LLT-Heis-1}
N^{2}\mu^{*N}(f) \rightarrow c_{\mu} \int_{G} f dx
\end{equation}

where $c_{\mu}>0$ only depends on $\mu$, and where $dx$ refers to the Lebesgue measure (or Haar measure) on $G$.

As in the  abelian case, a uniform version of the LLT is also available. For $r>0$, set $D_r:x \mapsto (x_1 r, x_2 r, x_3 r^2)$.
There exists a smooth probability measure $\nu$ on $G$ such that for every  $f\in C_{c}(G)$, for $N\geq1$,

\begin{equation} \label{LLT-Heis-2}
\mu^{*N}(f)= D_{\sqrt{N}}(\nu)(f) + o_{\mu,f}(1)N^{-2}
\end{equation}

where the term $o_{\mu,f}(1)$ goes to $0$ uniformly in the multiplicative translates of $f$, i.e. for the family of test functions $f_{g,h}: x \mapsto f(gxh)$ where $g,h\in G$. This was shown in [21] (one sided uniformity only) where the two-sided uniformity required an additional Cramér assumption on $\mu$, an assumption subsequently lifted in [29].

Note that <ref> implies that $\nu$ is unique and coincides with the limiting measure in the central limit theorem, in other terms $D_{\frac{1}{\sqrt{N}}}(\mu^{*N}) \rightarrow \nu$ weakly (see [28, 9]). Combining (<ref>)  and (<ref>) we see that  $c_{\mu}=u(0)$, where $u$ is the (smooth) density of $\nu$ with respect to $dx$. This density is nowhere-vanishing.  In fact $\nu$ corresponds to the law of the value at time $t=1$ of a hypoelliptic diffusion $(B^{(1)}_t,B^{(2)}_t,L_t)$ on $G$, where $(B^{(1)}_t,B^{(2)}_t)$ is a planar Brownian motion and $L_t$ is its Lévy area (see e.g. [36]). A well-known formula due to P. Lévy [62] allows to compute the Fourier transform of $\nu$ (see <cit.>, <cit.>), from which it follows that:

$$c_\mu= \frac{1}{4} \frac{1}{|\det{\Cov_\mu(x_1,x_2)}|^{\frac{1}{2}}}$$

where $\Cov_\mu(x_1,x_2)$ is the covariance matrix of the pair $(x_1,x_2)$ distributed according to $\mu_{ab}$. Note that this is insensitive to the mean of the central coordinate $\E_\mu(x_3)$.

The proof of the LLT for centered walks on the Heisenberg group obtained in [21] relied on Fourier methods combined with certain spectral gap estimates for infinite dimensional irreducible unitary representations of the Heisenberg group. In a recent breakthrough [29], Diaconis and Hough came up with a new idea that allowed them to bypass these estimates. They exploit the exchangeability of the random product under permutations of the variables in order to produce further smoothing in the central coordinate. This yields the desired decay of the Fourier transform.
With this new “path-swapping” technique they could obtain a new proof of the LLT for the Heisenberg group with improved (i.e. double-sided) uniformity. Quite remarkably, Hough [49] was able to further these ideas and managed to prove the LLT for non-lattice centered walks on arbitrary nilpotent Lie groups. He also obtained the uniform LLT under the additional assumption that $\mu$ satisfies a strong form of aperiodicity (Cramér condition).

In the first part of this paper,  we push the Diaconis-Hough path-swapping method   one step further and show how it can be adapted to prove the local limit theorem on nilpotent Lie groups for non-centered (we also say biased) walks as well. Our result is the first of this kind, as all previous works on the nilpotent LLT assume the random walk to be centered.   It is worth pointing out  that the study of random walks on a non-abelian nilpotent  Lie group does not reduce to  the centered case. More precisely, after recentering, such a random walk may be seen as a process with centered but time-dependent driving measure. There are at least two significant consequences. First,  the distribution $\mu^{*N}$ spreads at a faster rate around its average than  in the centered case. In fact, the large scale geometry of $\mu^{*N}$ is not captured by the descending central series (as for centered walks) but by some other filtration with smaller ideals,  which depends on the increment average.  We will need to work both with the descending central series (to exploit the bracket structure of $G$) and with this new filtration. The second novelty appearing in the non-centered case is that the limiting measure in the CLT is typically not of full support (even though it is absolutely continuous). This is a new phenomenon that came as a surprise in our previous work [9] and forces some changes in the very formulation of the LLT in the non-centered case. In fact, the recentering sequence $g_n$ needs to be fine-tuned to remain within the range of the support of the limiting measure for the LLT to be meaningful, see <Ref> below.

The limit theorems proven in this paper are the following.  <Ref> is the local limit theorem for non-lattice nilpotent random walks. It  allows the driving measure to be non-centered. It also provides two-sided moderate uniformity without a Cramér assumption, which is new even for centered walks, and useful in applications. We then deduce <Ref>, which answers affirmatively the question raised above at <Ref> for non-lattice driving measures. For centered walks, we also manage to show a uniform  limit theorem for ratios, <Ref>. Such estimates are new even in the Heisenberg case.  The proof of Theorems <ref> and  <ref>   unifies both the centered and the non-centered cases and is valid for any non-lattice  probability measure on $G$ with a finite moment of high enough order, say $2(\dim G)^2$.  This moment condition also improves on [21, 29, 49]. A truncated version of the uniform LLT is also established under a  moment of order $2+\eps$ (<Ref>).

The second part of the paper is devoted to two applications of our LLT for biased walks. The first, <Ref>, gives a new proof of the Choquet-Deny theorem for measures with finite moments on nilpotent Lie groups. This asserts that bounded $\mu$-harmonic functions are constant, a result proven by Guivarc'h in his thesis [42] which extends the classical Choquet-Deny theorem [26] on abelian groups. The second application, <Ref>, concerns  the equidistribution of unipotent random walks on homogeneous spaces. In an unpublished chapter of the second author's thesis <cit.>, a probabilistic Ratner's theorem was shown asserting that $\mu^{*n} * \delta_x$ weakly converges towards Ratner's invariant measure in any finite volume homogeneous space of a connected Lie group, under the assumption that $\mu$ is a finitely supported symmetric measure whose support generates a unipotent subgroup. Results of this type have since known a number of extensions to non-unipotent settings, in particular to subgroups whose Zariski-closure is semisimple, as in the breakthrough work of Benoist-Quint [11] (see also [8]). As a consequence of our LLT, we can now prove the equidistribution of random walks driven by  an arbitrary, possibly biased, probability measure with finite third moment and supported on a unipotent subgroup.

*    *

We now give a detailed presentation of  our main results. In the paper  $G$ will denote  a simply connected nilpotent Lie group  and $\kg $ its Lie algebra. The step (or nilpotency class) of $\kg$ is the smallest integer $s$ such that $G^{[s+1]}=\{1\}$, where $G^{[i]}:=[G,G^{[i-1]}]$ is the central descending series. In the sequel $G$ will be identified with $\kg$ via the exponential map $\exp: \kg \to G$. This allows to see the Lie product $*$ as a composition law on $\kg$.

Abusing notation we will continue to denote by $*$ the associated convolution product of measures on $\kg$.  Observe that $*$ is  in fact a polynomial map on $\kg$ as follows from the Campbell-Hausdorff formula [32]:

$$x*y = x+y + \frac{1}{2}[x,y] + \frac{1}{12}[x,[x,y]] +\frac{1}{12}[y,[y,x]]+\dots $$

which contains only finitely many non-vanishing terms as $G$ is assumed nilpotent.

Given a probability measure $\mu$ on $(\kg,*)$ we consider the random walk  $S_N=X_1* \ldots * X_N$ where the $X_i$'s are independent with law $\mu$. We denote by $\mu_{ab}$ the projection of $\mu$ onto the abelianization $\kg_{ab}:=\kg/[\kg,\kg]$. Throughout the text, we assume that $\mu_{ab}$  has finite first moment and is not supported on an affine hyperplane. We also set $\XXab=\E(\mu_{ab}) \in \kg_{ab}$. Where further moment or aperiodicity assumptions are required they will be specified.

§.§  The central limit theorem

Before discussing the local limit theorem, we need to recall the central limit theorem (or CLT for short) for the random product $S_{N}$, in particular inasmuch as its formulation in the non-centered case requires the introduction of a subtle rescaling as we discovered in [9]. The large scale behaviour of $S_N$ on the Lie group is captured by a certain decreasing sequence of ideals of $\kg$ whose definition depends only on $\XXab=\E(\mu_{ab}) \in \kg_{ab}$,

$$\kg=:\kg^{(1)} \supseteq \kg^{(2)} \supseteq \cdots  \supseteq \kg^{(t+1)} = \{0\}, $$

where $[\kg^{(i)},\kg^{(j)}]\subseteq \kg^{(i+j)}$ for all $i,j$. In  the centered case, i.e. when $\XXab=0$, the filtration coincides with the descending central  series, i.e. $\kg^{(i)}=\kg^{[i]}:=[\kg, \kg^{[i-1]}]$ and $t=s$. In general, the filtration is defined by induction, setting $\kg^{(0)} =\kg^{(1)}=\kg$ and $\kg^{(i)}=  [\kg, \kg^{(i-1)}] + [\XXab, \kg^{(i-2)}]$. Note that $\kg^{[i]}\subseteq \kg^{(i)} \subseteq \kg^{[i']}$ where $i'=\lfloor i/2 \rfloor +1$, in particular $\kg^{(2)}=\kg^{[2]}$ and $\kg^{(2s)}=0$. We call $(\kg^{(i)})_{i\leq 2s-1}$ the weight filtration associated to $\XXab$.

It is shown in [9] that for any choice of vector subspaces $\km^{(i)}\subseteq \kg$ such that $\kg^{(i)}= \km^{(i)} \oplus \kg^{(i+1)}$, the projection of $S_{N}$  to $\km^{(i)}$ (modulo the sum of the other $\km^{(j)}$) spreads at a rate $N^{i/2}$ around its average. Renormalizing accordingly, the law converges weakly and we obtain the central limit theorem (CLT). More precisely, let
\begin{align*}
\pi^{(i)}: \kg &\to \km^{(i)}\\ x &\mapsto \pi^{(i)}(x)=x^{(i)}.
\end{align*}

be the coordinate projections induced by the decomposition $\kg=\oplus_{i} \km^{(i)}$. Set $\XX=\E_{\mu}(x^{(1)})$ and observe $\XX$ corresponds to $\XXab$ via the identification $\km^{(1)}\equiv \kg_{ab}$ induced by $\pi^{(1)}$. Introduce an associated one-parameter family of dilations $(D_{r})_{r>0}$  by setting

\begin{equation} \label{dilation}
D_r:\kg \to \kg, \,\,\,D_{r}(x)= \sum_{i \ge 1} r^{i}x^{(i)}.
\end{equation}

The  central limit theorem  <cit.> states that if $\mu$ has finite second moment for the weight filtration, then we have the  convergence in law

\begin{equation}\label{CLT}
\DilsN(\mu^{*N} * \delta_{-N \XX}) \to \nu
\end{equation}

where $\nu$ is a certain probability measure on $\kg$ with a smooth density. Here, one says that $\mu$ has finite $m$-th moment for the weight filtration if the map $x\mapsto \|x^{(i)}\|^{m/i}$ is in $L^1(\mu)$ for any $i$ (we take some norm, say a Euclidean norm, on each $\km^{(i)}$). As $\kg^{[i]}\subseteq \kg^{(i)}$ and the inclusion may be strict, this condition is  weaker that the ordinary notion of moment on a nilpotent Lie group defined in terms of the central descending series.

The CLT also involves a description of the limiting measure $\nu$ as the value at time  $t=1$ of a certain explicit hypoelliptic diffusion process $(\sigma_{t})_{t\in \R^+}$ on $\kg$. The infinitesimal generator of $(\sigma_{t})_{t\in \R^+}$ is time-dependent, and left invariant for a new group structure  $*'$ on $\kg$, inherited from the choice of subspaces $\km^{(i)}$. This new group structure is defined by  the formula

\begin{equation}\label{*'-product}
a*'b=\lim_{t \to +\infty} D_{1/t}(D_t(a)*D_t(b)).
\end{equation}

With this new Lie structure, $(\kg,*')$ is again an $s$-step nilpotent Lie group and dilations act as automorphisms. The semigroup  $(\sigma_{t})_{t>0}$ is then  defined (as in   [50]) by way of its infinitesimal generator

\begin{align}
\mathscr{L}_{t}=\frac{1}{2}\sum_{i=1}^{q}  \left(\exp(ta_{\XX}) E_{i}\right)^2+ \exp(ta_{\XX})B_\mu
\end{align}

where $E_{i}, B_\mu$ are left invariant vectors fields on $(\kg, *')$ such that the $E_i$'s  form a basis of $\km^{(1)}$ in which the covariance matrix of $\mu_{ab}$ is the identity,  and $B_\mu= E_\mu(x^{(2)}) \in \km^{(2)}$. Here
$a_{\XX}:\kg \to \kg$ is the nilpotent operator defined by

\begin{equation}\label{axdef}a_{\XX}(y)=\pi^{(i+2)}([\XX,y])\, \textnormal{ for } y \in \km^{(i)}.\end{equation}

The properties of the limiting measure $\nu=\sigma_{1}$ are studied in [93] in the centered case and [9] in  general. It is proven that $\nu=v(x)dx$ where $v$ is a smooth and rapidly decaying (Schwartz) function on $\kg$. The  measure $\nu$ is an ordinary Gaussian distribution on $\kg$ (in the Euclidean sense) if and only if $[\XXab, \kg^{[i]}]=[\XXab, \kg^{[i+1]}]$ for every $i\leq s$, which corresponds exactly to situations where the eigenvalues of the renormalization $D_{\frac{1}{\sqrt{n}}}$ are odd negative powers of $\sqrt{n}$, see <cit.>.  Usually there is no explicit formula for $v(x)$ nor its Fourier transform (the case of the Heisenberg group <cit.> is an exception). However, satisfactory estimates hold when $\nu$ arises from a centered walk: there exist $A, \alpha>0$ such that

$$\alpha \exp(-A |x|^2) \leq v(x)\leq A\exp(-\alpha |x|^2) $$

where $|x|$ denotes the distance to the origin for some (any) left-invariant Riemannian metric on $(\kg,*)$.
Apart from the centered case, or when the group is abelian or of nilpotency class $2$ and certain exceptions (such as the full group of unipotent upper triangular matrices), it is usually the case that the support of the density $\nu$ is a proper closed subset of $\kg$, see <cit.>. This came as a surprise to us.

§.§ The non-centered local limit theorems

We are now in a position to state the main results of this paper.

Let $\dmu $ be the integer:
$$\dmu = \sum_{i \ge 1} \dim \kg^{(i)},$$
where $\kg^{(i)}$ is the weight filtration introduced above.

We call $\dmu$ the homogeneous dimension of the $\mu$-random walk, or more precisely of the weight filtration  induced by $\XXab$. Note that $\dmu$ is related to the determinant of the dilations:  $\det D_{r}=r^{\dmu }$. When $\mu$ is centered, $\dmu $ coincides with the homogeneous dimension of $\g$, namely $d_\kg=\sum_{i \ge 1} \dim \kg^{[i]}$, which controls the volume growth of balls in $G$ by the Bass-Guivarc'h formula [42, 22]. We further note that $\dmu  \ge d_\kg$ with equality if and only if the weight filtration coincides with the descending central series, i.e. $\kg^{(i)}=\kg^{[i]}$ for all $i$.

In the next theorem we  assume that the projection $\mu_{ab}$ is non-lattice; this means that its support is not contained in a translate of a proper closed additive subgroup of $\kg_{ab}$.

Let $G$ be a simply connected nilpotent Lie group with Lie algebra $\kg$.
Let $\mu$ be a probability measure on  $G$ with  finite $(\dmu +2)$-th moment for the weight filtration induced by $\XXab=\E(\mu_{ab})$, and such that $\mu_{ab}$ is non-lattice. Then for every continuous and compactly supported function $f$ on $\kg$,  for all $N\geq 1$,

\begin{align} \label{non-centered-LLT}
|(\mu^{*N} *\delta_{-N\XX}) (f) -(\DilN \nu)(f)| =o_{\mu, f}(N^{-\frac{1}{2} \dmu }),
\end{align}

where $\nu$ is the limit measure from the central limit theorem (<ref>), and $\dmu $ is the homogeneous dimension of the $\mu$-random walk.

Moreover for $\eps_{0}>0$ small enough, the error rate in (<ref>) is uniform in all the bideviations $\delta_g * f * \delta_h : x\mapsto f(g*x*h)$ for $g,h\in \kg$, provided that either $g$ or $h$ belongs to

$$\{x\in \kg \,:\, \|x^{(i)}\| \leq N^{\frac{i}{2}+\eps_{0}}\} + [-N^{1+\eps_{0}}, N^{1+\eps_{0}}]X_{\mu}. $$

In [21] this result was proven for the Heisenberg group in the special case of a centered $\mu$ of compact support. As discussed above, Diaconis and Hough [29] gave a new proof in the centered Heisenberg case.  Later, Hough [49] obtained the LLT for centered walks on arbitrary nilpotent Lie groups. His result allows deviations of the test function $f$ only under the further assumption that $\mu_{ab}$ satisfies the Cramér condition, which is a strong form of aperiodicity  typically concerning spread out measures, or discrete measures with good diophantine properties (see [20, 5]). We allow deviations without Cramér assumption and this is important for our applications in <Ref>.

<Ref> is the first of its kind to allow the measure $\mu$ to be non-centered. The  proof is again via characteristic functions.  It is inspired by the path-swap technique used by Diaconis and Hough in [29, 49]. Since their method was tailored to work for the bracket structure of the descending central series, we will need extra work to obtain results related to the weight filtration (they usually do not coincide in the non-centered case).

We also note that we improve on the moment assumption compared to  [21, 29, 49] even in the centered case. This improvement is performed by truncating suitably the measure $\mu$. <Ref> will in fact be a corollary of a local limit theorem for a truncated version of $\mu$, namely <Ref>,  available under the sole  condition that $\mu$ has a finite moment of order $2+\eps$ for some $\eps>0$. This  truncated formulation will turn out useful  for applications later.

Remark. If $f$ is in $L^1(\kg)$ and its Fourier transform has compact support, we can improve the error term in the LLT from $N^{-\frac{1}{2}\dmu }$ to $N^{-\frac{1}{2}(\dmu +1-\eps)}$ provided $\mu$ has finite $(\dd+3)$-th moment (<Ref>).

Remark. In the centered Heisenberg case, the LLT proved by Diaconis and Hough  in [29] is uniform in all deviations $g,h\in G$, and also allows the measure to be lattice. We do not know whether this extra uniformity holds in the  setting of <Ref>. In the Heisenberg case, it relies on the observation that the $L^1$-norm  of the Fourier transform of a test function is constant under two-sided deviation, but this is not true in a general nilpotent group. The LLT for lattice measures on a general nilpotent Lie group is also still open. It requires a more delicate analysis that has not been worked out yet, even in the centered case (with the notable exception of centered walks on discrete nilpotent groups by work of Alexopoulos [4], see <Ref>).

The local limit theorem reduces the estimation of $\mu^{*N} *\delta_{-N\XX} (f)$ to that of $\DilN\nu(f)$. The latter is easier to understand, because the limit measure $\nu$ has a smooth and rapidly decaying density as we have discussed. However $\nu$ does not always have full support. In particular if the density of $\nu$ vanishes at the origin, the LLT in the  form of (<ref>) does not provide meaningful asymptotics for the decay of $\mu^{*N} *\delta_{-N\XX} (f)$, but only an upper bound in $o_{\mu, f}(N^{-\frac{1}{2}\dmu})$. One way to remedy this is to use the uniformity with respect to deviations in our LLT in order to change the recentering appropriately as follows.

Pick $Y \in \kg$ and set $g_N=-N\XX*-\DilN Y$. Recall that $\nu$ has a smooth density $v(x) \ge 0$ on $\kg$. We have:

Keep the assumptions of <Ref>. Then for every continuous and compactly supported function $f$ on $\kg$ we have:

$$N^{\frac{1}{2}\dmu  } \,(\mu^{*N}*\delta_{g_{N}})(f) \underset{N\to +\infty}{\longrightarrow} v(Y) \int_{\kg}f(x) dx.$$

In particular this answers positively the question asked in the introduction as the quantity in <Ref> converges to a non-trivial multiple of the Haar measure on $G$ provided we choose $Y$ such that $v(Y)>0$. The following variant can also be derived in the centered case:

Assume that $\mu_{ab}$ is centered, non lattice,  and $\mu$ has finite moment of order $m_{\mu}>d_{\kg}+2$ for the central filtration. For all non-zero $f\in C_{c}(\kg)$ with $f\geq0$, one has

$$\frac{\mu^{*N}(f)}{\DilN\nu(f)} \underset{N\to +\infty}{\longrightarrow} 1$$

and for any fixed $c>0$ sufficiently small, the convergence is uniform as we replace $f$ by the bitranslates $\delta_g * f * \delta_h$ with

$$g,h\in \{x\in \kg \,:\, \|x^{(i)}\|\leq c \,(N\log N)^{i/2}\}.$$

Due to the possible vanishing of the limiting measure, the assumption that the walk is centered seems unavoidable to guarantee uniformity in the ratio LLT.

§.§ Applications of the LLT

We now present two applications of our LLT (<ref>, <ref>).

The Choquet-Deny property.
Let $G$ be a locally compact group, and $\mu$ a probability measure on $G$. A function $f\in L^\infty(G)$ is said to be $\mu$-harmonic if for almost-every $x\in G$,

$$f(x)=\int_G f(xg)\,d\mu(g).$$

Note that every function $f$ that is right-invariant by the support of $\mu$, i.e. such that $f*\delta_g=f$ for every $g \in \supp(\mu)$, must be $\mu$-harmonic.

$G$ is said to have the Choquet-Deny property if the converse holds: for any measure $\mu$, any bounded $\mu$-harmonic function $f\in L^\infty(G)$ is right-invariant by the support of $\mu$. True for $\Z^d$ [14] and discrete nilpotent groups [33], this property is by now completely understood when the group $G$ is discrete, thanks to the groundbreaking recent work of Frisch, Hartman, Tamuz and Vahidi-Ferdowsi [34] who showed that a finitely generated group is Choquet-Deny if and only if it is virtually nilpotent.  For totally disconnected groups of polynomial growth, several characterizations of the Choquet-Deny property are given in [54]. The case of continuous groups however remains more subtle.  Guivarc'h in his thesis [42] established the property for all locally compact nilpotent groups under the assumption of a finite moment (of arbitrarily small but positive order) on the measure $\mu$, while the general case with no moment condition is still an open problem, despite [73].  Guivarc'h's proof relied on unitary representations. Assuming that $G$ is a nilpotent Lie group and $\mu$ has a moment of order $2+\eps$, we provide a different argument, based on our LLT.

Let $G$ be a simply connected nilpotent Lie group. Let $\mu$ be a probability measure on $G$ with a finite moment of order strictly greater than $2$ for the weight filtration induced by $\XXab=\E(\mu_{ab})$, and whose support generates a dense subgroup of $G$. Then bounded $\mu$-harmonic functions on $G$ are constant.

The condition that $G$ is simply connected is not essential here, as one may always lift $\mu$ and $\mu$-harmonic functions to the universal cover of $G$. It is however convenient to express our moment condition on $\mu$.

Probabilistic Ratner theorem.
Let $H$ be a real Lie group and $\Lambda \leq H$ a lattice (discrete subgroup with finite covolume). Let $U$ be a simply connected nilpotent Lie group and $\rho:U \to H$ a Lie group homomorphism such that $Ad \circ \rho$ is contained in a unipotent subgroup of $\GL(\kh)$, where $\kh=Lie(H)$ is the Lie algebra of $H$. Considering the resulting action of $U$ on $\Omega:=H/\Lambda$, Ratner established [71, 66] that the closure of every $U$-orbit is itself a homogeneous subspace of finite volume: for each $\w \in \Omega$, the stabilizer $L_{\w}=\{h\in H, \, h.\overline{U\w}=\overline{U\w}\}$ acts transitively on  $\overline{U\w}$ while preserving a probability measure on $\overline{U\w}$. This measure is unique and we denote it by  $m_{\w}$. Ratner's proof is based on an equidistribution theorem for one-parameter unipotent flows, that was subsequently extended to the  multi-parameter case by Shah [81]. Combining this with our LLT we obtain:

In the above setting, let $\mu$  be a probability measure on $U$ whose support generates a dense subgroup of $U$. Assume $\mu$ has a finite moment of order strictly greater than $2$ for the weight filtration induced by $\XXab=\E(\mu_{ab})$. Then for every  $\w\in \Omega$, we have the following weak-$\star$ convergence of probability measures:

$$\frac{1}{N}\sum_{k=0}^{N-1} \mu^k*\delta_{\w}\underset{N\to \infty}{\longrightarrow} m_{\w}. $$

If, additionally, $\mu$ is assumed centered and aperiodic, then $(\mu^{*N}*\delta_\w)_{N \ge 1}$ itself converges to $m_\w$.

We stress that the LLT is used not only through the Choquet-Deny property and its consequence   that $\mu$-stationary measures are invariant. Indeed, we also rely crucially on the LLT to establish the non-escape of mass of the above sequences of probability measures. It ensures that any limit measure  has mass $1$ and is absolutely continuous with respect to $m_\w$.

The  moment condition in  <Ref> may not be optimal. However the other assumptions are tight. If $\mu$ fails to be centered, or aperiodic, then easy examples show that the convergence of individual powers fails and only the Cesaro convergence holds (see <cit.>).

<Ref> generalizes the unpublished chapter <cit.>, where this result was proven by a different method (combining work of Guivarc'h and Varopoulos) in the special case of a symmetric and finitely supported measure $\mu$, as well as the case of centered walks on the Heisenberg group, treated in [21].

We also point out the analogous fact for measures supported on subgroups whose Zariski-closure is semisimple is a major result of Benoist-Quint [11] in the Cesaro case, while convergence of individual powers was recently established by the first named author in [8] in that setting, see also [17, 44].

§.§ Historical remarks

Random walks and limit theorems on non-commutative Lie groups have a long history. We refer the reader to the early book [40] and survey [79] for background on non-commutative analogues of the law of large numbers and central limit theorems, and more recent references such as [78, 19, 12, 63, 92]. The local limit theorem has been studied for compact groups in [56, 6, 16, 10], for the group of Euclidean rigid motions in [57, 43, 90, 64] and semisimple Lie groups in [15, 59]. On nilpotent Lie groups, the central limit theorem was studied in [89, 94, 28, 72, 13, 9]. Random walks on nilpotent covers of compact manifolds or finite graphs have also attracted a lot of attention, see [51, 52, 53], where certain limit theorems for centered and non-centered walks are also obtained.

A completely new method was brought forward by Varopoulos in the 80s (see e.g. [93]) and further developed by Alexopoulos culminating in [3, 2, 4]. This method provides very complete results on the asymptotics of convolutions powers $\mu^{*N}$ of a measure $\mu$ on a nilpotent group (either Lie or  discrete), including central and local limit theorems, even with the full Edgeworth expansion. However there are two caveats. The first is that this method requires the measure $\mu$ to have a density with respect to the Haar measure, an assumption which seems unavoidable. The second is that the limit theorems that have been obtained by this method always assumed the measure to be centered.

The method used in our paper however (as well as that of [21, 29, 49, 9]) is based on the analysis of characteristic functions and allows to handle probability measures without any assumption on their regularity or whether they are centered or not. While our results assume the measure to be non-lattice, it remains an open problem to handle lattice measures (even  centered ones) in general nilpotent groups.

§.§ The path-swap principle behind the proof of <Ref>

The approach to prove <Ref> relies on characteristic functions  as in the classical proof of the LLT for abelian groups.  Similarly to previous works, the main part of the proof consists in establishing super-polynommial decay of the Fourier transform outside of a shrinking domain around the origin: for every $Q\ge1$, $\gamma_{0}>0$,
\begin{equation}\label{foudecay}|\widehat{\mu^{*N}}(\xi)| \ll_{Q, \gamma_{0}} N^{-Q}\end{equation}
outside the range of  Fourier parameters $\xi \in \widehat{\kg}$ such that $\|\xi \circ D_{\sqrt{N}}\| \ll N^{\gamma_{0}}$.

In [21], a spectral gap estimate for the infinite dimensional irreducible unitary representations of the Heisenberg group was established for this purpose. In [29], Diaconis and Hough proposed a different method to prove $(\ref{foudecay})$ in the centered Heisenberg case, which is reminiscent of the well-known van der Corput trick in the proof of Weyl's equiditribution theorem for polynomials, where one differentiates the polynomial phase to produce cancellation in an exponential sum (see e.g. <cit.> for an example of its use on the context of equidistribution of ergodic flows on nilmanifolds). In  [29], this discrete derivative is replaced by a difference of permutations acting on the variables, which after iteration, generates some independence. This path-swap principle is also exploited by [49] to prove  (<ref>) for centered walks on arbitrary nilpotent groups, and we use it as well for <Ref>.

We now give a brief outline of this principle (see <Ref> for details). Given a subgroup $F\leq \Sym(N)$ we may write
$$\widehat{\mu^{*N}}(\xi) = \E_{\mu^{*N}}\big(e_\xi(\Pi(\xx))\big)= \E_{\mu^{*N}}\big(\E_{\sigma \in F}(e_\xi(\Pi(\sigma \xx)))\big)$$
where $e_\xi(x)=\exp(-2i\pi \xi.x)$ and $\Pi(\xx)$ is the Lie product of the variables $\xx \in \kg^N$ viewed as an element of the free Lie algebra on $N$ letters evaluated at $\xx$. By Cauchy-Schwarz:
$$|\widehat{\mu^{*N}}(\xi)|^2 \leq \E_{\mu^{*N}} \big( |\E_{\sigma \in F}(e_\xi(\Pi(\xx)))|^2\big)=\E_{\mu^{*N}} \big(\E_{\sigma,\tau \in F}(e_\xi(\Pi(\sigma \xx)-\Pi(\tau \xx)))\big).$$

The right hand side can be rewritten $\E_{\sigma,\tau \in F} \big(\E_{\mu^{*N}} (e_\xi((\sigma-\tau)\Pi(\xx)))\big)$ where $\sigma-\tau$ is viewed as an element of the group algebra $\R[\Sym(N)]$ acting on the free Lie algebra.

We can then iterate this procedure $s-1$ times, averaging successively over commuting subgroups $F_1,\ldots,F_{s-1}$. If  $F=F_1\times \ldots \times F_{s-1}$,

$$|\widehat{\mu^{*N}}(\xi)|^{2^{s-1}} \leq \E_{\sigma,\tau \in F} \big(\E_{\mu^{*N}} (e_\xi(A_{\sigma,\tau}\Pi(\xx)))\big)$$

where $A_{\sigma,\tau}=\prod_1^{s-1}(\sigma_i-\tau_i) \in \R[F]$ and $\sigma=(\sigma_1,\ldots,\sigma_{s-1}) \in F$.

When $N=N'(2s-1)k$ for some $k,N'$, we may split the $N$ variables $x_i$ into $N'$ consecutive blocks $(B_{j})_{j\leq N'}$ of length $(2s-1)k$. We see each block as a concatenation of $(2s-1)$ sub-blocks of length $k$. We then let $F_{i}\simeq (\Z/2\Z)^{N'}$ be the group generated by the  transpositions  exchanging the two sub-blocks at distance $i$ from the central sub-block in some $B_{j}$. As the $F_{i}$ have disjoint support, they do commute. The combinatorial properties of the Lie product on a nilpotent group imply that:
$$A_{\sigma,\tau}\Pi(\xx)=\sum_{j=1}^{N'} A_{\sigma,\tau}\underline{u}_j,$$
where each $\underline{u}_j$ is a Lie polynomial involving only variables $x_i$ whose indices $i$ belong to the block $B_{j}$. It is therefore a sum of $N'$ uncorrelated terms. And hence:

$$|\widehat{\mu^{*N}}(\xi)|^{2^{s-1}} \leq \E_{\sigma,\tau \in F} \prod_{j=1}^{N'}\E_{\mu^{*N}} (e_\xi(A_{\sigma,\tau}\underline{u}_j))= \big( \E_{\sigma,\tau} \E_{\mu^{k(2a-1)}} (e_\xi(A_{\sigma,\tau}\underline{u}_1) \big)^{N'}.$$

Furthermore the choice of $F$ will ensure that $A_{\sigma,\tau}\underline{u}_1$ is a sum of $s$-brackets (i.e. pure commutators of length $s$), hence belongs to the center of $\kg$. Choosing $k \sim N^\eps$ and $N'\sim N^{1-\eps}$, this will eventually lead to the desired super-polynomial decay estimate $(\ref{foudecay})$.

§.§ Organization of the paper
In <Ref>, we  set up the notations to be used in the rest of the paper and we recall from [9] several techniques of truncation, graded replacement and  Lindeberg replacement, that help  estimate the Fourier transform of the random product. <Ref> is devoted to the proof of the uniform local limit theorem. In <Ref>, we prove the local limit theorem with variable recentering or controlled deviations. Finally, in <Ref>, we apply the uniform local limit theorem to prove the Choquet-Deny property on connected nilpotent Lie groups, and the equidistribution of unipotent walks on finite volume homogeneous spaces.

§.§ Acknowledgements
We are grateful to  Yves Benoist, Peter Friz and Laurent Saloff-Coste for very helpful discussions.

§ NOTATION AND PRELIMINARY BACKGROUND

The goal of this preliminary section is to collect the terminology and notation used throughout the paper, and to recall useful results from [9] on which we will build the proof of the LLT.

§.§ Set-up and notation

§.§.§   The ambient space and its bias extension

Let $(\kg, [.,.])$ be a nilpotent Lie algebra with associated Lie product $*$, and $\Xab \in \kg/[\kg,\kg]$. The   weight filtration of $\kg$ associated to $\Xab$ is  the nested sequence of ideals defined recursively by setting $\kg^{(0)}=\kg^{(1)}=\kg$ and for $i\ge 1$

\begin{equation*}
\kg^{(i+1)}=[\kg,\kg^{(i)}]+[\Xab,\kg^{(i-1)}].
\end{equation*}

It  is well-defined (independently of the choice of representative of $\Xab$ in $\kg$), and satisfies  for $i,j \ge 1$,
\begin{equation}\label{nestedg}
[\kg^{(i)},\kg^{(j)}] \subseteq \kg^{(i+j)}.
\end{equation}

For each  $i\leq 2s-1$, we choose once and for all a vector subspace $\km^{(i)}$ in $\kg$ such that

$$\kg^{(i)} = \km^{(i)}\oplus \kg^{(i+1)} $$

It follows that $\kg$ is the (vector space) direct sum

$$\kg=\km^{(1)} \oplus \dots \oplus \km^{(2s-1)}$$

We call such a direct sum (with choices of $\km^{(i)}$) a weight decomposition of $\kg$. We  may identify $\km^{(1)}$ with $\kg/[\kg,\kg]$. We denote by  $\Xm$  the unique element of $\km^{(1)}$ projecting to  $\Xab$.

It is useful to introduce an extension of $\kg$, realized by adding to $\kg$ an independent copy $\chi$ of $\Xm$. Given a random walk on $(\kg,*)$, this procedure will allow to distinguish between the influence of the average of the increment and the variations around this average.   If  $\Xab=0$, we just set $\tkg=\kg$, $\chi=0$. Otherwise, we define $\tkg$ as the Lie algebra direct sum:

$$\tkg:=\kg \oplus \R$$

and set $\chi=(\Xm,1)$. In particular, $\tkg=\kg\oplus \R \chi$ (as vector spaces) and $[\Xm, .]=[\chi,.]$. We call $\tkg$ the bias extension of $\kg$.   We extend the weight filtration and weight decomposition to $\tkg$ by setting : $\tkm^{(i)}=\km^{(i)}$ if $i \neq 2$, and $\tkm^{(2)}=\km^{(2)}+ \R\chi$, then $\tkg^{(i)}=\oplus_{j\geq i} \tkm^{(j)}$. We thus get the direct sum decomposition:

\begin{equation}\label{tilde-dec} \tkg = \km^{(1)} \oplus (\km^{(2)} \oplus \R\chi) \oplus \km^{(3)} \oplus \cdots \oplus \km^{(t)}
\end{equation}

The large time behavior of a random walk on $\kg$ with increment average $\Xab$ is governed by a new graded Lie bracket on $\kg$, or more generally on $\tkg$. It is defined by setting

\begin{equation} \label{bracket-gr}
[x,y]'= \pi^{(i+j)}([x,y])
\end{equation}

if $x \in \tkm^{(i)}, y \in \tkm^{(j)}$ and $\pi^{(k)}: \tkg \to \tkm^{(k)}$ denotes the linear projection modulo the other $\tkm^{(k')}$, $k' \neq k$. Notice that the vector space $\kg$ is an ideal of $(\tkg, [.,.]')$ and that $\Xm$ is central for $[.,.]'$. The   Lie product associated to $[.,.]'$ will be denoted by  $*'$. The operator $a_{\Xm}$ used in the introduction corresponds to $[\chi, .]'$.

Finally, we will make use of dilations as defined in the Introduction. They are a $1$-parameter  family of diagonal maps $(D_{r})_{r>0}$ on $\tkg$ defined by the formula $D_r(x)=\sum_{i}r^{i}x^{(i)}$. They are automorphisms of $(\tkg, [.,.]')$.

§.§.§ Conventions: norm, Lebesgue measure,  dual, Fourier transform
We choose once and for all a basis $(e_j^{(i)})_{1\leq j \leq q_i}$ of each $\km^{(i)}$, and we add $e_0^{(2)}=\chi$ to $\tkm^{(2)}$. We denote by $\|.\|$ the associated Euclidean norm on $\kg$ and $\tkg$ making that basis orthonormal. The associated Lebesgue measure will be denoted by $dx$. Notice that $dx$ is also a left and right invariant Haar measure (for both $*$ and $*'$).

We also fix a choice of left invariant Riemannian metric on $(\kg, *)$ and denote by $\norm{.}$ the induced distance to $0$.

We endow the dual space $\dkg$ of linear forms on $\kg$ with the operator norm and the Lebesgue measure $d\xi$ associated to the dual basis of the $(e_j^{(i)})_{i,j}$. Linear forms $\xi\in \dkg$ will be identified with their extension to $\tkg$ by setting $\xi(\chi)=0$.

Given $\xi \in \dkg$, we set

$$e_{\xi} : \kg \rightarrow \C, x\mapsto e^{-2i\pi \xi(x)} $$

and we define the Fourier transform of  a measure $\sigma$ or an integrable function $f$ on $\kg$ by

$$\widehat{\sigma}(\xi)= \int_{\kg} e_{\xi}(x)d\sigma(x) \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\hf(\xi)= \int_{\kg} e_{-\xi}(x) f(x)dx.$$

If $f$ is continuous and $\hf$ is integrable, the Fourier inversion formula states that

\begin{align*}
\sigma (f)&=\int_{\dkg} \hf(\xi) \,\widehat{\sigma}(\xi) d\xi.
\end{align*}

§.§.§ Polynomial maps and bigrading of the product

Every polynomial function $P$ from $\tkg$ to $\R$ may be written canonically as a sum of monomials, where a monomial is a product of coordinate linear forms ${e_j^{(i)}}^*$ associated to the basis ${e_j^{(i)}}$. We define the degree of $P$ by the usual rules and the convention that the degree of ${e_j^{(i)}}^*$ is equal to $i$. One may check that this definition does not depend on the choice of basis ${e_j^{(i)}}$ fixed above, nor  on the choice of weight decomposition $(\km^{(i)})$.

Write $\xx$ for the tuple $\xx=(x_1,\ldots,x_N)$. The polynomial map
\begin{align*}\Pi: \tkg^N &\rightarrow \tkg,\\ \xx &\mapsto \Pi(\xx):=x_{1}*x_{2}*\dots*x_{N}\end{align*}
will be of particular interest. Note that it can be written as a combination of iterated brackets. More precisely,  set  $L^{[r]}$ the $r$-bracket defined recursively by $L^{[1]}(x)=x$ and

$$L^{[r]}(x_{1}, \dots, x_{r})  =[L_{r-1}(x_{1}, \dots, x_{r-1}), x_{r}].$$

Then we have the Campbell-Hausdorff formula (see [32]) on every  Lie algebra of step at most $s$:

$$\Pi(\xx) = \sum_{r=1}^s \sum_{ \underset{ \,1\leq n_{1}< \dots < n_{t}<\infty}{r_{1}+\dots+r_{t}= r}, \,r_{i}\neq 0}  L_{\rr}(x_{n_{1}}^{\otimes r_{1}}, \dots, x_{n_{t}}^{\otimes r_{t}})$$

where $\underline{r}=(r_{1}, \dots, r_{t})$ and  $L_{\rr}$ is a linear combination with rational coefficients of terms, each of which is an $r$-bracket $L^{[r]}$ of some permutation of the $r$ variables involved. The notation $x_{n_i}^{\otimes r_i}$ is short for the $r_i$-tuple $(x_{n_i},\ldots,x_{n_i})$.

We will write the above product as a sum of polynomial maps capturing the different growth rates of a walk on $\kg$ and keeping track of the bracket structure. More precisely, we denote by $\Pi^{[a,b)}(\xx)$ the sum obtained formally from $\Pi(\xx)$ as follows.  Write each variable $x_{k}$ as a formal sum of “weight” variables $x_{k}=\oplus x^{(i)}_{k}$. Then expand  the multi-brackets using multilinearity and keep only the terms involving only the $a$-brackets whose variables $x^{(i_{1})}_{k_{1}}, \dots, x^{(i_{a})}_{k_{a}}$ satisfy $i_{1}+\dots+i_{a}=b$. In particular
We also use the notation $\Pi^{[a]}(\xx)=\oplus_{b}\Pi^{[a,b)}(\xx)$, $\Pi^{(b)}(\xx)=\oplus_{a}\Pi^{[a,b)}(\xx)$. Note that $\Pi^{(b)}(\xx)$ has weight $b$ in the sense that its image under any linear form on $\tkg$ is a polynomial of degree $b$.

§.§.§ The driving measure and its bias extension

Let $\mu$ be a probability measure on $\kg$. Denote by $\mu_{ab}$ the projection of $\mu$ to $\kg/[\kg, \kg]=:\kg_{ab}$. We assume that $\mu_{ab}$ has a finite first moment and write $\overline{X}=\E(\mu_{ab})$ for the expectation of $\mu_{ab}$. We also set $\mu^{(i)}:=\pi^{(i)}_{\star}\mu$, where the subscript $\star$ added to a map denotes the push-forward by that map. We set $\Xm= \E(\mu^{(1)})$ and notice $\Xm$ projects onto $\overline{X}$ under the abelianization map.

To prove the LLT, we will replace the measure $\mu$ by its bias extension $\tmu$, defined as the image of $\mu$ under the map $\kg\rightarrow \tkg, x \mapsto (x,1)$. Notice that for all $N\geq 1$, one has

\begin{align} \label{formule-ext}
\tmu^{*N} = \mu^{*N}-N\Xm+N\chi =\mu^{*N}*-N\Xm*N\chi
\end{align}

where the notation $ \mu^{*N}-N\Xm+N\chi $ stands for the image of $\mu^{*N}$ under the map $\tkg\rightarrow \tkg, x\mapsto x-N\Xm+N\chi$, and similarly for $\mu^{*N}*-N\Xm*N\chi$. The role of $\tmu$ is to convert the drift $\Xm \in \km^{(1)}$ into  $\chi\in \tkm^{(2)}$ while  keeping  the fluctuations around the drift in $\R\Xm \subseteq \km^{(1)}$. This separation aims to reflect the fact that the drift will contribute to  higher commutators with weight $2$ whereas the fluctuation around the mean will contribute  with weight $1$.

In proving the local limit theorem, we will  suppose that $\mu$ is aperiodic (or non-lattice). This means that the support of $\mu$ is not contained in a coset of some proper subgroup, i.e. some $x*H$ where $x\in \kg$, $H$ is a proper subgroup of $(\kg, *)$. For simply connected nilpotent Lie groups, it is equivalent to ask that $\mu_{ab}$ is non-lattice, or that $|\widehat{\mu_{ab}}(\xi)| <1$ for all $\xi \in \dkg_{ab}\smallsetminus \{0\}$.

We also need moment conditions.  For $m>0$, we say that $\mu$ has finite $m$-th moment for the weight filtration if for all $b\leq 2s-1$, one has
$$\int_{\kg} \|x^{(b)}\|^{m/b}d\mu(x)<\infty. $$

§.§.§ Moderate deviations

Given $N\geq 1$, $g,h\in \tkg$, we denote by $g*\tmu^{*N}*h$, or $\tmu^{*N}_{g,h}$ for short, the image of $\tmu^{*N}$ under the map $\tkg\rightarrow \tkg, x\mapsto g*x*h$. Setting $p: \tkg:=\kg\oplus \R \rightarrow \kg$ the projection to the first factor (which is a Lie algebra homomorphism), we have $p_{\star}\tmu^{*N}_{g,h}=\mu^{*N}_{p(g),p(h)}$.

We shall prove  estimates for $\tmu^{*N}_{g,h}$ uniformly in a certain range of moderate deviations: given  $\delta_{0} \in [0, 1)$, we set

$$\DN(\delta_{0})=\{x \in \tkg  \,:\, \forall i\leq 2s-1,\,\, \|x^{(i)}\| \leq  N^{i/2+s^{-1}\delta_{0}}\},$$
where $x^{(i)}=\pi^{(i)}(x)$ is the coordinate projection with respect to the grading $(\tkm^{(b)})_{b}$ of $\tkg$. It is worth pointing out that $\DN$ behaves well with respect to the product $*$ :  There exists $C>0$ such that for all $\delta_{0}>0$, for all large enough $N \geq 1$,

$$x,y\in \DN(\delta_{0}) \implies x*y\in \DN(C \delta_{0}).$$

§.§.§ Asymptotic notation

In the rest of the text, we use the standard Vinogradov notation  $\ll$ and Landau notations $O(.)$, $o(.)$. Unless otherwise stated, the implied constants will depend only on the initial data $\kg, \Xab, \mu, (\km^{(b)})_{b}, (e^{(i)}_{j})_{i,j}, \norm{.}$, and the decomposition  $\kg=\oplus \km^{[a,b)}$ introduced later in <Ref>.  In some statements, we will also use the notation $r_{1} \lll r_{2}$ to mean that the conclusions that follow are true up to choosing $r_{1}\leq c r_{2}$  where $c>0$ is a (small) constant depending only the initial data. Subscripts are added to indicate any additional dependency (for instance, on the parameter $\delta_{0}$ controlling the range of deviations). Furthermore $\N$ denotes the non-negative integers and $\R^+$ the non-negative reals.

§.§ The three reductions

To prove the local limit theorem <Ref> announced earlier, we perform a series of reductions on the random product  $X_{1}*\dots *X_{N}$ that help estimate its Fourier transform. First, we see that after a suitable truncation, the random product has  moment estimates similar to what can be proven in the case where $\mu$ has finite moment for every order. Second, we  note that the product structure $*$ can be replaced, up to a small error, by its graded counterpart $*'$ for which dilations act as automorphisms. Third, we show that $\mu$ can be replaced by any other driving measure  that share the same moments up to order $2$. Those reductions  were already used in [9] in order to prove the central limit theorem.

The notations $(\kg, *, \tkg, \Xab, (\km^{(b)})_{b\leq 2s-1}, \Pi^{[a,b)},  *', \mu, \tmu, \DN(\delta_{0}))$ are those of  <Ref>.  Throughout <Ref> we assume that $\mu$ has finite second moment for the weight filtration induced by $\Xab$.

§.§.§ Truncation and moment estimates

The proof of <Ref> relies on moment estimates that help control the possibly pathological behavior of the walk.  In the case where $\mu$ has finite moment for every order, these estimates concern  $\mu^{*N}$ directly  <cit.>. Under the weaker assumption that $\mu$ has finite second moment, we are still able to obtain such estimates but for a truncated version of $\mu^{*N}$  <cit.>. We recall here these estimates.

The truncation $T_{N}\tmu$  is the image of $\tmu$ under the map

\begin{align*}T_{N}: \tkg & \rightarrow \tkg,\\ x=\oplus x^{(b)} & \mapsto \oplus T_{N}x^{(b)}\end{align*}

T_{N}x^{(b)} = \left\{
\begin{array}{ll}
x^{(b)}  & \mbox{if } \|x^{(b)} \|\leq N^{b/2} \\
0 & \mbox{if } b\geq 2 \text{ and } \|x^{(b)} \|> N^{b/2}\\
c_{N} & \mbox{if }  b= 1 \text{ and } \|x^{(1)} \|> N^{1/2}
\end{array}
\right.

and $c_{N}:= - \tmu({\|x^{(1)} \|> N^{1/2}})^{-1}\E_{\tmu}(x^{(1)} \1_{{ \|x^{(1)} \|\leq N^{1/2}}})$ is chosen so that $T_{N}\tmu$ is centered in the $\km^{(1)}$-coordinate.

The   error induced when replacing  the measure $\tmu$ by its truncation $T_{N} \tmu$ is controlled via the following lemma.

[Cost of truncation]
Let $q \in [2, +\infty)$. If $\mu$ has finite $q$-th moment then for all $N\geq 1$, $(N_{i})_{i\leq N}\in \R_{>0}^N$,

$$\| \tmu^{\otimes N}-  \bigotimes_{i=1}^N T_{N_{i}}\tmu \| =o\left(\sum_{i=1}^N N_{i}^{-\frac{q}{2}}\right) $$

where $\|.\|$ is the norm in total variation.

Notice that
$$\tmu \{T_{N_{i}}x\neq x\} \leq \tmu \{\max_{b} \|x^{(b)}\|^{1/b}>N_{i}^{1/2}\} =o(N_{i}^{-\frac{q}{2}})$$

so $\bigotimes_{i=1}^N T_{N_{i}}$ is the identity on a subset of $\tkg^N$ of $\tmu^{\otimes N}$-measure $(1-o(\sum_{i=1}^N N_{i}^{-\frac{q}{2}}))$.

Given a monomial function $M:\tkg^{t}\rightarrow \R$ of degree $b\ge 1$, see <Ref>, we define its statistics $M^{\infty}$  on $\tkg^{\N^*}$  by

$$M^{\infty}(\xx)=\sum_{1\leq n_{1}<\dots <n_{t}<\infty}M(x_{n_{1}}, \dots, x_{n_{t}}).$$

For every $N\geq1$, we identify $\tkg^N$ as the subset of $\tkg^{\N^*}$ made of sequences that vanish after the  first $N$ terms.  In particular, $M^{\infty}$ is well-defined on $\tkg^N$.

Assume that $\mu$ has a finite second moment. For  $N\geq 1$, $(N_{i})_{i\leq N}\in  [0, N]^N$,  $m\geq 1$ an integer,

$$\E_{\bigotimes_{i=1}^N T_{N_{i}}\tmu}\left[{M^{\infty}}(\xx)^{2m} \right]\ll_{m} N^{bm},$$
and in particular for $\delta_{0}\geq 0$, $g,h\in \DN(\delta_{0})$, $\yy=(g,\xx,h)$,

$$\E_{\xx\sim \bigotimes_{i=1}^N T_{N_{i}}\tmu }\left[\|\Pi^{[a,b)}(\yy)\|^{2m} \right]\ll_{m} N^{(b+2\delta_{0})m }$$

This is <cit.>.

§.§.§ Graded replacement

We bound the error induced when replacing $*$ by the graded product $*'$ for which the dilations $(D_{r})_{r>0}$ act as automorphisms.  The result is formulated  in terms of Fourier transforms.

For  a linear form $\xi$ on $\tkg$, $N\geq 1$, $(N_{i})_{i\leq N}\in [ 0, N]^N$, $g,h\in \DN(\delta_{0})$, $\yy=(g,\xx,h)$,

$$\big|\E_{\xx \sim \bigotimes_{i=1}^N T_{N_{i}}\tmu } [e_{\xi} (\Pi(\yy) )-e_{\xi} (\Pi'(\yy) )] \big| \ll \,\| \DilN \xi \| \,N^{-1/2+\delta_{0}}.$$

This is <cit.>.

§.§.§ Lindeberg replacement

We bound the error induced when replacing the driving measure with another measure that coincides with it up to order $2$. This means that the two measures have the same evaluation on polynomials of degree at most $2$ for the weight filtration. The result is formulated  in terms of Fourier transforms.

Let $\eta$ be a probability measure on $\kg$ with finite second moment  for the weight filtration and that coincides with $\mu$ up to order $2$. For a linear form $\xi$ on $\tkg$, $N\geq 1$, $(N_{i})_{i\leq N}\in [ 0, N]^N$, $g,h\in \DN(\delta_{0})$, $\yy=(g,\xx,h)$,

$$\big|\E_{\xx \sim \bigotimes_{i=1}^N T_{N_{i}}\tmu } [e_{\xi} (\Pi'(\yy) )]-\E_{\xx \sim \bigotimes_{i=1}^N T_{N_{i}}\teta } [e_{\xi} (\Pi'(\yy) )]\big| \ll_{\eta} \,(\| \DilN \xi\|+\| \DilN \xi\|^3)N^{3 \delta_{0}}\sum_{i=1}^N N^{-1}_{i} \,\eps_{N}$$

where $\eps_{N}=o_{\eta}(1)$. More generally, assuming $\mu$, $\eta$ both have finite $m$-th moment for some $m\in [2,3]$, we may take $\eps_{N}=o_{\eta}(N^{-\frac{m}{2}+1})$ if $m<3$ and $\eps_{N}=O_{\eta}(N^{-1/2})$ if $m=3$.

We will apply this proposition to a sequence  $(N_{i})$ that is gradually increasing and such that $\sum_{i=1}^N N^{-1}_{i}=O(1)$, so the right-hand side will be $O_{\eta}(1)(\| \DilN \xi\|+\| \DilN \xi\|^3)N^{3s \delta_{0}}\,\eps_{N}$.

This is <cit.>.

§ UNIFORM LOCAL LIMIT THEOREM

This section is dedicated to the proof of the uniform local limit theorem, namely <Ref>.    The argument is based on the Fourier inversion formula and the analysis of characteristic functions. It exploits the reduction techniques recalled in <Ref> but the treatment requires a new and crucial step: the domain reduction.

We use the notations $(\kg, *, \tkg, \Xab, \dd, (\km^{(b)})_{b\leq 2s-1}, \Xm, \chi, *')$ from   <Ref>. We let $\mu$ be a probability measure on $\kg$, with finite moment of order $m_{\mu}$ for the weight filtration   where $m_{\mu}>2$, and such that $ \mu_{ab}$ has  expectation $\Xab$ and is non-lattice. In particular $\E(\mu^{(1)})=\Xm$. We will  use the bias extension $\tmu=\mu-\Xm+\chi$  on $\tkg$, the notations for deviations  $\tmu^{*N}_{g,h}$, $\DN(\delta_{0})$ from <Ref> where $N\geq 1$, $g,h\in \tkg$, as well as the truncation $T_{N}\tmu$  introduced in <Ref>. We set $p:\tkg\rightarrow \kg, x\oplus t\chi \mapsto x+t\Xm$ and notice it is a morphism for $*$.

We will deduce <Ref> from a more general theorem, <Ref> below, that deals with truncated random products and works under a weaker moment assumption. <Ref> will then follow from <Ref> by applying <Ref>.  <Ref>  will be useful in the applications given later in <Ref>. In order to state <Ref>, we need to set up some additional notation: we fix a (small) parameter $\gamma_{0} \in (0,1)$ to be specified below and set for $a\in \llbracket 1, s\rrbracket$,

$$\gamma_{a}= (32s)^{-a}\gamma_{0}\1_{a<s} \,\,\, \,\,\,  \,\,\, \,\,\, \,\,\,\,\,\,N_{a}=\lfloor N^{1-\gamma_{a}}\rfloor   \,\,\, \,\,\, \,\,\,\,\,\, \,\,\, \,\,\,\kappa_{N,a}=(T_{N_{a}} \tmu)^{\otimes N_{a}-N_{a-1}}.$$

We then define

\begin{align*}
&\Upsilon^N=\Pi_{\star}(\kappa_{N,1}\otimes \dots \otimes \kappa_{N,s})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&&\Theta^N=p_{\star}(\Upsilon^N)\\
& \Upsilon^N_{g,h}=g*\Upsilon^N*h   \,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,&&\thetaNgh=p_{\star}(\Upsilon^N_{g,h})
\end{align*}

So $\Upsilon^N$ is a measure on $\tkg$, which is the image under the Lie product of the $s$ measures $\kappa_{N,a}$, each one of which is a product of several copies of the truncated measure $T_{N_{a}} \tmu$. And $\Theta^N$ is the projection of $\Upsilon^N$ onto $\kg$. We say that $\Theta^N$ is  the  gradual truncated approximation of $\mu^{*N}$. This truncation will play a role later in the proof of <Ref>.

We are now ready to state <Ref>. The standing assumption on $\mu$ is that it has a finite moment of order $m_\mu>2$ for the weight filtration (associated to $\Xab$). Note that this is weaker than the assumption made in <Ref>.

Keep the  above notations.
Then for every $f \in C_{c}(\kg)$ and every $0\leq \delta_{0}\lll \gamma_{0} \lll1$, $N\geq 1$, $g,h\in \tkg$ one of which lies in $\DN(\delta_{0})$, we have

$$|\thetaNgh (f) -(\DilN \nu*N\Xm)(f)| =o_{f, \gamma_{0}}(N^{-\frac{\dd}{2}}).$$

We start with an overview of the proof. Let $f :\kg\rightarrow \R$ be a continuous integrable test function. We aim to estimate $\thetaNgh(f)$ for large $N$. If  $\hf$ is integrable, the Fourier inversion formula allows to write, for all $N\geq 1$, $g,h\in \tkg$,

\begin{align*}
\thetaNgh (f)&=\int_{\dkg} \hf(\xi) \,\widehat{\thetaNgh}(\xi) d\xi \nonumber \\
&=\int_{\dkg} \hf(\xi) \,\huNgh(\xi) e^{-2i\pi \xi\circ p(g^{(\chi)}+N\chi+h^{(\chi)})}d\xi
\end{align*}

where we identify $\xi$ with its extension to $\tkg$ defined by setting $\xi (\chi)=0$, and where $g^{(\chi)}, h^{(\chi)}$ denote the $\R\chi$ components  of $g,h$ in the decomposition <ref>.

Assume that $\hf$ is supported on  the ball $B_{\dkg}(0, R)$ of radius $R>0$. Then

\begin{align*}
\thetaNgh  (f) =\int_{B_{\dkg}(0, R)} \hf(\xi) \,\huNgh(\xi) e^{-2i\pi \xi\circ p(g^{(\chi)}+N\chi+h^{(\chi)})}d\xi.
\end{align*}

In what follows we will use the aperiodicity (non-lattice) assumption  on $\mu$ together with a path-swapping argument inspired by that of Diaconis-Hough [29] to restrict the integration domain even further to a small neighborhood of the origin, namely to the reduced domain

$$\UN(\gamma_{0}):=\{\xi \in \dkg  \,:\, \forall b\leq 2s-1,\,\,\|\xi_{|\kg^{(b)}}\| \leq  N^{-b/2+\gamma_{0}} \}.$$

[Domain reduction]propositiondomainreduction
For $\delta_{0} \lll \gamma_{0}$, we have for all $Q>0$, $N\geq 1$, $g,h\in \DN(\delta_{0})$, and $\xi \in B_{\dkg}(0,R) \smallsetminus \UN(\gamma_{0})$,

$$ |\huNgh(\xi)| \ll_{ R, \gamma_{0},Q} N^{-Q}.$$

Hence, this allows us to write:

\begin{align*}
\thetaNgh  (f) =\int_{\UN(\gamma_{0})} \hf(\xi) \,\huNgh(\xi) e^{-2i\pi \xi\circ p(g^{(\chi)}+N\chi+h^{(\chi)})}d\xi  \,&+\|f\|_{L^1}O_{R, \gamma_{0},Q}( N^{-Q}). \end{align*}

On the reduced domain, we may  replace $*$ by its graded counterpart $*'$ (<Ref>) up to an error $O(N^{-1/2+O(\gamma_{0})})$, then replace  $\mu$ by any order $2$ approximation $\eta$ with finite moments of all orders (<Ref>) up to an error $O_{\eta}(N^{-\frac{1}{2}\min(1,m_{\mu}-2)+O(\gamma_{0})})$. The total error, once integrated on $\UN(\gamma_{0})$, will be of order $O_{\eta, \gamma_{0}}\left(N^{ -\frac{1}{2}(\dd +\min(1, m_{\mu}- 2))+O(\gamma_{0}) } \right)$.
We will then choose   $\eta$ so that  $\teta^{*'N}=D_{\sqrt{N}}\nu*'N\chi$ for all $N$,  and infer the following local limit theorem with power saving and moderate bideviations for the test function $f$.

[LLT with power saving]propositionTLLFouriercompact
Let $R>0$. For every integrable function $f\in C^{\infty}(\kg)$ with Fourier transform supported on $B_{\dkg}(0,R)$, and every  $\delta_{0}\lll \gamma_{0}$,  $N\geq 1$, $g,h\in \DN(\delta_{0})$,    we have

$$|\thetaNgh  (f) - (\DilN \nu * N\Xm )_{p(g),p(h)}(f) | =  \|f\|_{L_{1}} O_{R, \gamma_{0}}\left(N^{ -\frac{1}{2}(\dd +\min(1, m_{\mu}- 2))+O(\gamma_{0}) } \right). $$

By $L^1$-approximation of the test function by functions with compactly supported Fourier transform, and using the fact that the family of deviations authorized in <Ref> does not depend on the radius $R$ bounding the support of the Fourier transform, we deduce:

[LLT with moderate bideviations]propositionTLLfcompact
For every continuous function with compact support $f\in C_{c}(\kg)$, for  $ \delta_{0} \lll \gamma_{0}\lll 1$, $N\geq 1$, $g,h\in \DN(\delta_{0})$,

$$|\thetaNgh   (f) - (\DilN \nu * N\Xm )_{p(g),p(h)}(f) | =   o_{f, \gamma_{0}}(N^{-\frac{\dd}{2}}).$$

Finally the proof of <Ref> will be complete if we manage to authorize one of the deviations to be arbitrary. For this it is sufficient to check that in that situation the terms $\Theta^N_{p(g),p(h)}(f)$, $(\DilN \nu * N\Xm )_{p(g),p(h)}(f)$ will be negligible compared to $N^{-\frac{\dd}{2}}$. This is the role of the next lemma.

For every continuous function with compact support  $f\in C_{c}(\kg)$, for  $\gamma_{0}\in [0,1]$, for $\delta_{0} \ggg \delta_{1}>0$,  $N\geq 1$, $g,h \in \tkg$ with $p(g)\notin p(\DN(\delta_{0}))$, $h\in \DN(\delta_{1})$

$$|\Theta^N_{p(g),p(h)}  (f)|=   o_{f, \delta_{0}, \delta_{1}}(N^{-\frac{\dd}{2}})  \,\, \, \,\, \, \,\, \, \,\, \, \,\, \, |(\DilN \nu * N\Xm )_{p(g),p(h)}(f) | =   o_{f, \delta_{0}, \delta_{1}}(N^{-\frac{\dd}{2}}) $$

and this also holds if we switch the conditions on $g,h$.

This concludes the (overview of the) proof of <Ref>.

§.§ Path-swap

The main part of the argument is the domain reduction, namely  <Ref>. For this we will use  a path-swap technique inspired by the work of  Diaconis and Hough in [29, 49]. We take advantage of the fact that the law of the product $X_1*\ldots*X_N$ is invariant under permutations of the $X_i$ by averaging over a certain subset of permutations in $Sym(N)$, in order to generate smoothness in the higher commutators. This is akin to a kind of discrete derivative and its use in combination to the Cauchy-Schwarz inequality as effected below can be viewed as a kind of non-abelian analogue (for commutator calculus) of the van der Corput trick, which is ubiquitous in nilpotent dynamics [61, 48, 39]. In what follows the average is taken over a certain elementary abelian $2$-subgroup of $Sym(N)$.  In this subsection, as well as in  [29, 49], the path-swap technique is tailored to deal with the central filtration of $\kg$. Later, we will need extra work to obtain information regarding the weight filtration instead.

Fix $a\in \{2, \dots, s\}$.  The group  $F= (\Z/2\Z)^{a-1}$ acts on  sequences of $2a-1$ elements $\uu=(u_{-a+1}, \dots,u_{0}, \dots ,u_{a-1})$ as follows. If $(\overline{e}_{i})_{1\leq i\leq a-1}$ is the standard basis of $F=(\Z/2\Z)^{a-1}$  we make $\overline{e}_i$ act on $\uu$ by swapping $u_{i}$ and $u_{-i}$. This defines an action of $F$. For example,

$$(1,1,0).(u_{-3}, u_{-2}, u_{-1}, u_{0}, u_{1},u_{2},u_{3})=(u_{-3}, u_{2}, u_{1}, u_{0}, u_{-1},u_{-2},u_{3}) $$

Let $\uu_{1}\dots \uu_{N'}$  be a concatenation of $N'$ sequences of $2a-1$ elements. One can see every element of $F$ as a column vector, so that $F^{N'}\equiv M_{(a-1), N'}(\Z/2\Z)$. For $\sigma \in F^{N'}$, we write $\sigma^j \in F$ for the $j$-th column, $\sigma_{i}$ for the $i$-th row, and  $\sigma_i^j$ for the $(i,j)$ coordinate. In particular $\sigma= [\sigma_1,\ldots,\sigma_{a-1}]^t$, where the superscript $t$ denotes the transpose of the matrix. The element $\sigma$ permutes the letters of $\uu_{1}\dots \uu_{N'}$ by letting the $j$-th column $\sigma^j$ act on the $j$-th block $\uu_{j}$.

Let $k, N'\geq 1$, $N= N'(2a-1)k$. Every finite sequence $x_{1}, \dots, x_{N}$
can be seen as a concatenation of $N'$ blocks, each of which is made of $(2a-1)$ sub-blocks  of length $k$. More precisely,

$$x_{1}\dots x_{N}= \uu_{1}\dots \uu_{N'} $$

$ \uu_{j}=(\uu_{j,-a+1}, \dots,  \uu_{j,0}, \dots\uu_{j,a-1}) $ and

\begin{align*}
\uu_{j,-a+l}
=  (x_{(j-1)(2a-1)k +(l-1)k+1}, \dots, x_{(j-1)(2a-1)k +lk} ).
\end{align*}

Let $\kappa$ be a probability measure on $\tkg^N$ such that $\Pi_{\star}\kappa$ and $\Pi_{\star}\sigma_{\star}\kappa$ have same distribution for any $\sigma\in F^{N'}$ (for instance $\kappa$ could be the product of $N$ copies of the same law on $\tkg$). For $\xi \in \dkg$, we can write

\begin{align*}
\widehat{\Pi_{\star}\kappa}(\xi)&= \int_{\tkg^N} e_{\xi}(\Pi(\xx)) \,\,d\kappa(\xx)\\
&=\int_{\tkg^N}\mathbb{E}_{\sigma_{a-1}}\dots \mathbb{E}_{\sigma_{1}} e_{\xi}(\Pi(\sigma. \xx)) \,\,d\kappa(\xx)
\end{align*}

where the $\sigma_{i}$'s refer to independent random rows, chosen with the uniform probability measure on $(\Z/2\Z)^{N'}$.

Applying the Cauchy-Schwarz inequality, followed by Fubini's theorem, we obtain

\begin{align*}
& \leq\int_{\tkg^N}\mathbb{E}_{\sigma_{a-1}}\dots |\mathbb{E}_{\sigma_{1}} e_{\xi}(\Pi(\sigma. \xx))|^2 \,\,d\kappa(\xx)\\
& =\int_{\tkg^N}\mathbb{E}_{\sigma_{a-1}}\dots \mathbb{E}_{\sigma_{1}, \tau_{1}} e_{\xi}\left(\Pi(\sigma. \xx) - \Pi([\tau_{1},\sigma_{2},\dots, \sigma_{a-1}]^t. \xx)\right) \,\,d\kappa(\xx)\\
& =\int_{\tkg^N} \mathbb{E}_{\sigma_{1}, \tau_{1}}\mathbb{E}_{\sigma_{a-1}}\dots  \mathbb{E}_{\sigma_{2}} e_{\xi}\left(\Pi(\sigma. \xx) - \Pi([\tau_{1},\sigma_{2},\dots, \sigma_{a-1}]^t. \xx)\right) \,\,d\kappa(\xx)
\end{align*}

where $\tau_{1}$ is a new independent row variable with uniform distribution on $(\Z/2\Z)^{N'}$.  We can repeat the same process and obtain:

\begin{align*}
|\widehat{\Pi_{\star}\kappa}|^{2^{a-1}}\leq \int_{\tkg^N} \mathbb{E}_{\sigma_{1}, \tau_{1}}\dots  \mathbb{E}_{\sigma_{a-1}, \tau_{a-1}} e_{\xi}\left( (A_{\sigma, \tau} \Pi).\xx)\right) \,\,d\kappa(\xx)
\end{align*}

(A_{\sigma, \tau} \Pi). \xx= \sum_{\eps \in \{0,1\}^{a-1}}   (-1)^{|\eps|} \Pi \circ  (\sigma\tau)_{\eps}\,. \,\xx

and where given $\eps=(\eps_{1}, \dots, \eps_{a-1})$, we set $|\eps|=\sum \eps_{i}$ and the notation $( \sigma\tau)_{\eps}$ refers to the matrix whose $i$-th row is $\sigma_{i}$ if $\eps_{i}=0$ and $\tau_{i}$ otherwise.  The next lemma, inspired by  <cit.> and <cit.>, states that the operator $A_{\sigma, \tau}$ kills the components $\Pi^{[a']}(\xx)$ for $a'<a$ and desynchronizes the contributions of different blocks for the component $\Pi^{[a]}(\xx)$. The contribution of each block is also made explicit.

Let $k, N'\geq 1$, $a\in \{2, \dots, s\}$,  $N=N'(2a-1)k$ and $\sigma, \tau \in F^{N'}$.  For all $\xx=(x_{1}, \dots, x_{N}) = \uu_{1} \dots \uu_{N'}=(\uu_{j,l})_{1\leq j\leq N', |l|\leq a-1}$ in $\tkg^N$.

(i) If $a' \in \{1, \dots, a-1\}$, then

$$(A_{\sigma, \tau} \Pi^{[a']}).\xx=0$$

\begin{align*}
(A_{\sigma, \tau}\Pi^{[a]}).\xx
& =\sum_{j=1}^{N'} (A_{\sigma^j, \tau^j} \Pi^{[a]}).\uu_{j}
\end{align*}

where $A_{\sigma^j, \tau^j}$ corresponds to the operator defined when we only consider the $j$-th column for $\sigma$ or $\tau$. Moreover,

\begin{align*}(A_{\sigma^j, \tau^j} \Pi^{[a]}).\uu_{j} =0
\end{align*}

unless $\sigma^{j}_{i}\neq \tau^{j}_{i} \text{ for all } i \in \llbracket 2, a\rrbracket$, in which case

\begin{align*}
(A_{\sigma^j, \tau^j} \Pi^{[a]}).\uu_{j} =   (-1)^{|\sigma^{j}|} L^{[a]}(\overline{\uu}_{j,0}, \overline{\uu}_{j,1}-\overline{\uu}_{j,-1}, \dots,  \overline{\uu}_{j,a-1}-\overline{\uu}_{j,-a+1})
\end{align*}

where $L^{[a]}$ is the  $a$-bracket defined in <Ref>, $\overline{\uu}_{j, l}$ stands for the sum of the terms contained in the sub-block  $\uu_{j,l}$.

Remark. If $\yy =\xx_{0}\oplus \xx \oplus \xx_{1}$ where $\xx_{0}$ and $\xx_{1}$ are prefix and suffix of finite length concatenated to  $\xx$, we can always perform this average on the permutations affecting only the block decomposition of   $\xx$ and obtain via Cauchy-Schwarz the the corresponding inequality, with some $A_{\sigma, \tau}\Pi(\yy)$. It follows from <Ref> that $A_{\sigma, \tau}\Pi^{[\leq a]}(\yy) = A_{\sigma, \tau}\Pi^{[\leq a]}(\xx)$, in other words the contributions of the prefix and the suffix to the components involving at most $a$ iterated brackets are erased by the averaging. To see this, just add some zero coordinates so that $\xx_{0}$ and $\xx_{1}$ have length which are divisible by $(2a-1)k$, then apply the lemma to the permutations $\sigma', \tau'$ on $\yy$ extending $\sigma, \tau$ by acting trivially outside of $\xx$.

Recall the formula for the product

$$\Pi(\xx) =\sum_{r=1}^s\sum_{ \underset{ \,1\leq l_{1}< \dots < l_{t}\leq N}{r_{1}+\dots+r_{t}= r}, r_{i}\neq 0}  L_{\rr}(x_{l_{1}}^{\otimes r_{1}}, \dots, x_{ l_{t}}^{\otimes r_{t}})$$

It follows that

$$\Pi((\sigma\tau)_{\eps}.\xx) =\sum_{r=1}^s \sum_{\underset{r_{i}\neq 0}{r_{1}+\dots+r_{t}= r}} \sum_{l_{1}, \dots, l_{t}\leq N} \1_{(\sigma\tau)_{\eps} l_{1}< \dots <(\sigma\tau)_{\eps} l_{t}} L_{\rr}(x_{l_{1}}^{\otimes r_{1}}, \dots, x_{ l_{t}}^{\otimes r_{t}})$$

A key observation is that for fixed $r,\rr, l_{1}, \dots, l_{t}$, we have

$$ \sum_{\eps \in \{0,1\}^{a-1}}   (-1)^{|\eps|} \1_{(\sigma\tau)_{\eps} l_{1}< \dots <(\sigma\tau)_{\eps} l_{t}}=0$$
as long as at least one of the $a$ different types of sub-blocks is avoided by the subset of subscripts $l_i$. Here the sub-block $u_i$ of size $k$ in $\underline{u}=(u_{-a+1},...,u_0,...,u_{a-1})$ is said to be of type $|i| \in [0,a]$. In particular, if $t\leq a-1$, or if $t=a$ and two $l_{i}$ belong to a same type of sub-block. Indeed, one sees directly in this case that the relative order $((\sigma\tau)_{\eps} l_{i})_{i\leq t}$ does not depend on the choice of $\sigma$ or $\tau$ for a particular row coordinate.  The sum is also null if $t=a$ and the $l_{i}$ meet different (large) blocks $\uu_{j}$. Indeed, one may assume there is exactly one subscript $l_{i}$ per type of sub-block. Consider a block $\uu_{j_{0}}$ which meets the $l_{i}$'s but not via $\uu_{j_{0},0}$.  Call $l_{0}\in \{1, \dots, a-1\}$ the smallest integer such that $\uu_{j_{0}, \pm l}$ meets the $l_{i}$'s. We observe that the relative order of $((\sigma\tau)_{\eps} l_{i})_{i\leq t}$ does not depend on the choice of $\sigma_{j_{0}}$ or $\tau_{j_{0}}$. Hence, the sum vanishes as well.

This discussion justifies that for $a'<a$,

$$(A_{\sigma, \tau} \Pi^{[a']}). \xx=0$$

\begin{align*}
(A_{\sigma, \tau}\Pi^{[a]}).\xx
& =\sum_{j=1}^{N'} (A_{\sigma^j, \tau^j} \Pi^{[a]}).\uu_{j}
\end{align*}

We are led to analyse the alternating sum on a single block. We assume $N'=1$, in particular, $\xx= \uu_{-a+1}\dots \uu_{0}\dots \uu_{a-1}$ has length $N=(2a-1)k$.
Write $\mathfrak{C}$ the set of characters  $\cc : F\rightarrow \{-1, 1\}$ of $F$. We have a unique decomposition

$$\Pi(\xx) = \sum_{\mathfrak{C}} \Pi_{\mathfrak{c}}(\xx) $$

where $ \Pi_{\cc}(\xx) $ satisfies for all $\sigma\in F$,

$$ \Pi_{\cc}(\sigma. \xx) =\cc(\sigma) \Pi_{\cc}(\xx)$$

For each $ \Pi_{\cc}$, we can rewrite

$$A_{\sigma, \tau}\Pi_{\cc}(\xx)= \sum_{\eps \in \{0,1\}^{a-1}}   (-1)^{|\eps|} \Pi_{\cc} \circ  (\sigma\tau)_{\eps}\,. \,\xx =\prod_{i=1}^{a-1}\left( \cc(\sigma_{i})-\cc(\tau_{i})\right) \Pi_{\cc}(\xx)$$

which is $0$ except perhaps in the case where $\cc$ is the character $\cc_{0}$ such that $\cc_{0}(\overline{e}_{i})=-1$ and  $\sigma_{i}\neq\tau_{i}$ for all $i$. Hence,

(A_{\sigma, \tau} \Pi). \xx= \left\{
\begin{array}{ll}
0  & \mbox{if } \exists j, \, \sigma_{j}=\tau_{j} \\
(-1)^{|\sigma|}2^{a-1} \Pi_{\cc_{0}}( \xx) & \mbox{otherwise.}
\end{array}
\right.

To conclude,  we need to check the formula

\begin{align} \label{eigenproj}
\Pi^{[a]}_{\cc_{0}}(\xx)\,= \,2^{-a+1} L^{[a]}( \overline{\uu}_{0}, \, \overline{\uu}_{ 1}-\overline{\uu}_{ -1}, \,\dots,\,  \overline{\uu}_{a- 1}-\overline{\uu}_{ -a+1} )
\end{align}

We argue by induction on $a$. For $a=2$, the equality is straightforward. Suppose it holds for $a-1\geq 2$. Set $\zz=\uu_{-a+2}\dots \uu_{a-2}$. Using the Campbell-Hausdorff formula, we see that

\begin{align*}
\Pi(\xx)= \Pi(\uu_{-a+1})*\Pi(\zz) * \Pi(\uu_{a-1})=\frac{1}{2}[\Pi(\zz),\, \overline{\uu}_{a-1}-\overline{\uu}_{-a+1}] +I(\xx) + B(\xx)
\end{align*}

where $I(\xx)= \overline{\uu}_{-a+1}+\Pi(\zz)+\overline{\uu}_{a-1} + \frac{1}{12}\ad(\Pi(\zz))^2( \overline{\uu}_{a-1}+\overline{\uu}_{-a+1})$ is invariant by permutation of $\uu_{-a+1}$ and $\uu_{a-1}$,
and $B(\xx)$ involves multibrackets with at least two occurrences of  variables in $\uu_{-a+1}$ or $\uu_{a-1}$.  Swapping suitable pairs $\uu_{l}$, $\uu_{-l}$, we see that  $I^{[a]}_{\cc_{0}}(\xx)$ and $B^{[a]}_{\cc_{0}}(\xx)$ must be identically null.
It follows that

\begin{align*}
\Pi^{[a]}_{\cc_{0}}(\xx)=\frac{1}{2}[\Pi^{[a-1]}_{\cc_{0}}(\zz), \,\overline{\uu}_{a-1}-\overline{\uu}_{-a+1}]
\end{align*}

This justifies the induction step, whence the formula (<ref>).

<Ref> is used to show the following upper bound.

Let $a\in \{2, \dots, s\}$ and $\xi \in \dkg$ vanishing on $\kg^{[a+1]}$.  For each integer  $N\geq 1$ of the form  $N = kaN'$ where $k,N'\geq 1$,   and  all finite sequences  $\xx_{0}, \xx_{1}$ in $\tkg$, for any probability measure $\omega$ on $\tkg$, we have

$$\big| \E_{\xx_{0} \oplus \omega^{\otimes N} \oplus \xx_{1}}\left[e_{\xi}\left(\Pi(\yy)  \right)\right] \big|^{2^{a-1}}  \leq \left( 1-\frac{1}{2^{a-1}}+ \frac{1}{2^{a-1}} \E_{(\omega_{ab}^{+k})^{\otimes a}}\big[\cos(2\pi \xi L^{[a]} (v_{1}, \dots, v_{a})    )\big] \right)^{N'} $$

where $\omega_{ab}$ is the projection of $\omega$ on $\km^{(1)}+\R\chi$, and $\omega^{+k}_{ab}$ is the distribution of the sum of $k$ independent variables  with law  $\omega_{ab}$.

We apply the path swapping technique on block permutations of the component $\xx$ varying with law $\kappa=\kappa_0^{\otimes N'}$ where $\kappa_0=\delta^{\otimes (a-1)k}_0\otimes \omega^{\otimes ak}$. We have artificially added $(a-1)kN'$ dummy variables equal to the identity. Then  from <Ref> and the remark following it, and from the condition $\xi_{|\kg^{[a+1]}}=0$ we obtain

\begin{align*}
\big| \E_{\xx_{0} \oplus \omega^{\otimes N} \oplus \xx_{1}}\left[e_{\xi}\left( \Pi(\yy) \right)\right]  \big|^{2^{a-1}}
&=\big| \E_{\kappa}\left[e_{\xi}\left( \Pi( \xx_{0}\oplus \xx \oplus \xx_{1}) \right)\right]  \big|^{2^{a-1}}\\
& \leq \int_{\tkg^N} \mathbb{E}_{\sigma, \tau} e_{\xi}\left( (A_{\sigma, \tau} \Pi).(\xx_{0}\oplus \xx \oplus \xx_{1})\right) \,\,d\kappa(\xx)\\
& = \int_{\tkg^N} \mathbb{E}_{\sigma, \tau} e_{\xi}\left( (A_{\sigma, \tau} \Pi^{[a]}).\xx\right) \,\,d\kappa(\xx)\\
& =  \mathbb{E}_{\sigma, \tau} \int_{\tkg^N}e_{\xi}\left( (A_{\sigma, \tau} \Pi^{[a]}).\xx\right) \,\,d\kappa(\xx)\\
& =  \mathbb{E}_{\sigma, \tau} \prod_{j=1}^{N'} \int_{\tkg^{ka}}e_{\xi}\left( (A_{\sigma^j, \tau^j} \Pi^{[a]}).\uu\right) \,\,d\kappa_0(\uu)\\
& = \left( \mathbb{E}_{\sigma^1, \tau^1} \int_{\tkg^{ka}}e_{\xi}\left( (A_{\sigma^1, \tau^1} \Pi^{[a]}).\uu\right) \,\,d\kappa_0(\uu)\right)^{N'}\\
\end{align*}

where the last equation uses the independence of the column coordinates $\sigma^1$, $\tau^1$,..., $\sigma^{N'}$, $\tau^{N'}$. By <Ref>, the integral that we have obtained is equal to $1$ for a proportion $1-\frac{1}{2^{a-1}}$ of possibilities for $(\sigma^1, \tau^1)$; and for the other scenarios, it is equal to

$$ \int_{\tkg^{ka}} e_{\xi} \left((-1)^{|\sigma^{1}|}  L^{[a]}(v_{1}, \dots, v_{a})\right) \,d(\omega^{+ k})^{\otimes a}(v) $$

As $\xi$ vanishes on $\kg^{[a+1]}$, we may replace $\omega^{+ k}$ by its projection $\omega^{+ k}_{ab}$ without affecting the value of the integral.  The result follows by noticing that on the scenarios we consider, $|\sigma^{1}|$ is even or odd with the same probability.

To prove the reduction of the integration domain (<Ref>), we will apply   <Ref> to a fixed $\xi$ and some integer $k$, chosen to guarantee the right hand side in <Ref> is uniformly bounded away from  $1$. The existence of such  $k$ is justified by the next proposition. It is the main result of the section, and the only one that will be used in the rest of the proof of <Ref>. We set $\kg^{[a, b)}=\kg^{[a]}\cap \kg^{(b)}$.

Let $R>0$. There exist  constants $C,c>0$ depending only on $R$ and the initial data $(\kg, \Xab, \mu, (\km^{(b)})_b, (e^{(i)}_j)_{i,j})$ such that the following holds. Let $a\in \{1, \dots, s\}$, $\xi \in B_{\dkg}(0,R)$ vanishing on  $\kg^{[a+1]}$, and $k\geq 1$ the smallest integer for which there exists $b\geq a$ satisfying $\|k^{\frac{b}{2}}\,\xi_{|\kg^{[a,b)}}\|\geq 1$. Then, for $N'\geq 1$,  $N = kaN'$, $N_{0}\geq N$, $\xx_{0}, \xx_{1}$ finite sequences in $\tkg$, we have

$$\big| \E_{\xx_{0} \oplus (T_{N_{0}}\tmu)^{\otimes N} \oplus \xx_{1}}\left[e_{\xi}\left( \Pi(\yy)  \right)\right] \big|^{2^{a-1}} \leq C (1-c  )^{N'} $$

Let us begin with the case $a=1$, which is elementary and helps build an intuition for the case $a\geq 2$. By assumption, $\xi \in B_{\dkg}(0,R)$ is a linear form vanishing on the derived algebra $\kg^{[2]}$, and $k\geq 1$ is the smallest integer for which $\|k^{\frac{1}{2}}\,\xi\|\geq 1$. The quantity to bound can be rewritten

\begin{align*}
\big| \E_{\xx_{0} \oplus (T_{N_{0}}\tmu)^{\otimes N} \oplus \xx_{1}}\left[e_{\xi}\left( \Pi(\yy)  \right)\right] \big|= \big| \widehat{(T_{N_{0}}\tmu)_{ab}}(\xi) \big|^N= \big| \E_{(T_{N_{0}}\tmu)^{+ k}_{ab}} \left[e_{k^{\frac{1}{2}}\xi}(k^{-\frac{1}{2}}x) \right] \big|^{N'}
\end{align*}

Hence, it is sufficient to show that for $N'$ large enough depending on the initial data,

$$\big| \E_{(T_{N_{0}}\tmu)^{+ k}_{ab}} \left[e_{k^{\frac{1}{2}}\xi}(k^{-\frac{1}{2}}x) \right] \big| \leq 1-c'$$

where $c'>0$ depends on $R$ and the initial data. According to <Ref> and the inequality $N_{0}\geq \max(N',k)$, we may  remove the truncation for that purpose.
Now, let $r$ be a parameter depending only on the initial data, and that we will specify below. The aperiodicity assumption on $\mu$ implies that $|\widehat{\mu_{ab}}(\xi)|^k$ is uniformly bounded away from $1$ if $r\leq |\xi|< R$. This yields the result for this window of parameters $\xi$. But as $r$ tends to $0$, the distribution $k^{-\frac{1}{2}} \mu_{ab}^{+k}$ converges to a Gaussian distribution, whose Fourier transform outside any fixed neighborhood of $0$ is uniformly bounded away from  $1$. Thus we may choose $r$ small enough so that $|\E_{\mu^{+ k}_{ab}} \left[e_{k^{\frac{1}{2}}\xi}(k^{-\frac{1}{2}}x) \right] \big| $ is strictly less than  $1$ uniformly in $|\xi|\leq r$, which concludes the case $a=1$.

We now deal with the case $a\geq 2$. Applying <Ref> with $\omega=T_{N_{0}}\tmu$, and arguing as above to remove the truncation, it is sufficient to  show that

$$\E_{(\tmu_{ab}^{+k})^{\otimes a}}\big[\cos(2\pi \xi L^{[a]} (v_{1}, \dots, v_{a})\big] \leq 1-c'$$

where $c'>0$ depends on $R$ and the initial data.

The condition that the subgroup generated by $\mu_{ab}$ is dense in $\kg/\kg^{[2]}$ implies that for every $\xi\in \dkg$ such that $\|\xi_{|\kg^{[a]}}\|= 1$ that vanishes on $\kg^{[a+1]}$, there is a positive proportion of tuples $(v_{2}, \dots, v_{a})$ chosen via $\tmu_{ab}^{\otimes a-1}$ such that
$ v_{1}\mapsto \xi L^{[a]}(v_{1}, v_{2}, \dots, v_{a})$ is a non-zero linear form. By compactness, there exists $r_{1}=r_{1}(\mu)>0$ such that for all such $\xi$, for a $\tmu_{ab}^{\otimes a-1}$-proportion $r_{1}$ of tuples $(v_{2}, \dots, v_{a})$, the linear form $ v_{1}\mapsto \xi L^{[a]}(v_{1}, v_{2}, \dots, v_{a})$ has its norm in the interval $[r_{1}, r_{1}^{-1}]$.

Let $r>0$ be a parameter depending only on the initial data and to be specified below. Let $\xi \in B_{\dkg}(0,R)$ and $k\geq 1$  as in the statement of <Ref>.

If  $\|\xi_{|\kg^{[a]}}\|\geq r$, the above discussion combined with the aperiodicity of $\mu$ implies that

$$\E_{(\tmu_{ab}^{k})^{\otimes a}}\big[\cos(2\pi \xi L^{[a]} (v_{1}, \dots, v_{a})\big] \leq 1-c'$$

for some constant $c'>0$ depending only on $(\mu, R, r)$.

Hence, we need to consider the case $\|\xi_{|\kg^{[a]}}\|<r$. We write  $v_{i}=v_{i}'\oplus k\chi$ where $v'_{i}\in \km^{(1)}$ and expand the $a$-bracket to obtain

$$L^{[a]} (v_{1}, \dots, v_{a}) =\sum_{b=a}^{2a-1}L^{[a, b)}_{k\chi}(v'_{1}, \dots, v'_{a}) $$

where $L^{[a, b)}_{k\chi}(v'_{1}, \dots, v'_{a})$ is the sum of the terms of degree $b$, i.e. with exactly  $b-a$ occurrences of $k\chi$. Observing that

\begin{align*}
\frac{1}{k^{\frac{b}{2}}} L^{[a, b)}_{k\chi}(v'_{1}, \dots, v'_{a})=  L^{[a, b)}_{\chi}(\frac{v'_{1}}{k^{1/2}}, \dots, \frac{v'_{a}}{k^{1/2}})
\end{align*}

we can reformulate as

\begin{align*}
\xi L^{[a]} (v_{1}, \dots, v_{a}) &=\sum_{b=a}^{2a-1}k^{\frac{b}{2}}\xi\, L^{[a, b)}_{\chi}(\frac{v'_{1}}{k^{1/2}}, \dots, \frac{v'_{a}}{k^{1/2}}).
\end{align*}

We notice that for small $r$, the parameter $k$ must be large, so each term $L^{[a, b)}_{\chi}(\frac{v'_{1}}{k^{1/2}}, \dots, \frac{v'_{a}}{k^{1/2}})$ behaves like the density of $L^{[a, b)}_{\chi}(\mathscr{N}_{1}, \dots, \mathscr{N}_{a})$ where the $\mathscr{N}_{i}$'s are independent centred normal distributions on $\km^{(1)}$ with the same covariance matrix as $\mu_{ab}$. Moreover, the linear forms $k^{\frac{b}{2}}\xi_{|\kg^{[a,b)}} $ applied to the various terms have norm less than $2$, and at least one of these norms  is greater than $1$. <Ref> then follows by choosing $r$ small enough and applying <Ref> below.

There exists a constant $c'_{1}=c_{1}(\mu)>0$ such that, for  $a\in \{2, \dots, s\}$, for all linear forms  $(\varphi_{b}\in \widehat{\kg^{[a,b)}})_{a\leq b\leq 2a-1}$ satisfying  $\varphi_{b| \kg^{[a+1,b)}}=0$ and $1\leq \max_{b} \|\varphi_{b}\|\leq 2$, one has

$$\E_{\mathscr{N}_{1}\otimes \dots \otimes \mathscr{N}_{a}}\left(\cos\left(2\pi \sum_{b=a}^{2a-1} \varphi_{b}L^{[a, b)}_{\chi}(x_{1}, \dots, x_{a})\right)\right) <1-c'_{1}$$

Fix $a$. The condition on the norms of the $\varphi_{b}$'s defines a compact subset of  $\prod_{b\geq a}\widehat{\kg^{[a,b)}}$. The expectation we aim to bound varies continuously with respect to the $\varphi_{b}$'s. So it is sufficient to check the lemma for a fixed family  $(\varphi_{b})_{b\geq a}$. This boils down to showing that the sum $\sum_{b=a}^{2a-1} \varphi_{b}L^{[a, b)}_{\chi}(x_{1}, \dots, x_{a})$ is not identically zero on  $(\km^{(1)})^{\otimes a}$. To see this, we call $b_{0}$ the greatest integer such that $\varphi_{b_{0}}\neq 0$. In particular, there must exist some $a$-bracket $L^{[a]}(\vv)\in \kg^{[a,b_{0})}$ such that $\varphi_{b_{0}}(L^{[a]}(\vv))\neq 0$. As $\varphi_{b_{0}}$ vanishes on $\kg^{[a+1]}$, we must have $\vv \in (\km^{(1)}\cup \R \chi)^a$. The number of occurrences of variables in $\R \chi$ is at least $b_{0}-a$, and, up to replacing some of them by their analogue in $\R \Xm$, we may suppose that this number is exactly $b_{0}-a$. Now let $\xx\in (\km^{(1)})^{\otimes a}$ be defined by $x_i= v_i \1_{v_i \notin \R \chi} $. We get

$$\sum_{b=a}^{2a-1} \varphi_{b}L^{[a, b)}_{\chi}(x_{1}, \dots, x_{a})= \varphi_{b_{0}}(L^{[a]}(\vv)) \neq 0.$$

This concludes the proof.

§.§ Reducing the domain of integration

We apply <Ref> obtained in the previous section to show
<Ref>, which guarantees that $|\huNgh(\xi)|$ is rapidly decreasing uniformly in $g,h\in \DN(\delta_{0})$ and  $\xi \in B_{\dkg}(0,R)\smallsetminus \UN(\gamma_{0})$. We recall that for  $\gamma_{0}>0$, $\delta_{0}\geq 0$, we write

$$\UN(\gamma_{0})=\{\xi \in \dkg  \,:\, \forall b\leq 2s-1,\,\,\|\xi_{|\kg^{(b)}}\| \leq  N^{-b/2+\gamma_{0}} \},$$

$$\DN(\delta_{0})=\{x \in \tkg \,:\, \forall b\leq 2s-1,\,\, \|x^{(b)}\| \leq  N^{b/2+s^{-1}\delta_{0}}\}.$$

The proof uses a partition of the variable $\xx$ into two pieces $\xx=\xx_{a}\oplus \xx_{a+1}$ where the length of $\xx_{a}$ is  $N_{a}:=\lfloor N^{1-\gamma_{a}}\rfloor$. We will need the following lemma to guarantee that the mass of $\Pi^{[a,b)}(\xx)$ is essentially due to the contribution of $\Pi^{[a,b)}(\xx_{a+1})$, i.e. to the brackets that only contain variables from $\xx_{a+1}$. We also authorize moderate deviations. The fact that $\Upsilon^N$ truncates  gradually the variables in $\tmu^{* N}$ plays a role in the proof of the next lemma.

Let $a\in\{1, \dots, s\}$. Decompose every $\xx\in \tkg^N$  as a concatenation  $\xx=\xx_{a}\oplus \xx_{a+1}$ where $\xx_{a}$ has length $N_{a}$. Then for $b\geq a$, $N\geq 1$, $g,h\in \DN(\delta_{0})$, $m\geq1$,

$$\E_{\kappa_{N,1}\otimes \dots \otimes \kappa_{N,s} }\left[ \|\Pi^{[a,b)}(g,\xx,h) -\Pi^{[a,b)}(g,\xx_{a+1},h)\|^{2m} \right] \ll_{m} N^{m(b +2\delta_{0})}\left( \frac{N_{a}}{N}\right)^m.$$

Note first that $\Pi^{[a,b)}(g,\xx,h)$ is a finite formal linear combination (depending on $a,b$ but not $N$) of statistics in $\xx$ associated to  monomials of degree at most $b$, and such that each statistics of degree $b' \leq b$ has its coefficients depending on $g,h$ and bounded above by
$$O(1)\sum_{i_{1}+\dots+j_{l}=b-b' } \|g^{(i_{1})}\|\dots \|g^{(i_{k})}\| \,\|h^{(j_{1})}\|\dots \|h^{(j_{l})}\|$$

if $b'<b$, and $O(1)$ otherwise.

In view of the condition $g,h\in \DN(\delta_{0})$,  the proof of <Ref>  boils down to showing that for a statistics $M^\infty$ induced by a monomial $M:\tkg^t\rightarrow \R$ of  degree $b'$ (see <Ref>),  we have

\begin{align} \label{tronc1}
\E_{\kappa_{N,1}\otimes \dots \otimes \kappa_{N,s} }\left[ |M^\infty(\xx) - M^{\infty}(\xx_{a+1})|^{2m} \right] \ll_{m} N^{mb'}\left( \frac{N_{a}}{N}\right)^m.
\end{align}

Observe that the difference above can be rewritten as

$$M^\infty(\xx) - M^{\infty}(\xx_{a})=\sum_{\substack{n_{1}<\dots <n_{t}\\ n_{1}\leq N_{a}}} M(x_{n_{1}},\dots, x_{n_{t}}).$$

Given $\alpha\in \N^{\dim\tkg}$,  denote by $x\mapsto x^\alpha$  the corresponding monomial on $\tkg$ (say for some ordering of the basis $e^{(i)}_{j}$ fixed in <Ref>) and write $d(\alpha)$ its degree for the weight filtration. Expanding the power $2m$, we can write  for $N\geq 1$, $\xx\in \tkg^N$,

\begin{align*}
(M^\infty(\xx) - M^{\infty}(\xx_{a}))^{2m} &= \sum_{l=1}^{2mb} \sum_{\aalpha \in \mathcal{A}^1_{l}}  \sum_{\nn\in \mathcal{B}_{\aalpha}}  C_{\aalpha} x_{n_{1}}^{\alpha_{1}}\dots  x_{n_{l}}^{\alpha_{l}}
\end{align*}

where   $\sup_{\aalpha}|C_{\aalpha}| =O_{m}(1)$ and

$$\mathcal{A}^{1}_{l} =\left\{\aalpha \in (\N^{\dim\tkg})^l \,:\, \sum_{i=1}^l d(\alpha_{i})=2mb \,\text{ and }\,\inf_{i}d(\alpha_{i})\geq 1 \right\}$$

$$\mathcal{B}_{\aalpha} =\left\{\nn \in \llbracket 1, N\rrbracket^l \,:\,  \sum_{i=1}^l d(\alpha_{i})\1_{n_{i}\leq N_{a}} \geq 2m \right\}$$

Then by linearity and independence,

\begin{align*}
\E_{\kappa_{N,1}\otimes \dots \otimes \kappa_{N,s} }\left[(M^\infty(\xx) - M^{\infty}(\xx_{a}))^{2m}\right] &= \sum_{l=1}^{2mb} \sum_{\aalpha \in \mathcal{A}^1_{l}}  \sum_{\nn\in \mathcal{B}_{\aalpha}}  C_{\aalpha}\E_{T_{N(n_{1})}\tmu}( x_{n_{1}}^{\alpha_{1}})\dots  \E_{T_{N(n_{l})}\tmu}(x_{n_{l}}^{\alpha_{l}})
\end{align*}

where $N(n_{i}) = N_{j}$ if $n_{i} \in \{N_{j-1}+1, \dots, N_{j} \}$.

The terms for which some $x_{i}^{\alpha_{i}}$ has degree $1$ satisfy $\E_{\tmu}(x_{i}^{\alpha_{i}})=0$, hence the $\aalpha$ that contribute belong to

$$\mathcal{A}^{2}_{l}=\left\{\aalpha \in \mathcal{A}^{1}_{l} \,:\, \inf_{i}d(\alpha_{i})\geq 2 \right\}.$$

This forces $l$ to belong to $\{1, \dots, mb\}$:

$$\E_{\kappa_{N,1}\otimes \dots \otimes \kappa_{N,s} }\left[(M^\infty(\xx) - M^{\infty}(\xx_{a}))^{2m}\right]= \sum_{l=1}^{mb} \sum_{\aalpha \in \mathcal{A}^1_{l}}  \sum_{\nn\in \mathcal{B}_{\aalpha}} C_{\aalpha}\E_{T_{N(n_{1})}\tmu}( x_{n_{1}}^{\alpha_{1}})\dots  \E_{T_{N(n_{l})}\tmu}(x_{n_{l}}^{\alpha_{l}}).$$

Now notice that for $d(\alpha)\geq 2$, $N'\geq 1$,

$$\E_{T_{N'}\tmu}(|x^\alpha|)= \E_{\tmu}\left(|(T_{N'}x)^\frac{2\alpha}{d(\alpha)}|^{\frac{d(\alpha)}{2}-1} |(T_{N'}x)^\frac{2\alpha}{d(\alpha)}| \right)=O(N'^{\frac{d(\alpha)}{2}-1})$$

because $|T_{N'}x^\frac{2\alpha}{d(\alpha)}|=O(N')$ and  $ \E_{\tmu}(|(T_{N'}x)^\frac{2\alpha}{d(\alpha)}|)\leq  \E_{\tmu}(|x^\frac{2\alpha}{d(\alpha)}|)+o(1)$ is bounded independently of $N'$. In particular, for $n_{i}\leq N_{a}$ one has $\E_{T_{N(n_{i})}\tmu}( |x_{n_{i}}^{\alpha_{i}}|)\leq N_{a}^{\frac{d(\alpha_{i})}{2}-1}$, while for $n_{i}\geq N_{a}+1$ we have $\E_{T_{N(n_{i})}\tmu}( |x_{n_{i}}^{\alpha_{i}}|)\leq N^{\frac{d(\alpha_{i})}{2}-1}$. We deduce that for $l\leq mb$, $\aalpha \in \mathcal{A}^2_{l}$, $n\in \mathcal{B}_{\aalpha}$ one has

$$\E_{T_{N(n_{1})}\tmu}( x_{n_{1}}^{\alpha_{1}})\dots  \E_{T_{N(n_{l})}\tmu}(x_{n_{l}}^{\alpha_{l}}) = O_{m}(1)N^{mb} \left(\frac{N_{a}}{N}\right)^{m} N^{-|\{i:n_{i}\leq N_{a}\}|}_{a}N^{-|\{i:n_{i}> N_{a}\}|}.$$

The result follows as the parameters $l, \aalpha$ belong to a bounded set independent of $N$, and because
$\sum_{\nn\in \llbracket 1, N\rrbracket^l} N^{-|\{i:n_{i}\leq N_{a}\}|}_{a}N^{-|\{i:n_{i}> N_{a}\}|}=O_{l}(1)$.

We now turn to the proof of <Ref>. We are going to use a decomposition $\kg=\oplus_{a\leq b}\km^{[a,b)}$, that fits both the central descending series $(\kg^{[a]})_{a\leq s}$ and weight filtration $(\kg^{(b)})_{b\leq 2s-1}$, i.e. such that $\kg^{[a,b)}:=\kg^{[a]}\cap \kg^{(b)}=\oplus_{a'\geq a, b'\geq b}\km^{[a',b')}$ for all $a\leq b$.  This is always possible: for each $a\leq b$ find  a subspace $\km^{[a,b)}$  so that

$$\kg^{[a,b)}=\km^{[a,b)}\oplus \left( \kg^{[a+1, b)}+\kg^{[a, b+1)}\right) $$

We  shall also consider the projection $p^{[a,b)} : \kg\rightarrow \kg^{[a,b)}$ modulo the other $\km^{[a',b')}$.

and for $\xi \in \dkg$, we set  $\xi^{[a,b)}=\xi\circ p^{[a,b)}$, and $\xi^{[a]}= \xi^{[a,a)}$. In view of <ref>, it is useful to notice the following inequalities

$$\beta \|\xi^{[a,b)}\| \leq \|\xi_{|\kg^{[a,b)}}\| \leq \|\xi^{[a,b)}\| $$

for some small constant  $\beta>0$ depending on the choices of $(\km^{[a',b')})_{a',b'}$.

Recall the choice of parameters $\gamma_{a}$ defined at the start of <Ref>. We set
$ \eps_{a}:=\frac{1}{8}\gamma_{a-1}+2s\gamma_{a}$.
We will use the property (whose verification is straightforward) that for $\delta_{0}\lll \gamma_{0}$ and $1\leq a\leq s-1$, one has

$$\frac{1}{4s}\eps_{a}> \gamma_{a} > 4(\eps_{a+1} +\delta_{0}),$$

while for $a=s$, we still have $\frac{1}{4s}\eps_{s}> \gamma_{s}=0$. From now on, we fix such $\delta_{0}$.

It is sufficient to check the proposition where  $\UN(\gamma_{0})$ is replaced by

$$\UN' =\{\xi \in \dkg : \,\forall a, \forall b,\,\|\xi^{[a,b)}\|\leq N^{-\frac{b}{2}+\eps_{a}}\}$$

because $\UN'\subseteq \UN(\gamma_{0})$ due to $\sup_{a}\eps_{a}\leq \gamma_{0}$.

Let $\xi \in \dkg$ such that $\xi \notin \UN'$.
We must upper bound the norm of

$$\huNgh(\xi)=\E_{g\otimes \kappa_{N,1}\otimes\dots \otimes \kappa_{N,s} \otimes h}\left[  e_{\xi}(\Pi(\yy))\right]$$

uniformly in $g,h \in \DN(\delta_{0})$ and independently of $\xi$.

Let $a\in \{1, \dots, s\}$ be maximal such that  $\|\xi^{[a,b)}\|> N^{-\frac{b}{2}+\eps_{a}}$ for some $b$. Write $\xi= \xi^{[a+1]}\oplus \xi'$ the decomposition of $\xi$ corresponding to $\kg=\kg^{[a+1]}\oplus \bigoplus_{\substack{\,\, a'< a, b'}}\km^{[a',b')}$. Write also the variable  $\yy$ as  $\yy=(g,\xx,h)$ where $\xx$ varies with law $\tmu^{\otimes N}$.
The first step of the proof is to use that $a$ is maximal to get rid of the component $\Xi(\yy) :=\xi^{[a+1]}\Pi(\yy)$ in the formula for $\huNgh(\xi)$ above, and thus reduce to the situation where $\xi$ vanishes on $\kg^{[a+1]}$. To do so, we use a truncation argument which consists in removing the dependence on a prefix of $\xx$ in $\Xi(\yy)$. The length of this prefix must be small enough to ensure that its deletion from $\Xi(\yy)$ is negligible, but also large enough so that we can apply to it the path swapping method from the previous section.

To carry out this plan,  we decompose the variable $\xx\in \tkg^N$ as  $\xx=\xx_{a}\oplus \xx_{a+1}$ where $\xx_{a}$ has length $N_{a}:=\lfloor N^{1-\gamma_{a}} \rfloor$, and write $\yy_{a+1}=(g,\xx_{a+1},h)$ the variable obtained by removing from $\yy$ the component $\xx_{a}$.
We also fix a parameter $m\geq 1$ and set $T_{2m-1}(t)= \sum_{j}\frac{(-2i\pi t)^j}{j!} $ the Taylor expansion up to order $2m-1$ of $t\mapsto \exp(-2i\pi t)$, so that

$$|T_{2m-1}(t)- \exp(-2i\pi t)| \leq \frac{(2\pi t)^{2m}}{(2m)!}$$

We now perform the removal of $\xi^{[a+1]}$ for the brackets involving variables outside $\yy_1$. Setting $\ukappaN=\kappa_{N,1}\otimes \dots\otimes \kappa_{N,s}$ we have

\begin{align*}
\huNgh(\xi)
&= \E_{g\otimes \ukappaN \otimes h}\left[  e_{\xi'}(\Pi(\yy)) e_{\xi^{[a+1]}}(\Pi(\yy))\right]\\
&= \E_{g\otimes \ukappaN \otimes h}\left[   e_{\xi'}(\Pi(\yy)) e_{\xi^{[a+1]}}(\Pi(\yy_{1})) \exp\left(-2i\pi (\Xi(\yy) -\Xi(\yy_{1}))\right)\right]\\
&= \E_{g\otimes \ukappaN \otimes h}\left[  e_{\xi'}(\Pi(\yy)) e_{\xi^{[a+1]}}(\Pi(\yy_{1})) T_{2m-1}(\Xi(\yy) -\Xi(\yy_{1}))\right]\\
&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+O_{m}(1)\E_{g\otimes \ukappaN \otimes h}(\|\Xi(\yy) -\Xi(\yy_{1})\|^{2m})
\end{align*}

Let us check that the error term is negligible for a good choice of $m$. Using the triangle inequality for the $L^{2m}$ norm, the equality $\xi^{[a+1]}\circ p^{[a',b')}=\xi^{[\max(a+1, a'),b)}$, the maximality of $a$, and <Ref>, we may write

\begin{align*}
&\E_{g\otimes \ukappaN \otimes h}(\|\Xi(\yy) -\Xi(\yy_{1})\|^{2m})\\
&\ll_{m} \sum_{a', b'} \E_{g\otimes \ukappaN \otimes h} (\| \xi^{[a+1]} . \Pi^{[a',b')}(\yy) -\xi^{[a+1]}  . \Pi^{[a',b')}(\yy_{1})\|^{2m})\\
&\leq  \,\,\, \,\,\,  \sum_{a', b'} \|\xi^{[\max(a+1, a'),b')}\|^{2m} \E_{g\otimes \ukappaN \otimes h} (\|  \Pi^{[a',b')}(\yy) - \Pi^{[a',b')}(\yy_{1})\|^{2m})\\
&\ll_{m} \,\sum_{a', b'\leq 2s-1} N^{-b'm+2\eps_{a+1}m} \,N^{(b'+2\delta_{0})m}\left( \frac{N_{a}}{N}\right)^m \\
&\ll \,\,\, \,\,\,  \,  N^{(2\eps_{a+1}+2\delta_{0} -\gamma_{a})m}  \\
&\leq \,\,\, \,\,\,  \,  N^{-\frac{m}{2}\gamma_{a}}
\end{align*}

because $\gamma_{a}\geq 4(\eps_{a+1}+\delta_{0})$.   Then we choose  $m=m(Q, \gamma_{a})$ so that $\frac{m}{2}\gamma_{a}>Q$. For this choice of parameter the error term  becomes negligible.

We now bound the remaining expectation using the path-swapping technique on the variable $\xx_{a}$, which has been (essentially) freed from $\xi^{[a+1]}$. Consider $g,h$ as fixed, and decompose the polynomial $T_{2m-1}(\Xi(\yy) -\Xi(\yy_{1}))$ as

$$T_{2m-1}(\Xi(\yy) -\Xi(\yy_{1}))= \sum_{M}c_{M}(g,h, \xi)M(\xx)$$

where $M$ varies in the set of monomials on $\tkg^{N}$ whose degree (in the classical sense) is at most  $(2m-1)s$, and $c_{M}(g,h, \xi)\in \C$ is a coefficient whose norm is bounded by  $(NR)^{O_{m}(1)}$, uniformly in $g,h\in \DN(\delta_{0})$, $\|\xi\|\leq R$.

Consider one of these monomials $M$. We  apply the path-swapping technique of the previous section to a subsequence of variables in $\xx$ which does not appear in $\xx_{a-1}$, $\xx_{a+1}$, nor $M(\xx)$.
We set $k=k(a,\xi)\geq 1$  the sub-block length prescribed for $\xi'$ by <Ref>, and then $N'_{aa}:= \lfloor \frac{N_{a}-N_{a-1}}{2mska}\rfloor$.
As the number of distinct variables in $M(\xx)$ is bounded by $2ms$, we may consider a decomposition  $\xx_{a}=\xx_{a-1}\oplus \zz_{0} \oplus \zz \oplus \zz_{1}$ where $\zz$ is a sequence of variables which do not appear in $M(\xx)$ and of length  $N_{aa}=ka N'_{aa}$. Then we write

\begin{align*}
\E_{g\otimes \ukappaN \otimes h}\left[  e_{\xi'}(\Pi(\yy)) e_{\xi^{[a+1]}}(\Pi(\yy_{1})) M(\xx)\right] \,=\, \E_{\xx_{a\pm 1},\zz_{0},\zz_{1}}\left[ M(\xx) e_{\xi^{[a+1]}}(\Pi(\yy_{1})) \E_{\zz} [e_{\xi'}(\Pi(\yy))]\right]
\end{align*}

<Ref> implies that

\begin{align*}
\big| \E_{\zz} [e_{\xi'}(\Pi(\yy))] \big| & \ll_{R} (1-c)^{\frac{1}{2^{a-1}} N'_{aa}}
\end{align*}

where $c>0$ only depends on $R$ and the initial data. To justify that all this makes sense, we  check that for $N$ large enough, the number of blocks  $N'_{aa}$ is non zero, and is even  bounded below by a power $N$. It comes down to showing that $k$ is not too large. As  $\|\xi^{[a,b)}\|> N^{-\frac{b}{2}+\eps_{a}}$ for some $b$, and $\xi^{[a,b)}= \xi^{'[a,b)}+\xi^{[a+1,b)}$, we can see directly that $\|\xi'_{|\kg^{[a,b)}}\|\geq \beta \|\xi'^{[a,b)}\|\geq  \frac{1}{2}\beta N^{-b/2+\eps_{a}}$ as long as $N$ is large enough (depending on $\eps_{a}, \eps_{a+1}$). Hence, we have  $k\leq 2\beta^{-1} N^{1-\frac{1}{s}\eps_{a}}$, so $N'_{aa}\geq N^{\frac{1}{s}\eps_{a}-2\gamma_{a}}> N^{\frac{1}{2s}\eps_{a}}$ due to  $\frac{1}{s}\eps_{a} >4 \gamma_{a}$, and up to assuming $N$ larger than a constant depending on $(\gamma_{a-1}, \gamma_{a},m, s, \beta)$.

We infer that

\begin{align*}
\huNgh(\xi)
& = \sum_{M} c_{M}\E_{g\otimes \ukappaN\otimes h}\left[  e_{\xi'}(\Pi(\yy)) e_{\xi^{[a+1]}}(\Pi(\yy_{1})) M(\xx)\right]  +O_{m}(N^{-Q})\\
& = O_{R, \gamma_{a-1}, \gamma_{a}, m, \beta}(1) \,(1-c)^{\frac{1}{2^{a-1}} N^{\frac{\eps_{a}}{2s}}}  \sum_{M} c_{M} +O_{m}(N^{-Q}).
\end{align*}

Using that the number of coefficients $c_{M}$ is at most polynomial in $N$ and that each of them is bounded by $(NR)^{O_{m}(1)}$, we may write $\sum_{M} |c_{M}| =(NR)^{O_{m}(1)}$. Finally, since the constants $m$, $\beta$ can be prescribed from the beginning as  functions of $(\gamma_{a})_{1\leq a\leq s}$, $Q$ and the initial data, and since  $(\gamma_{a})_{1\leq a\leq s}$ only depends on $\gamma_{0}$, we finally have the desired  bound

\begin{align*}
|\huNgh(\xi)|= O_{R, \gamma_{0}, Q}( N^{-Q}).
\end{align*}

§.§ Local limit theorem with power saving

We use the previous section to prove <Ref>, which is  a local limit theorem with power saving and moderate deviations  for integrable functions whose Fourier transform has compact support.

As we observed in the proof overview at the beginning of <Ref>, the Fourier inversion formula implies that for   $N\geq 1$, $g,h \in \tkg$,

\begin{align*}
\thetaNgh (f)&=\int_{B_{\dkg}(0,R)} \hf(\xi)\, \huNgh(\xi) e_{\xi,N,g,h} \,d\xi
\end{align*}

where $e_{\xi,N,g,h}= e^{-2i\pi \xi\circ p(g^{(\chi)}+N\chi+h^{(\chi)})}$, and  $g^{(\chi)}, h^{(\chi)}$ are the  $\R\chi$ components of $g,h$ in the decomposition $(\ref{tilde-dec})$.

We fix $0<\delta_{0}\lll \gamma_{0}$  such that the conclusion of  <Ref> holds true.
Supposing $g,h\in \DN(\delta_{0})$,  <Ref> applied to $Q=\frac{1}{2}(\dd+1)$, combined with the inequality $\|\hf \|_{L^\infty}\leq \|f\|_{L^1}$, then yields

\begin{align*}
\thetaNgh (f)&=\int_{\UN(\gamma_{0})} \hf(\xi)\, \huNgh(\xi) e_{\xi,N,g,h}\,d\xi \, +\, \|f\|_{L^1}O_{R, \gamma_{0}}\left(N^{-\frac{1}{2}(\dd+1)}\right).
\end{align*}

By <cit.>, there exists a probability measure $\eta$ on $\kg$, which has finite moment of all orders, coincides with $\mu$ up to order $2$, and is linked to the limit measure $\nu$ in the central limit theorem via the relation $\teta^{*'N}=\DilN \nu*'N\chi$ for all $N$. On the reduced domain  $\UN(\gamma_{0})$, we successively apply  Propositions <ref>, <ref> to $\eta$, <ref> to $\eta$, in order to replace $\uNgh$ by $g*'\DilN \nu*' N\chi*'h $, or even by $(\DilN \nu * N\chi)_{g,h}:=g*\DilN \nu* N\chi*h $ thanks to <Ref> below. This gives

\begin{align*}
\thetaNgh (f)&=\int_{\UN(\gamma_{0})} \hf(\xi)\, \widehat{(\DilN \nu * N\chi)_{g,h}}(\xi)\, e_{\xi,N,g,h}\,d\xi  \,+\, \|f\|_{L^1}O_{R, \gamma_{0}}\left(N^{ -\frac{1}{2}(\dd +\min(1, m_{\mu}- 2))+O(\gamma_{0}) } \right)\\
&=\int_{\UN(\gamma_{0})} \hf(\xi)\, \widehat{(\DilN \nu * N\Xm)_{p(g),p(h)}}(\xi)\,d\xi \,\,+\|f\|_{L^1}O_{R, \gamma_{0}}\left(N^{ -\frac{1}{2}(\dd +\min(1, m_{\mu}- 2))+O(\gamma_{0}) } \right).
\end{align*}

To make the term $(\DilN \nu * N\Xm)_{p(g),p(h)}(f)$ appear, we must check that the integral on the complementary domain $\dkg \smallsetminus \UN(\gamma_{0})$ is also negligible. This is proven in <Ref> below.

Once  <Ref> is established, Fourier inversion formula and the inequality $\|\hf \|_{L^\infty}\leq \|f\|_{L^1}$ imply that

$$|\thetaNgh (f)-(\DilN \nu*N\Xm)_{p(g),p(h)}(f)| =  \|f\|_{L^1}O_{R, \gamma_{0}}\left(N^{ -\frac{1}{2}(\dd +\min(1, m_{\mu}- 2))+O(\gamma_{0}) } \right)$$

which concludes the proof.

For the proof of <Ref> to be complete we need to check the next two lemmas.
[Back to $*$]  For $N\geq 1$, $\gamma_{0}>\delta_{0}\geq 0$, $g,h\in \DN(\delta_{0})$, $\xi \in  \UN(\gamma_{0})$, we have

$$\big|   \widehat{(g*\DilN \nu*N\chi*h)}(\xi)    -   \widehat{(g*'\DilN \nu*'N\chi*'h)}(\xi)      \big|  \ll_{\gamma_{0}}  N^{-1/2+2\gamma_{0}}.$$

Set  $\yy=(g,\DilN (x), N\chi, h)$ where $x\in \tkg$, and observe that

\begin{align*}
&\big|   \widehat{(g*\DilN \nu*N\chi*h)}(\xi)    -   \widehat{(g*'\DilN \nu*'N\chi*'h)}(\xi)      \big|  \\
&= \big|\E_{x\sim \nu}[ e_{\xi}(\Pi(\yy))-e_{\xi}(\Pi'(\yy)) ]\big|\\
&\ll \E_{x\sim \nu}[|\xi .\Pi(\yy))- \xi.\Pi'(\yy) ]\big|\\
&\ll \|\xi_{|\kg^{(b+1)}}\| \sum_{b\leq 2s-1} \E_{x\sim \nu}\left(\|\Pi^{(b)}(\yy)\|\right)
\end{align*}

where the last line uses the triangle inequality and the fact that $\Pi^{(b)}(\yy) - \Pi'^{(b)}(\yy)$ is the projection of $\Pi^{(b)}(\yy) $ to $\tkg^{(b+1)}$ parallel to $\tkm^{(b)}$.

Hence, it is sufficient  to show that

$$\E_{x\sim \nu} \left[\|\Pi^{(b)}(\yy) \| \right]  \ll_{\nu} N^{b/2+\delta_{0}}.$$

This bound follows from the equality $\Pi^{(b)}(\yy)= N^{b/2} \Pi^{(b)}(\zz)$ where $\zz=(\DilsN g, x, \chi, \DilsN h)$ and the remark that  $\|\Pi^{(b)}(\zz)\|\ll N^{\delta_{0}} P(x)$ where $P$ is some polynomial.

[Domain completion]  Up to choosing $\delta_{0}\lll\gamma_{0}$, we have for $N\geq 1$,  $g,h\in \DN(\delta_{0})$,

$$ \int_{\dkg\smallsetminus \UN(\gamma_{0})} |\widehat{(\DilN \nu * N\chi)_{g,h}}(\xi)| \,d\xi \,=\,O_{\gamma_{0}}(N^{-\frac{\dd +1}{2}}).$$

As $N\chi *h\in \DN(O(\delta_{0}))$,  it is enough to check that for $\delta_{0}\lll\gamma_{0}$, $N\geq 1$, $g,h\in \DN(\delta_{0})$,

\begin{align*}
\int_{\dkg\smallsetminus \UN(\gamma_{0})} |\widehat{(\DilN \nu)_{g,h}}(\xi)| d\xi  =O_{\gamma_{0}}(N^{-\frac{\dd+1}{2}}).
\end{align*}

Replacing the variable $\xi$ by $\DilN \xi:=\xi\circ \DilN $, this can be rewritten:

\begin{align*}
\int_{\dkg\smallsetminus \DilN \UN(\gamma_{0})} |\widehat{\sigma_{N,g,h}}(\xi)| d\xi  =O_{\gamma_{0}}(N^{-1/2})
\end{align*}

where $\sigma_{N,g,h}:= p_{\kg}\left(\DilsN  ( g * \DilN \nu *h)\right)$, with $p_{\kg}: \tkg\rightarrow \kg, x\oplus t\chi\mapsto x$. Moreover,  a direct computation yields the inclusion $B_{\dkg}(0,N^{\gamma_{0}})\subseteq {\DilN }.\UN(\gamma_{0})$. Hence, it is sufficient to check that

\begin{align*}
\int_{\dkg\smallsetminus B_{\dkg}(0,N^{\gamma_{0}})} |\widehat{\sigma_{N,g,h}}(\xi)| d\xi  = O_{\gamma_{0}}(N^{-1/2}).
\end{align*}

We now give an explicit formula for the Radon-Nikodym derivative of $\sigma_{N,g,h}$ with respect to $dx$. By <cit.>, we can write $\nu=v(x) dx$ where $v : \kg\rightarrow \R_{+}$ is a smooth function whose derivatives (at all orders) decay rapidly.
For $y\in \kg$, we also write $L_{y}, R_{y}, S_{y}: \kg \rightarrow \kg$ for the operators of left multiplication, right multiplication, or sum, by $y$, i.e. $L_{y}(x)=y*x$, $R_{y}=x*y$, $S_{y}(x)=x+y$. A direct computation gives

$$\sigma_{N,g,h}= v_{N,g,h}(x)dx$$

where $v_{N,g,h}= v\circ s_{N,g,h}$ with

$$s_{N,g,h}= \DilsN  \circ L_{-p(g)} \circ R_{-p(h)} \circ S_{p(g^{(\chi)}+h^{(\chi)})} \circ \DilN. $$

Let $e_{1}, \dots, e_{d}$ ($d=\dim\kg$) be an enumeration of the basis $(e^{(i)}_{j})$ of $\kg$ chosen in <Ref>. For each $i\in \{1, \dots, d\}$, denote by $\partial_{i}$ the derivation on $\kg$ with respect to the vector $e_{i}$ (for the addition).  Using integration by parts, we see that for all $k\geq 0$, $\xi \in \dkg\smallsetminus \{0\}$,

$$|\widehat{v_{N,g,h}}(\xi)| \ll_{k} \frac{1}{\|\xi\|^k} \sum_{i=1}^d \|\partial^k_{i} v_{N,g,h} \|_{L^1}.$$

Hence, as $ \widehat{\sigma_{N,g,h}} = \widehat{v_{N,g,h}}$,

\begin{align*}
\int_{\dkg\smallsetminus B_{\dkg}(0,N^{\gamma_{0}})} |\widehat{\sigma_{N,g,h}}(\xi)| d\xi & \,\ll_{k}\,  \int_{\dkg\smallsetminus B_{\dkg}(0,N^{\gamma_{0}})} \frac{1}{\|\xi\|^k} d\xi \,\sum_{i=1}^d \|\partial^k_{i} v_{N,g,h} \|_{L^1}\\
& \,\ll\, \frac{1}{N^{\gamma_{0}(k-d)}}\,\sum_{i=1}^d \|\partial^k_{i} v_{N,g,h} \|_{L^1}.
\end{align*}

To conclude, it is enough to show that for all $\delta_{0}\lll\gamma_{0}$, there exists $k=k(\gamma_{0})$  (large) such that for  $N\geq1$, $g,h\in \DN(\delta_{0})$,

$$\sum_{i=1}^d \|\partial^k_{i} v_{N,g,h} \|_{L^1}=O_{k} (N^{\gamma_{0}(k-d) -1/2}).$$

In order to estimate the $k$-th derivative of $v_{N,g,h}$ and its $L^1$-norm, we will need more information on the term $s_{N,g,h}$. The crucial observation is that $s_{N,g,h}(x)$ can be realised by dividing by a power of $\sqrt{N}$ some coefficients of a polynomial $P$ in the variables $(\DilsN g, \DilsN h, x)$ and of total degree (in the classical sense) equal to $s$. To justify this, observe first that one can simply delete the projections $p(\cdot)$ in the definition of $s_{N,g,h}$, i.e.  $s_{N,g,h}(x)=\DilsN \left( (-g)* (\DilN (x)+ g^{(\chi)}+h^{(\chi)}) *(-h) \right)$. Then notice that this expression would be a polynomial in the variables  $(\DilsN g, \DilsN h, x)$ if $*$ were replaced by the graded product $*'$ (for which $\DilN $ is an automorphism), and that the error induced by this substitution is responsible for the supplementary terms that we mentioned. The polynomial $P$ that we obtain only depends on the initial data fixed in <Ref>.  Hence, as $\DilsN \DN(\delta_{0}) \subseteq B_{\tkg}(0, O(N^{s^{-1}\delta_{0}}))$,  we have

$$\|s_{N,g,h}(x)\| \ll N^{\delta_{0}}\max(1, \|x\|^s).$$

All this also holds for the inverse transformation  $s_{N,g,h}^{-1} : \kg \rightarrow \kg$, and in particular,

$$\|s_{N,g,h}^{-1}(x)\| \ll N^{\delta_{0}}\max(1, \|x\|^s).$$

We now prove the bound for $\|\partial^k_{i} v_{N,g,h} \|_{L^1}$ that we announced, for a fixed subscript $i\in \{1, \dots, d\}$.  Given a $d$-tuple $\alpha=(\alpha_{1}, \dots, \alpha_{d})\in \N^d$, write $|\alpha|=\sum_{j}|\alpha_{j}|$ and $\partial^{\alpha}= \partial^{\alpha_{1}}_{1}\dots \partial^{\alpha_{d}}_{d}$.  Arguing by induction, it is easy to check that $k\geq 0$,

$$ \partial^k_{i} v_{N,g,h}(x)  =\sum_{|\alpha|\leq k} P^{(\alpha)}_{N,g,h}(x) \,(\partial^{\alpha} v) \circ s_{N,g,h}(x)

where $P^{(\alpha)}_{N,g,h}(x)$ can be realised by dividing by a power of $\sqrt{N}$ some coefficients of a polynomial $P^{(\alpha)}$ in the variables $(\DilsN g, \DilsN h, x)$ and of total degree (in the classical sense) bounded above by $ks$. Moreover the coefficients of $P^{(\alpha)}$ are determined by the initial data fixed in <Ref>, and  $i,k, \alpha$. We infer an upper bound for the $L^1$-norm

\begin{align*}
\|\partial^k_{i} v_{N,g,h} \|_{L^1} &\leq  \sum_{|\alpha|\leq k} \int_{\kg} |P^{(\alpha)}_{N,g,h}(x) \,(\partial^{\alpha} v) \circ s_{N,g,h}(x)| \,dx\\
&\ll_{i, k}  N^{\delta_{0}k}\sum_{|\alpha|\leq k} \int_{\kg}  \max (1, \|x\|^{ks}) \,|(\partial^{\alpha} v) \circ s_{N,g,h}(x)| \,dx\\
&=  N^{\delta_{0}k}\sum_{|\alpha|\leq k} \int_{\kg}  \max (1, \|s_{N,g,h}^{-1}(x)\|^{ks}) \,|\partial^{\alpha} v (x)| \,dx
\end{align*}

where the last line uses that $s_{N,g,h}$ preserves the Lebesgue measure on $\kg$. From the above-mentioned bound on $s_{N,g,h}^{-1}(x)$, we conclude that

\begin{align*}
\|\partial^k_{i} v_{N,g,h} \|_{L^1}
&\ll_{i, k}
N^{\delta_{0}k(1+s)}\sum_{|\alpha|\leq k} \int_{\kg}  \max (1, \|x\|^{ks^2}) \,|\partial^{\alpha} v (x)| \,dx.
\end{align*}

Since the integral on the right-hand side is bounded, we may sum over  $i$ and conclude that for all $k\geq 0$, $N\geq 1$, $g,h\in \DN(\delta_{0})$,

\begin{align*}
\sum_{i=1}^d \|\partial^k_{i} v_{N,g,h} \|_{L^1}&=O_{k} (N^{\delta_{0}k(1+s)})\\
&=O_{k} (N^{\gamma_{0}(k-d) -1/2})
\end{align*}

up to choosing $\delta_{0}<\frac{\gamma_{0}}{2(1+s)}$  and $k=k(\gamma_{0})$ large enough so that  $\delta_{0}k(1+s)<\gamma_{0}(k-d) -1/2$.
This finishes the proof of <Ref>.

§.§ Approximation by band-limited functions

To convert <Ref> into a local limit theorem dealing with any compactly supported continuous function (<Ref>), we will need the following approximation result. It is certainly well-known, but in the absence of a convenient reference, we include a proof.

Let $f\in C_{c}(\kg)$.
For all $\eps>0$, there exist $f^-, f^+\in C^\infty(\kg)$ which are integrable, bounded, and satisfy

$\bullet$ $f^-\leq f\leq f^+$.

$\bullet$ $\widehat{f^-}$ and $\widehat{f^+}$ are compactly supported.

$\bullet$ $\|f^+ - f^-\|_{L^1} <\eps$.

Remark. The conclusion of the proposition is still valid if we only require $f$ to be a limit in $L^1$ of functions  $f^-\leq f\leq f^+$ with compactly supported Fourier transform. This is a direct consequence of <Ref>. Hence the local limit theorem <ref> proven below  will also be valid for this class of functions.

Let $\mathscr{C}$ be the cone of functions $\rho\in C^\infty(\kg)$ such that  $\rho\geq 0$, $\rho \in L^1(\kg)$ and $\hrho$ is compactly supported. We first show that $\mathscr{C}$ is non trivial, and contains  some functions decreasing fast, others decreasing slowly.

* For every $k\geq 2$, there is $\rho\in \mathscr{C}$ with $\|\rho\|_{L^1}=1$ and

$$0<\rho(x)\leq 1/\|x\|^k.$$

* For all $\eps>0$, there is $\tau\in \mathscr{C}$ such that $\|\tau\|_{L^1}<\eps$ and

$$\tau(x)\geq 1/\|x\|^{2 \dim \kg} $$

outside of a compact set.

Let $\kg \simeq \R^d$. It is enough to prove the result for $d=1$, since we may then consider the product $f(x_1)\cdot \ldots \cdot f(x_d)$ over distinct variables, where $f=\rho$ or $\tau$.  Let $(a_{n})_{n\in \Z}\in (\R_{\geq 0})^\Z$ be non identically zero with finite total sum. Let $p\geq 1$. We define the function  $\phi : \R\rightarrow \R$ by setting

$$\phi(x)= \sin^{2p}(\pi x)\sum_{n\in \Z}\frac{a_{n}}{(x-n)^{2p}}.$$

Then $\phi$ is smooth, $\phi \geq 0$, $\|\phi\|_{L^1}\leq C_{p}\sum_{n\in \Z} a_{n}$ for some $C_{p}>0$. The Paley-Wiener theorem ensures that $\widehat{\phi}$ has compact support. Hence $\phi\in \mathscr{C}$.
For the first item, we choose $p=k/2$, $a_{n}=\1_{n=0}$, and we set

$$\rho_{1}(x) =\phi(x) +\phi(x+1/2)    \,\,\,\,\, \,\,\,\,\, \,\,\,\,\, \,\,\,\,\, \,\,\,\,\, \,\,\,\,\,\rho= \frac{1}{\|\rho_1\|_{L^1}}\rho_1.$$

For the second item, we first notice that $\phi_{|[n-\frac{1}{2}, n+\frac{1}{2}]}\geq c_{p}a_{n}$  for some constant $c_{p}>0$. We choose $p=1$, $a_{n}=c^{-1}_{1}|n|^{-3/2} \1_{|n|\geq R}$ where $R>0$ is large enough so that
$\|\phi\|_{L^1}\leq C_{1}\sum_{n\in \Z} a_{n}<\eps$. We can then take $\tau=\phi$.

We may assume $f\geq 0$ and non zero. We fix a Euclidean ball $K=B(0,R_0)$  of $\kg$ containing the support of $f$, and a positive integrable function $\rho \in \mathscr{C}$ as in the first item of <Ref> for $k=2(\dim \kg +1)$. Given $t>0$, we set $\rho_{t}(x)=t^{\dim \kg}\rho(tx)$ the  $t$-contraction of $\rho$. We may choose $t$ large enough $(t=t(f, \eps))$ so that

$\bullet$  $ f-\eps \leq f* \rho_{t} \leq f+\eps $ on $K$

$\bullet$ $ \|(f*\rho_{t}) \1_{\kg\smallsetminus K}\|_{L^1}\leq \eps$.

We are going to show the existence of $\tau_{1}, \tau_{2}\in \mathscr{C}$ such that  $\tau_{1}> \eps \1_{K}$, $\tau_{2}> (f*\rho_{t}) \1_{\kg\smallsetminus K}$, and  $\|\tau_{i}\|_{L^1} = O_{K}(\eps)$. We then set

$$f^-=f*\rho_{t} - \tau_{1}-\tau_{2}\,\,\,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\, \,\,  f^+=f*\rho_{t} +\tau_{1}$$

to obtain <Ref>.

The component $\tau_{1}$ can be constructed by fixing at the beginning of the proof  $\phi\in \mathscr{C}$ such that $\phi>\1_{K}$, then setting $\tau_{1}=\eps \phi$.

For the component $\tau_{2}$, we start by observing that $f*\rho_{t}$ decays rapidly at infinity:

\begin{align*}
f*\rho_{t}(x)= &\int_{\kg} \rho_{t}(x+s) f(-s)ds\\
&\leq \|f\|_{L^\infty} \sup_{x+K}\rho_{t}\\
&= t^{\dim \kg}\|f\|_{L^\infty} \sup_{tx+tK}\rho.\\
\end{align*}

There exists $C=C(f,K,\rho, t)>R$ such that for $\|x\|\geq C$, we have

$$\sup_{tx+tK}\rho \leq \frac{1}{ t^{\dim \kg}\|f\|_{L^\infty} \|x\|^{2\dim \kg +1}}$$

which implies for $\|x\|\geq C$,

\begin{align*}
f*\rho_{t}(x)\leq   \|x\|^{-2\dim \kg -1}.
\end{align*}

By <Ref> (second item), there exists $\tau\in \mathscr{C}$ such that $\|\tau\|_{L^1}< \eps$ and $\tau > f*\rho_{t}(x)$ outside of a compact set $K_1=K_1(f,K,\rho,t) \supseteq K$. It remains to  upper bound $f*\rho_{t}$ in the intermediate  zone  $L:=K_1\smallsetminus K$. To do so, we fix $\phi'\in \mathscr{C}$ such that $\phi'\geq \1_{L}$, and observe that for  large enough $t'=t'(f, \rho, t, L, \phi')$

$$(f*\rho_{t})_{|L}< (f*\rho_{t})_{|L}*\rho_{t'}+\eps \frac{1}{\|\phi'\|_{L^1}} \phi'.$$

We conclude the construction of $\tau_{2}$ by setting

$$ \tau_{2}= \tau + (f*\rho_{t})_{|L}*\rho_{t'}+\eps \frac{1}{\|\phi'\|_{L^1}} \phi'.$$

§.§ End of proof

We   combine  Propositions <ref> and <ref> to deduce Theorems <ref> and  <ref>.

We start with the case of moderate deviations:

Fix $\gamma_{0}>0$ small enough depending only on the initial data  so that the  $O(\gamma_{0})$ appearing as an exponent in the bound of  <Ref> is  less than $\frac{1}{4}\min(1, m_{\mu}-2)$. Hence, the total error term in this proposition can be bounded by $\|f\|_{L^1}o_{R, \gamma_{0}}(N^{-\frac{\dd}{2}})$. Choose $\delta_{0}>0$ small enough compared to $\gamma_{0}$ so that we may apply <Ref>.

Let $f\in C_{c}(\kg)$. Let  $\eps>0$ and $f^-\leq f\leq f^+$ be  $\eps$-approximations of $f$ as in <Ref>. For $g,h\in \DN(\delta_{0})$, we infer from  <Ref> that

\begin{align*}
&\thetaNgh(f)- (\DilN \nu*N\Xm)_{p(g)p(h)}(f)\\
& \leq \thetaNgh(f^+)- (\DilN \nu*N\Xm)_{p(g),p(h)}(f^-)\\
&= (\DilN \nu*N\Xm)_{p(g),p(h)}(f^+)- (\DilN \nu*N\Xm)_{p(g),p(h)}(f^-) +o_{f, \gamma_{0}, \eps}(N^{-\frac{\dd}{2} })\\
&= (\DilN \nu)(\psi) +o_{f,  \gamma_{0},\eps}(N^{-\frac{\dd}{2} })
\end{align*}

where $\psi(x)= (f^+ - f^-)(p(g)*x*N\Xm* p(h))$. Fourier inversion formula yields

\begin{align*}
(\DilN \nu)(\psi)&= \int_{\dkg}\,\hpsi( \xi) \,\hnu(\DilN \xi)\, d\xi\\
&= N^{-\frac{\dd}{2}}\int_{\dkg}\hpsi(\DilsN  \xi) \,\hnu(\xi)\,  d\xi\\
&\leq N^{-\frac{\dd}{2}} \|\psi\|_{L^1}\|\hnu\|_{L^1}
\end{align*}

$$\|\psi\|_{L^1}= \|f^+ -f^-\|_{L^1}\leq \eps $$

we must have

\begin{align*}
\thetaNgh(f)- (\DilN \nu*N\Xm)_{p(g),p(h)}(f) & \leq N^{-\frac{\dd}{2}} \eps \|\hnu\|_{L^1}   +o_{f, \gamma_{0}, \eps}(N^{-\frac{\dd}{2} })\end{align*}

One can argue in the same way, reversing the roles of  $f^+$ and $f^-$, to obtain a lower bound, so in the end

\begin{align*}
\big|  \thetaNgh(f)- (\DilN \nu*N\Xm)_{p(g),p(h)}(f) \big| & \leq N^{-\frac{\dd}{2}} \eps \|\hnu\|_{L^1}    +o_{f,  \gamma_{0},\eps}(N^{-\frac{\dd}{2} })\end{align*}

This implies

$$\limsup_{N\to +\infty} \sup_{g,h\in \DN(\delta_{0})}N^{\frac{\dd}{2}} |\thetaNgh (f)- (\DilN \nu*N\Xm)_{p(g),p(h)}(f)| \leq \eps \|\hnu\|_{L^1}$$

As $\eps$ is arbitrary, the limit must be zero, which is exactly  <Ref>.

To conclude the proof of <Ref>, it is now sufficient to check that when exactly one of the deviations is non moderate, the terms  $\Theta^{N}_{p(g),p(h)}(f)$ and $(\DilN \nu*N\chi)_{p(g),p(h)}(f)$ are both negligible compared to $N^{-\dd/2}$, and this holds uniformly in $g,h$. This is the content of the next lemma.

Let $K\subseteq \kg$ be a compact set containing the support of $f$. We may replace $f$ without loss of generality by $\1_{K}$. For any $g,h\in \tkg$, one has

\begin{align*}
\Theta^{N}_{p(g),p(h)}(K)=\Upsilon^{N}_{g,h}(p^{-1}(K))=\Upsilon^N((-g)*p^{-1}(K)*(-h))
\end{align*}

Let $\delta_{0}>\delta_{1}>0$ and assume $p(g)\notin p(\DN(\delta_{0}))$, $h\in \DN(\delta_{1})$. If $\delta_{1}\lll \delta_{0}$, then for large $N$ (depending on $K$, $\delta_{0}$), we have

$$K* p(\DN(\delta_{1}))*p(\DN(\delta_{1}))  \subseteq p(\DN(\delta_{0}))$$

and this implies

$$(-g)*p^{-1}(K)*(-h)\subseteq \tkg\smallsetminus \DN(\delta_{1}).$$

Indeed, we would otherwise have $k'*(-h)*x=g$ where $k'\in p^{-1}(K)$, $x\in \DN(\delta_{1})$, hence $K* p(\DN(\delta_{1}))^{*2}\ni p(g)$, which contradicts our assumption on $g$.

We deduce that for $N$ large, any $g,h$ as above,

$$\Theta^{N}_{p(g),p(h)}(K) \leq\Upsilon^N(\tkg\smallsetminus \DN(\delta_{1})).$$

By the moment estimate <Ref>, it follows that $\Theta^{N}_{p(g),p(h)}(K)$ must be bounded by $o_{K, \delta_{0}, \delta_{1}}(N^{-\frac{\dd}{2}})$. The other cases are similar.

Combine <Ref> and <Ref>.

By <Ref>, one has in total varation

$$\|\Theta^{N}_{p(g),p(h)}- \mu^{*N}_{p(g),p(h)}\| =  o(N^{1-\frac{1-\gamma_{0}}{2}m_{\mu} })$$

which is negligible compared to $N^{-\dd/2}$ as long as $\gamma_{0}$ is chosen small enough. Hence the LLT for $\Theta^{N}_{p(g),p(h)}$, namely <Ref>, implies that for $\mu^{*N}_{p(g),p(h)}$, i.e. <Ref>.

§ LOCAL LIMIT THEOREM FOR RATIOS

In this section we prove <Ref> and <Ref>.
We keep the notations $(\kg, *, \tkg, \Xab, \dd, (\km^{(b)})_{b\leq 2s-1}, \Xm, \chi, *')$ from   <Ref>. We assume that $\mu$ is an aperiodic probability measure on $\kg$ with finite $(\dd+2)$-th moment for the weight filtration induced by $\Xab$, and expectation $\E(\mu_{ab})=\Xab$. In particular $X=\E(\mu^{(1)})$. We recall that $\nu=v(x)dx$ denotes the limit density in the central limit theorem  (<ref>) for the $\mu$-walk on $(\kg,*)$.

Our argument relies  on the uniform local limit theorem established in <Ref> and a good control of the limit density $\nu$. Accordingly, all the estimates of this section are still valid if we only assume $\mu$ to have a finite moment of order strictly greater than $2$ and replace $\mu^{*N}$ by its truncated approximation $\Theta^N$, see  <Ref>. We leave this verification to the reader.

§.§ Comparison with Lebesgue

We prove <Ref>. It follows from the following more general result.

For every $f\in C_{c}(\kg)$, $C>0$ $N\geq 1$, $g,h\in \DilN (B_{\tkg}(0,C))$ we have

$$ (\mu^{*N}*-N\Xm)_{p(g), p(h)}(f) = v(x_{N,g,h}) N^{-\dd/2}\int_{\kg}f dx \,\,+\,\, o_{ f, C}(N^{-\dd/2})$$

where $v$ is the density of the limit measure $\nu=v(x)dx$, and

$$x_{N,g,h}= p\left(\DilsN(g)\right)^{-1} *p\left(\DilsN(h)^{-1}\right)$$

for $p:\tkg\rightarrow \kg$  the projection $x\oplus t\chi\mapsto x+t\Xm$.

Remark. Recall from <cit.> that $v(0)$ may vanish. In particular, the sequence $ (\mu^{*N}*-N\Xm)(f)$ may be $o_{ f, C}(N^{-\dd/2})$. This never happens in the centered case  (i.e. $\Xab=0$) for which $v>0$ (see <Ref>).

By <Ref>,

\begin{align*}
(\mu^{*N}*-N\Xm)_{p(g), p(h)}(f) = (\DilN \nu)_{p(g), p(h)}(f)+o_{f,C}( N^{-\dd/2})
\end{align*}

Let $\gamma_{0} <1/2$. Set $e_{\xi,N,g,h}= e^{-2i\pi \xi\circ p(g^{(\chi)}+h^{(\chi)})}$ and $g_{N}=\DilsN g$, $h_{N}=\DilsN h$. Applying Fourier inversion formula, <Ref> and <Ref>, we obtain

\begin{align*}
(\DilN \nu)_{p(g), p(h)}(f) &=\int_{\dkg} \hf(\xi) \widehat{(\DilN \nu)_{p(g), p(h)}}(\xi) d\xi\\
&=\int_{\dkg} \hf(\xi) \widehat{(\DilN \nu)_{g, h}}(\xi) e_{\xi,N,g,h} d\xi\\
&=\int_{\UN(\gamma_{0})} \hf(\xi) \widehat{(\DilN \nu)_{g, h}}(\xi) e_{\xi,N,g,h} d\xi   + \|f\|_{L^1}O_{\gamma_{0}}(N^{-\frac{\dd+1}{2}})\\
&= \int_{\UN(\gamma_{0})} \hf(\xi) \widehat{(g*'\DilN \nu *'h)}(\xi) e_{\xi,N,g,h} d\xi   + \|f\|_{L^1}O_{\gamma_{0}}(N^{-\frac{\dd+1}{2}+O(\gamma_{0}) })\\
&= \int_{\UN(\gamma_{0})} \hf(\xi) \widehat{(\DilN (g_{N}*'\nu *'h_{N}))}(\xi) e_{\xi,N,g,h} d\xi   + \|f\|_{L^1}O_{\gamma_{0}}(N^{-\frac{\dd+1}{2}+O(\gamma_{0}) })\\
&= \int_{\UN(\gamma_{0})} \hf(\xi) \widehat{(\DilN (g_{N}*\nu *h_{N}))}(\xi) e_{\xi,N,g,h} d\xi   + \|f\|_{L^1}O_{\gamma_{0}}(N^{-\frac{\dd+1}{2}+O(\gamma_{0}) })\\
&=[\DilN (p(g_{N})*\nu *p(h_{N}))](f)+ \|f\|_{L^1}O_{\gamma_{0}}(N^{-\frac{\dd+1}{2}+O(\gamma_{0}) })
\end{align*}

Pursuing the computation

\begin{align*}
[\DilN (p(g_{N})*\nu *p(h_{N}))](f)
&= \int_{\kg} f\left(\DilN (p(g_{N})* x*p(h_{N}))\right) v(x)dx\\
&= N^{-\dd /2} \int_{\kg} f(x)\, v \left(p(g_{N})^{-1}*\DilsN x*p(h_{N})^{-1} \right) dx\\
&= v \left(p(g_{N})^{-1}*p(h_{N})^{-1} \right) N^{-\dd /2} \int_{\kg} f(x) dx  +r
\end{align*}

where the difference $r$ is bounded by

\begin{align*}
r \leq  N^{-\dd /2} \int_{\kg}  \big|v \left(p(g_{N})^{-1}*\DilsN x*p(h_{N})^{-1} \right) -  v \left(p(g_{N})^{-1}*p(h_{N})^{-1} \right)\big| \,\,|f(x)|dx
\end{align*}

whence is $o_{C}(N^{-\dd/2})$  by dominated convergence.
This concludes the proof.

§.§ Comparison with $\nu$

The goal of the section is to show <Ref>. We  assume that $\mu$ is centered, aperiodic, with finite moment of order $m_{\mu}>\dg+2$ for the central filtration, and we show that the asymptotic $\mu^{*N}(f) \sim (\DilN \nu)(f)$ implied by <Ref>, actually holds uniformly for a certain range of deviations, namely those in

$$W_{N}(c):=\{x\in \kg,\,|x^{(i)}|\leq c\left(N\log N\right)^{i/2} \}$$

for $c>0$ small.

The assumption that $\mu$ is centered is used to guarantee that the limit density $\nu=v(x)dx$ has a  Gaussian lower bound. Indeed, we record the following fact [93].

Assume $\Xab=0$. Let $\norm{.}$ denote the distance to $0$ for a left-invariant distance on $(\kg,*)$. There exists $A>1$ such that for all $x\in \kg$

$$A^{-1} e^{-A \norm{x}^2} \leq v(x) \leq A e^{-A^{-1} \norm{x}^2}$$

Using the  lower bound, we show

Assume $\Xab=0$. Let $\delta>0$ and $c\lll_{\delta} 1$. Let $f\in C^\infty(\kg)$ be a non-negative integrable function  with compactly supported Fourier transform, such that $f\neq 0$. For all $N\geq 1$ large enough,  $g,h \in W_N(c)$

$$(\DilN \nu)_{g,h}(f)\geq  N^{-\frac{\dg}{2} -\delta}  $$

Combining with <Ref>, we get

Assume $\Xab=0$, $m_{\mu}>\dg+2$ and $c\lll 1$. Let $f\in C^\infty(\kg)$ be a non-negative integrable function  with compactly supported Fourier transform, such that $f\neq 0$. For $N\geq 1$,  $g,h\in W_{N}(c)$,
we have

$$\frac{\mu^{*N}_{g,h}(f)}{(\DilN \nu)_{g,h}(f)} = 1 +o_{f}(1)$$

We will then deduce  <Ref>  by applying a refinement of the approximation technique used in the proof of the uniform local limit theorem.

We can write $(\DilN \nu)_{g,h}=N^{-\dg/2}w_{N,g,h}(x) dx$
$$w_{N,g,h}(x)=v\circ r_{N,g,h}(x) \,\, \,\, \,\, \,\, \,\, \,\,\,\,\,\,\,\,\,\,\,\,r_{N,g,h}(x)=\DilsN \left( g^{-1}*x *h^{-1} \right)$$

We need to show that for $c\lll_{\delta}1$, $N\geq 1$ large enough (possibly depending on $f$), $g,h\in W_{N}(c)$,

$$\int_{\kg} f(x) w_{N,g,h}(x)dx \geq N^{-\delta} $$

As $f$ is non-negative non-identically zero, it is enough to show that for $c\lll_{\delta}1$, any compact subset of $\kg$ is included in the set $\{w_{N,g,h} \geq N^{-\frac{\delta}{2}}\}$ for  $N$  large enough and $g,h\in W_{N}(c)$. To analyse this set, we use the lower bound in <Ref>

Consider $x$  such that $w_{N,g,h}(x) < N^{-\frac{\delta}{2}}$. Using  <Ref>, we infer

\begin{align*}
A^{-1} e^{-A \norm{r_{N,g,h}(x)}^2} < N^{-\frac{\delta}{2}}
\end{align*}

which can be rewritten

$$ \frac{\delta}{2A} \log N - \frac{\log A}{A} \leq \norm{r_{N,g,h}(x)}^2$$

so for $N\ggg_{A, \delta} 1$,

$$ \sqrt{\frac{\delta \log N}{3A }} \leq \norm{r_{N,g,h}(x)}$$

This discussion justifies the inclusion

$$\{w_{N,g,h} \geq N^{-\frac{\delta}{2}}\} \supseteq \left\{x\in \kg\,:\, \norm{r_{N,g,h}(x)} < \sqrt{\frac{\delta \log N}{3A }} \right\}$$

for any $N\ggg_{A, \delta} 1$.
However, by the ball-box principle,

$$\norm{r_{N,g,h}(x)} \ll \max_{i\leq s}\|(r_{N,g,h}(x))^{(i)}\|^{1/i}= N^{-1/2} \,\max_{i\leq s}\|(g^{-1}*x*h)^{(i)}\|^{1/i}$$

\begin{align*}
\{w_{N,g,h} \geq N^{-\frac{\delta}{2}}\}& \supseteq \left\{x\in \kg\,:\,\forall i\leq s, \,\|(g^{-1}*x*h^{-1})^{(i)}\| \lll_{A, \delta} (N\log N)^{i/2}  \right\}\\
\end{align*}

Noticing that for any $\delta>0$, we have $W_{N}(\delta)*W_{N}(\delta)\subseteq W_{N}(O(\delta))$, we conclude that for $c\lll_{A, \delta}1$, $N\ggg_{A, \delta} 1$, $g,h\in W_{N}(c)$,

\begin{align*}
\{w_{N,g,h} \geq N^{-1/4}\}& \supseteq W_{N}(c)
\end{align*}

As $A$ can be fixed as a function of the initial data, our conventions allow to remove it from the subscript, see <Ref>.

We can now conclude the proof of <Ref>.

Fix $\delta \in (0,1/2]$ such that $\mu$ has a finite moment of order $\dg+2+2\delta$. We choose $\delta$ is a function of $\mu$, so it does not appear in the subscripts of our Landau and Vinogradov notations (see <Ref>). Fix $c\lll 1$ as in <Ref> and  <Ref>.  Let $\eps>0$ and fix  $f^-\leq f \leq f^+$  an $\eps$-approximation of $f$ by functions with compactly supported Fourier transforms, as in <Ref>. As  $f$ is non negative and non identically zero, we have for $N\geq 1$, $g,h\in W_N(c)$

\begin{align*}
\frac{\mu^{*N}_{g,h}(f)}{(\DilN \nu)_{g,h}(f)}&\geq \frac{\mu^{*N}_{g,h}(f^-)}{(\DilN \nu)_{g,h}(f^+)} \\
&= \frac{(\DilN \nu)_{g,h}(f^-) +o_{f^-}(N^{-\frac{\dg}{2}-\delta}) }{(\DilN \nu)_{g,h}(f^+)} \\
&= \frac{(\DilN \nu)_{g,h}(f^-)}{(\DilN \nu)_{g,h}(f^+)} +o_{f^\pm}(1)
\end{align*}

where the second line uses the local limit theorem with power saving from <Ref>, and the last line uses <Ref>.

\begin{align*}
\frac{\mu^{*N}_{g,h}(f)}{(\DilN \nu)_{g,h}(f)}&\leq \frac{\mu^{*N}_{g,h}(f^+)}{(\DilN \nu)_{g,h}(f)} \\
&= \frac{\mu^{*N}_{g,h}(f^+)}{(\DilN \nu)_{g,h}(f^+)} \frac{(\DilN \nu)_{g,h}(f^+)}{(\DilN \nu)_{g,h}(f)} \\
&= (1+o_{f^+}(1)) \frac{(\DilN \nu)_{g,h}(f^+)}{(\DilN \nu)_{g,h}(f)}
\end{align*}

where the last line uses <Ref>.
To conclude, it is enough to prove that

$$\frac{(\DilN \nu)_{g,h}(f^-)}{(\DilN \nu)_{g,h}(f^+)}=1+O_{f}(\eps) +o_{f^{\pm}}(1)$$

Only the lower bound needs a proof. We can reformulate it as

\begin{align} \label{appII}
(\DilN \nu)_{g,h}(f^+-f^-)\leq \left(O_{f}(\eps)+o_{f^{\pm}}(1)\right)(\DilN \nu)_{g,h}(f^+)
\end{align}

To prove (<ref>), we  write as above $(\DilN \nu)_{g,h}=N^{-\dg/2}w_{N,g,h}(x) dx$
$$w_{N,g,h}(x)=u\circ r_{N,g,h}(x) \,\,\,\,\,\,\,\,\,\,\,\,r_{N,g,h}(x)=\DilsN (g^{-1}*x*h^{-1})$$

We also introduce the domain

$$\Omega_N :=\{x\in \kg\,:\, \|x\|\leq N^{\frac{1}{4s}}\}$$

For $c\lll1$, there exists $V>1$ such that for all $N\geq1$, $g,h \in W_N(c)$,

$$\frac{\sup_{\Omega_N}w_{N,g,h}}{\inf_{\Omega_N}w_{N,g,h}}\leq V $$

Let us see how the lemma allows to conclude. Write $\psi=f^+-f^-$.

\begin{align*}
(\DilN \nu)_{g,h}(f^+-f^-) &= N^{-\dg/2}\int_{\kg} \psi(x) w_{N,g,h}(x) dx\\
&=\underbrace{ N^{-\dg/2}\int_{\Omega_N} \psi(x) w_{N,g,h}(x) dx}_{T_N} \,+\,\underbrace{N^{-\dg/2}\int_{\kg\smallsetminus \Omega_N} \psi(x) w_{N,g,h}(x) dx}_{S_N}
\end{align*}

Asking that $c$ also satisfies <Ref>, and taking $N$ large enough so that $\int_{\Omega_N}f\geq \frac{\|f\|_{L^1}}{2}$, we can control the first term:

\begin{align*}
T_N &\leq N^{-\dg/2} \int_{\Omega_N} \psi(x) \sup_{\Omega_N}w_{N,g,h}\ dx\\
&\leq N^{-\dg/2} \|\psi\|_{L^1} V\inf_{\Omega_N}w_{N,g,h}\\
&\leq N^{-\dg/2} \|\psi\|_{L^1} V \frac{2}{\|f\|_{L^1}}\int_{\Omega_N} f(x) w_{N,g,h}(x)dx \\
&\leq \|\psi\|_{L^1} V \frac{2}{\|f\|_{L^1}}   (\DilN \nu)_{g,h}(f^+) \\
\end{align*}

As $\|\psi\|_{L^1}\leq \eps$, we finally have for $N\geq 1$, $g,h\in W_N(c)$,

\begin{align*}
T_N=  O_{f}(\eps) \, (\DilN \nu)_{g,h}(f^+)
\end{align*}

Let us now deal with $S_N$. Using that $w_{N,g,h}$ is bounded by $\sup v$, and that $f^+, f^-$ are rapidly decreasing, we obtain for every $Q>0$

\begin{align*}
S_N &\ll N^{-\dg/2}\int_{\kg\smallsetminus \Omega_N} \psi(x)  dx \\
&= O_{f^+, f^-, Q}(N^{-Q})
\end{align*}

Choosing $Q=\frac{\dg}{2}+1$ and using the lower bound $(\DilN \nu)_{g,h}(f^+)\geq  N^{-\frac{\dg}{2} -\delta}$ from <Ref> and our choice for $c$, we deduce that

\begin{align*}
S_N = o_{f^{\pm}}(1) (\DilN \nu)_{g,h}(f^+)
\end{align*}

This concludes the proof of <Ref> modulo <Ref>, which we prove below.

We need to show that for $c\lll1$, the maximum variation

$$\sup_{x,y\in \Omega_N}\big|\log(w_{N,g,h}(x)) -\log(w_{N,g,h}(y))\big|$$

is bounded  by some constant $V'>0$ independent of $N\geq1$, $g,h\in W_N(c)$. Enumerate as $e_1, \dots, e_d$ the basis $(e^{(i)}_j)$ of $\kg$ fixed in <Ref> and write $\partial_i$ for the derivation in the variable $e_i$ (in the classical sense). Fixing $i\in \{1, \dots, d\}$, it is sufficient to show that for $c\lll 1$, $x\in \kg$

$$|(\partial_i \log w_{N,g,h})(x)| \leq |P_{N,g,h}(x)|$$

where $P_{N,g,h}$ is a polynomial on $\kg$ of degree at most $s$, and whose coefficients are $O(\frac{(c\log N)^{O(s)}}{N^{1/3}})$ uniformly in $N\geq2$, $g,h\in W_N(c)$.

Let us analyse this derivative. We can rewrite

$$(\partial_i \log w_{N,g,h})(x) = (\partial_i r_{N,g,h})(x) \,\frac{\partial_i v }{v}(r_{N,g,h}(x))$$

We study each term in the product.

For the first term $\partial_i r_{N,g,h}$, recall that
$r_{N,g,h}(x)=\DilsN (g^{-1}*x*h^{-1})$. As in the proof of <Ref> (<Ref>), we can write $r_{N,g,h}$ as a sum

$$r_{N,g,h}(x)= Q_N(\DilsN g, \DilsN x, \DilsN h)$$

where   $Q_N$ is a polynomial on $\kg^3$ of degree at most $s$,  only[$Q_N$ also depends on the initial data $\kg$, $\mu$, etc. fixed once and for all in  <Ref>.] depending on $N$, and with coefficients $O(1)$.

As $\|\DilsN g\|,  \|\DilsN h\| =O( c(\log N)^{s/2})$, and as any term in $Q_N$ with non-zero $\partial_i$-derivative in $x$ must involve $\DilsN x$, we infer that $\partial_i r_{N,g,h}(x)$ is  a polynomial whose coefficients  are $O(\frac{(c\log N)^{O(s)}}{\sqrt{N}})$ uniformly in $N\geq2$, $g,h\in W_N(c)$.

For the second term $\frac{\partial_i v }{v}(r_{N,g,h}(x))$, we bound the numerator by a constant and use the lower bound on $v$ from <Ref> to get

$$|\frac{\partial_i v }{v}(r_{N,g,h}(x))|\ll A e^{ A \norm{r_{N,g,h}(x)}^2} $$

However,  we have for all $N\geq 1$, $x\in \Omega_N$, $g,h\in W_N(c)$ that

$$\norm{r_{N,g,h}(x)} \leq O(c) \sqrt{\log N}$$

It follows that

$$|\frac{\partial_i v }{v}(r_{N,g,h}(x))|\ll A N^{A O(c)^2} \ll N^{\frac{1}{6}}$$

up to taking $c\lll_A 1$. As $A$ can be fixed as a function of the initial data, our conventions allow to remove it from the subscript, see <Ref>.

The result follows by combining the discussions for each term.

§ CONSEQUENCES FOR CHOQUET-DENY AND EQUIDISTRIBUTION.

§.§ Bounded harmonic functions on nilpotent groups

In this paragraph, we prove <Ref>, namely the Choquet-Deny property. While doing so, we also establish the proposition below, which is an important consequence of our truncated uniform local limit theorem <ref>, and will be crucial in the next section.