# Signatures of the spin Hall effect in hot and dense QCD matter

Baochi Fu<EMAIL_ADDRESS>Center for High Energy Physics, Peking
University, Beijing 100871, China  Department of Physics and State Key
Laboratory of Nuclear Physics and Technology, Peking University, Beijing
100871, China  Collaborative Innovation Center of Quantum Matter, Beijing
100871, China  Long-Gang Pang<EMAIL_ADDRESS>Key Laboratory of Quark
and Lepton Physics (MOE) and Institute of Particle Physics, Central China
Normal University, Wuhan 430079, China  Huichao Song<EMAIL_ADDRESS>Department of Physics and State Key Laboratory of Nuclear Physics and
Technology, Peking University, Beijing 100871, China  Collaborative Innovation
Center of Quantum Matter, Beijing 100871, China  Center for High Energy
Physics, Peking University, Beijing 100871, China  Yi Yin<EMAIL_ADDRESS>Quark Matter Research Center, Institute of Modern Physics, Chinese Academy of
Sciences, Lanzhou 730000, China  University of Chinese Academy of Sciences,
Beijing 100049, China

###### Abstract

The spin Hall effect (SHE) is a generation of spin polarization for moving
spin carriers in materials under an external electric field and has been
observed in semiconductors, metals, and insulators at or below room
temperature. Recent theoretical analyses show that spin Hall current can be
induced by the baryon chemical potential gradient which plays the role of the
analogous electric field and which becomes sizable in the fireballs created in
heavy-ion collisions at beam energy of ${\cal O}(10)$ GeV. In this letter, we
study this important mechanism for spin polarization generation that has not
been systematically explored before and predict the signature of the SHE in
those collisions using a (3+1) D viscous hydrodynamic model MUSIC with AMPT
initial condition. We propose to use the second Fourier coefficients of the
net spin polarization of Lambda hyperon as sensitive probes to search for the
SHE. Those SHE observables show a qualitative difference in both the sign and
beam energy dependence for the situations with and without the SHE. Future
experimental observation of these distinct qualitative features would provide
strong evidence for the existence of the SHE in the hot and dense QCD matter
at trillions of degrees.

Introduction— The spin Hall effect (SHE) is a quantum phenomenon that
showcases the intriguing interplay between charge and spin degrees of freedom
and probes transport properties in quantum materials Dyakonov and Perel
(1971); Sinova  _et al._ (2014). For a class of materials under an external
electric field, the spin polarization of moving charge carriers can be
generated by the SHE due to the Berry curvature of the charge carrier and
relativistic spin-orbital coupling Murakami  _et al._ (2003). Since its first
detection in 2004 Kato  _et al._ (2004), the SHE has been observed in
semiconductors Kato  _et al._ (2004); Wunderlich  _et al._ (2005), metals Liu
_et al._ (2011) and insulators König  _et al._ (2007); see Ref. Sinova  _et
al._ (2014) for a review.

The materials mentioned above are at or below room temperature and are
microscopically described by QED. In contrast, in this letter, we will study
and predict the signatures of the SHE in the spin polarization of $\Lambda$
hyperon (and $\overline{\Lambda}$ anti-hyperon) produced in the extremely hot
and dense QCD matter created in relativistic heavy-ion collisions where the
spin carriers (quarks, gluons and/or hadrons) interact with each other through
strong interaction in media at temperature above $10^{12}$ K. Recently, the
observation of $\Lambda$ ($\overline{\Lambda}$) polarization at Relativistic
Heavy-ion Collider (RHIC) and Large Hadron Collider (LHC) Adamczyk  _et al._
(2017); Voloshin (2018); Adam  _et al._ (2018); Acharya  _et al._ (2020); Adam
_et al._ (2019); Acharya  _et al._ (2021) has inspired many studies on spin
dynamics of QCD many-body physics, (see Refs. Voloshin (2018); Becattini and
Lisa (2020) for reviews). While the ”global” (phase space averaged) spin
polarization is well-described by the effects of vorticity for a wide range of
beam energy, the differential polarization measurements of the azimuthal angle
dependence at top RHIC and LHC energies Adam  _et al._ (2019); Acharya  _et
al._ (2021) show sign opposite to the theory calculations based on thermal
vorticity effects Becattini and Karpenko (2018), see also Refs. Karpenko and
Becattini (2017); Xie  _et al._ (2016); Xia  _et al._ (2018); Wei  _et al._
(2019); Fu  _et al._ (2021a); Guo  _et al._ (2021); Sun  _et al._ (2021) for
examples of studies on differential spin polarization at BES energies. Newly-
discovered shear-induced polarization (SIP) Liu and Yin (2021a); Fu  _et al._
(2021b); Becattini  _et al._ (2021a) is found to be crucial for solving this
”polarization sign puzzle” Fu  _et al._ (2021b); Becattini  _et al._ (2021b).
Despite of those developments, the possible signature of the SHE has not been
fully explored until now (see Refs. Liu and Yin (2021b); Ryu  _et al._ (2021)
for previous investigation of the SHE).

Specifically, we shall consider the SHE induced by baryon chemical potential
gradient $\nabla\mu_{B}$ which becomes sizable in the fireballs created at
colliding energy of ${\cal O}(10)$ GeV. This phenomena as well as spin
polarized by electric field has been predicted and quantitatively described
within the framework of relativistic quantum kinetic equations Son and
Yamamoto (2013); Hidaka  _et al._ (2017, 2018); Hattori  _et al._ (2019); Liu
and Yin (2021a); Yi  _et al._ (2021) and linear response theory Liu and Yin
(2021b, a). Just as charge current will be induced by both chemical potential
gradient and electric field, the spin Hall current can be generated by
$\nabla\mu_{B}$ which plays an analogous role of the electric field 111 More
precisely speaking, as shown in Ref. Liu and Yin (2021b), the spin carriers’
response to both electric field and $\mu_{B}$ gradient are determined by the
same retarded correlator between stress-energy tensor and axial Wigner
function. . That is, for a spin carrier of momentum $\bm{p}$ with baryon
charge $q_{B}$, the induced spin polarization vector $\bm{P}$ is given by

$\displaystyle\bm{P}\propto-\bm{p}\times(q_{B}\nabla\,\mu_{B})\,.$ (1)

Eq. (1) shows that spin polarized by the SHE depends on momentum. Therefore
one should look for the SHE signature in phase space distribution of
polarization observables, i.e. the “local” spin polarization, rather than in
the global polarization. In fact, the recent model calculations confirm the
SHE contribution to the global spin polarization is small in magnitude Ryu
_et al._ (2021).

In this letter, we present the first quantitative prediction of the SHE
signals in differential spin polarization that can be tested experimentally at
RHIC beam energy scan (BES) energies. Using the freeze-out profiles generated
from MUSIC viscous hydrodynamics with AMPT initial conditions, we calculate
the spin polarization of $\Lambda$ and $\overline{\Lambda}$ hyperons along the
beam direction $\hat{z}$ and the out-plane direction $\hat{y}$ in Au+Au
collisions with the beam energy in the range $\sqrt{s_{NN}}=7.7$\- $200$ GeV.
We find that the SHE gives rise to significant separation for the differential
polarization between $\Lambda$ and $\overline{\Lambda}$. The second harmonics
of the net spin polarization can be utilized as unambiguous observables to
search for the SHE.

Method— In field theory, the phase space density of spin polarization of a
fermion/anti-fermion with mass $m$ and four-momentum $p^{\mu}$ is described by
the corresponding axial Wigner function ${\cal A}^{\mu}(x,p)$. To the first
order in the gradient of hydrodynamic variables, i.e., temperature $T$, flow
velocity $u^{\mu}$, chemical potential $\mu_{B}$, ${\cal A}^{\mu}$ for a
baryon-rich fluid is given by Liu and Yin (2021a, b) (see also Refs. Hidaka
_et al._ (2018); Yi  _et al._ (2021)):

${{\cal A}}^{\mu}(x,p)=\beta
f_{0}(x,p)(1-f_{0}(x,p))\varepsilon^{\mu\nu\alpha\rho}\times\Big{(}\underbrace{\frac{1}{2}p_{\nu}\partial_{\alpha}^{\perp}u_{\rho}}_{\text{vorticity}}-\underbrace{\frac{1}{T}u_{\nu}p_{\alpha}\partial_{\rho}T}_{\text{T-gradient}}-\underbrace{\frac{p^{2}_{\perp}}{\varepsilon_{0}}u_{\nu}Q^{\lambda}_{\alpha}\sigma_{\rho\lambda}}_{\text{SIP}}-\underbrace{\frac{q_{B}}{\varepsilon_{0}\beta}u_{\nu}p_{\alpha}\partial_{\rho}(\beta\mu_{B})}_{\text{SHE}}\Big{)},$
(2)

where $f_{0}(x,p)=(e^{(\varepsilon_{0}-q_{B}\mu_{B})\beta}+1)^{-1}$ is the
Fermi-Dirac distribution function with $\varepsilon_{0}=p\cdot u$ and
$\beta=1/T$. Note $q_{B}=1$ for baryons and $q_{B}=-1$ for anti-baryons. Here,
we define
$\partial^{\mu}_{\perp}\equiv\Delta^{\mu\nu}\partial_{\nu},p_{\perp}^{\mu}\equiv\Delta^{\mu\nu}p_{\nu}$
with $\Delta^{\mu\nu}=g^{\mu\nu}-u^{\mu}u^{\nu}$. We denote momentum
quadrupole by
$Q^{\mu\nu}=-p^{\mu}_{\perp}p^{\nu}_{\perp}/p^{2}_{\perp}+\Delta^{\mu\nu}/3$
and shear stress tensor by
$\sigma^{\mu\nu}=\partial^{(\mu}_{\perp}u^{\nu)}-(1/3)\Delta^{\mu\nu}(\partial\cdot
u)$.

Figure 1:  Left panels: the collision energy dependence of second Fourier
coefficients, $P^{{\rm net}}_{2,z}$ and $P^{{\rm net}}_{2,y}$, for
differential polarization of net $\Lambda$ hyperons along the beam and out of
plane direction, calculated in the “Lambda equilibrium” and “strange memory”
scenarios, see text. Right panels: the initial net baryon density $n_{B}$ on
the transverse plane $x-y$ at spatial rapidity $\eta_{s}=0$ and on the
reaction plane $x-\eta_{s}$ at $y=0$ in 20-50% Au+Au collisions at
$\sqrt{s_{NN}}=7.7,19.6,200$ GeV.

Eq. (2) includes all possible first order hydrodynamic gradient contributions
to the spin polarization in phase space that are allowed by symmetry, except
for the vorticity-quadrupole coupling which is found to be suppressed in
perturbative calculations Liu and Yin (2021a). Among those four independent
contributions, the first and second terms in the brackets represent the
vorticity-induced and T-gradient induced polarization respectively. The
combination of the two is proportional to the thermal vorticity, the effect of
which has been widely studied previously Becattini  _et al._ (2013); Becattini
and Karpenko (2018). The recently-discovered shear-induced polarization (SIP)
Liu and Yin (2021a); Fu  _et al._ (2021b); Becattini  _et al._ (2021a); Yi
_et al._ (2021) is shown in the third term. The SHE contribution, given by the
last term, is of potential importance for baryon-rich QCD matter. We shall
employ Eq. (2) as a basis for the subsequently systematic analysis of spin
polarization at BES energies.

To compute the spin polarization of the produced $\Lambda$ and
$\overline{\Lambda}$ hyperons on the freezeout-surface $\Sigma_{\mu}$ from the
axial Wigner function, we follow the widely-used prescription Becattini  _et
al._ (2013); Fang  _et al._ (2016):

$P^{\mu}(\bm{p})=\frac{\int{d\Sigma^{\alpha}p_{\alpha}{\cal
A}^{\mu}(x,\bm{p};m)}}{2m\int{d\Sigma^{\alpha}p_{\alpha}f_{0}(x,p)}}\,,$ (3)

where the factor $2$ in the denominator comes from the degeneracy of spin
$1/2$ fermions.

$\sqrt{s_{NN}}[{\rm GeV}]$ | $c_{N}$ | $\eta/s$ | $\tau_{0}{\rm[fm/c]}$ | $\epsilon_{\text{SW}}\ {\rm[GeV/fm^{3}]}$
---|---|---|---|---
7.7 | 1.8 | 0.16 | 2.0 | 0.35
19.6 | 1.8 | 0.12 | 1.2 | 0.45
27 | 1.8 | 0.10 | 1.0 | 0.50
62.4 | 1.9 | 0.08 | 0.6 | 0.60
200 | 1.9 | 0.08 | 0.4 | 0.65
Table 1: The subset of parameters used in hydrodynamic simulations, see text.

The hydrodynamic profile on the freeze-out surface is generated by 3+1
dimensional viscous hydrodynamics MUSIC. The initial energy-momentum tensor
$T^{\mu\nu}(x)$ and net baryon number density $n_{B}(x)$ are obtained from
AMPT model Lin  _et al._ (2005); He and Lin (2017) with Gaussian smearing at
proper time $\tau_{0}$ Pang  _et al._ (2012) after average over more than
$10^{5}$ events at each $\sqrt{s_{NN}}$. We have multiplied $T^{\mu\nu}$
obtained from AMPT by a phenomenological normalization factor $c_{N}$ which is
tuned to match with charged hadron production in most central collisions. The
hydrodynamic evolution ends when energy density reaches a fixed value
$\varepsilon_{\rm SW}$. Table 1 lists relevant parameters used in our
calculation of hydrodynamic profiles. Unless noted in Table. 1, the model set-
up for AMPT+MUSIC are the same as our previous studies Fu  _et al._ (2021a,
b).

Given significant uncertainties in the current understanding of the
hadronization of spin polarization and its subsequent hadronic evolution, we
consider two benchmark scenarios: the “Lambda equilibrium” and “strange
memory” Fu  _et al._ (2021b). The former assumes that the spin of $\Lambda$
immediately responds to the hydrodynamic gradients Pang  _et al._ (2016);
Karpenko and Becattini (2017); Fu  _et al._ (2021a). In contrast, the latter
supposes that the spin of $\Lambda$ inherits that of the strange quark and is
frozen after the hadronization Liang and Wang (2005); Gao  _et al._ (2020);
Liu  _et al._ (2020). In practice, we input $m=1.116\;\textrm{GeV},q_{B}=\pm
1$ and $m=0.3\;\textrm{GeV},q_{B}=1/3$ in Eq. (3) for the “Lambda equilibrium”
and “strange memory” scenario respectively, as we did in Ref. Fu  _et al._
(2021b). Clearly, those two scenarios represent two extremes, i.e., the spin
relaxation time of $\Lambda$ is very short $\tau_{{\rm spin}}\rightarrow 0$
and very long $\tau_{{\rm spin}}\rightarrow\infty$, respectively, with the
reality (possibly) sitting somewhere in between. However, as we shall see
below, the main feature of the SHE observables that we propose is robust in
both scenarios.

Results— To devise the SHE observables, we shall rely on two observations
followed from Eq. (2). First, as evident in this equation, the SHE
distinguishes itself from the other contributions there since its sign depends
on the baryon charge of the spin carrier. It is then natural to consider net
$\Lambda$ differential spin polarization $P^{{\rm net}}_{z,y}(\phi)\equiv
P_{z,y}(\phi)-\overline{P}_{z,y}(\phi)$ to isolate the SHE contribution. Here
$P_{z,y}(\phi)$ and $\overline{P}_{z,y}(\phi)$ is the polarization of
$\Lambda$ and $\overline{\Lambda}$ hyperons as a function of azimuthal angle
$\phi$ along the out-of-plane direction ($y$) and the beam direction ($z$).
Second, we notice that the SHE contribution is of a similar form to that of
the T-gradient contribution with $\partial_{\mu}T$ replaced by
$(q_{B}T^{2}/\varepsilon_{0})\partial_{\mu}(\beta\mu_{B})$. It has been
demonstrated in a number of studies Becattini and Karpenko (2018); Fu  _et
al._ (2021a, b) that the effect of T-gradient will lead to a characteristic
“$\sin(2\phi)$” and “$\cos(2\phi)$” pattern to $P_{z}(\phi)$ and $P_{y}(\phi)$
respectively. So we expect that the SHE would give rise to a similar
sine/cosine pattern in differential polarization, with the sign depending on
the typical direction of $\mu_{B}$ gradient on the freezeout surface.
Combining those observations, we designate the second harmonic component of
$P^{{\rm net}}_{z,y}(\phi)$ in the Fourier decomposition to characterize the
SHE signal:

$\displaystyle P^{{\rm net}}_{2,z}\equiv\langle P^{{\rm net}}_{z}(\phi)\sin
2\phi\rangle\,,P^{{\rm net}}_{2,y}\equiv-\langle P^{{\rm net}}_{y}(\phi)\cos
2\phi\rangle\,$ (4)

where $\langle...\rangle$ denotes the average over the $\phi$ angle.

Figure 2:  Differential spin polarization $P_{z}(\phi)$ and $-P_{y}(\phi)$
calculated in the “Lambda equilibrium” (labeled by
$\Lambda,\overline{\Lambda}$) and the “strange memory” (labeled by
$s,\bar{s}$) scenarios in 20-50% Au+Au collision at $\sqrt{s_{NN}}=$ 7.7 GeV.
Left panels: with and without the SHE contribution for $\Lambda$ and
$\overline{\Lambda}$ hyperons. Right panels: The individual contribution from
the vorticity, the T-gradient, the shear-induced polarization (SIP) and the
SHE terms in Eq. (2) to the polarization of $\Lambda$.

The main results of this letter are shown in Fig. 1 (left) where the proposed
SHE observables $P^{{\rm net}}_{2,z}$ and $P^{{\rm net}}_{2,y}$ as a function
of collision energy are plotted for both “Lambda equilibrium” and “strange
memory” scenarios. While the magnitude of those observables quantitatively
depends on the mass of spin carriers, we observe several distinctive traits
which are robust in both scenarios. Without the SHE, the second sine harmonics
of longitudinal net spin polarization $P^{{\rm net}}_{2,z}$ is negative at the
beam energy of ${\cal O}(10)$ GeV. With the SHE, it becomes positive and grows
as collision energy decreases. Turning to the SHE observable in $y$-direction,
we see a different non-monotonic collision energy dependence in situations
with and without the SHE. As beam energy goes down, $P^{{\rm net}}_{2,y}$
shows an interesting sign change, from negative to positive, in the presence
of the SHE. In contrast, this cosine coefficient would first increase to some
positive value and then drop down when the SHE is absent. In short, we find
that the proposed SHE observables show a marked difference in both sign and
beam energy dependence with and without the SHE. Those striking qualitative
features make us believe the predicted signatures provide crucial guidance for
the future search for the SHE.

Physically, the SHE converts the spatial baryon chemical potential
distribution into spin polarization in momentum space. Therefore, the
characteristics of the baryon density profiles and their beam energy
dependence would appear in observables sensitive to the SHE. To illustrate
this, in Fig. 1 (right), we present initial baryon density distribution, which
evolves hydrodynamically and can be converted to $\mu_{B}$ through EoS. As Eq.
(1) tells us, the spin vector polarized by the SHE is transverse to the
gradient of $\mu_{B}$. So, to appreciate the behavior of $P^{{\rm
net}}_{2,z}$, we need to look at the $n_{B}$ distribution in the $x-y$ plane.
We notice that the magnitude of $n_{B}$ becomes larger with decreasing beam
energy. Accordingly, we see the growing trends of $P^{{\rm net}}_{2,z}$ when
beam energy becomes smaller. Turning to the initial profile of $n_{B}$ in the
reaction plane $x-\eta_{s}$ at $y=0$, it transits from a double peak structure
to a single peak structure from $\sqrt{s_{NN}}=200$ GeV to $7.7$ GeV. This
transition is anticipated from examining baryon stopping process at different
collision energies Lin  _et al._ (2005); Videbaek (2009); Anticic  _et al._
(2011); Busza  _et al._ (2018); Mohs  _et al._ (2020). Such a qualitative
change in baryon density profile will flip the sign of baryon density gradient
along the longitudinal direction and is expected to induce the non-monotonic
behaviors of $P^{{\rm net}}_{2,y}$, as was confirmed quantitatively in Fig. 1
(left) 222 When computing SHE quantitatively, we need to evaluate
$\nabla(\mu_{B}/T)$ on the freezeout surface which is related to $n_{B}$
through $T$ profile and EoS. Therefore, Fig. (1) (right) is presented only for
illustrative purpose. Moreover, we notice that although $\nabla n_{B}$ can be
expressed in terms of a linear combination of $\nabla T$ and $\nabla\mu_{B}$,
the sign of $\nabla n_{B}$ is not always the same as that of $\nabla\mu_{B}$
because of non-trivial $\nabla T$ profile, see Ref. Fu  _et al._ for details.
.

It is worth emphasizing that even without the SHE, $P^{{\rm
net}}_{2,z},P^{{\rm net}}_{z,y}$ is generically non-zero at BES energies. This
is because there are other mechanisms for the spin polarization, such as the
SIP, with magnitude depending on $\mu_{B}$ through distribution function and
Pauli blocking factor $(1-f)$ in Eq. (2). An important factor within the
present model is that $\Lambda$ tends to freeze-out later than
$\overline{\Lambda}$ , meaning $\Lambda$ and $\overline{\Lambda}$ probe the
regimes with different hydrodynamic profiles. The resulting non-trivial non-
SHE contribution necessitates a quantitative description of differential spin
polarization by systematically including both the SHE and non-SHE effects, as
was done for the first time in the present letter.

To complement Fig. 1, we show $P_{z}(\phi),P_{y}(\phi)$ and
$\overline{P}_{z}(\phi),\overline{P}_{y}(\phi)$ at one representative beam
energy $\sqrt{s_{NN}}=7.7$ GeV for both “$\Lambda$ equilibrium” and “strange
memory” scenarios in Fig. 2 (left). This figure exemplifies several
characteristic features which are also present at other beam energies, see our
coming publication for details Fu  _et al._ . First, azimuthal angle
dependence of $P_{z,y}(\phi)$ ($\overline{P}_{z,y}(\phi)$) is mainly
characterized by $\cos(2\phi),\sin(2\phi)$ pattern. This supports choosing the
second harmonics of differential polarization to detect the SHE. Second, the
SHE leads to a sizable separation of $\Lambda$ and $\overline{\Lambda}$
differential polarization. Without the SHE, either the difference between them
is small, or the ordering is opposite to that with the SHE. For example, in
“$\Lambda$ equilibrium” scenario, the splitting between $P_{y}(\phi)$ and
$\overline{P}_{y}(\phi)$, shown in the bottom-left panel of Fig. 2 (left), is
small in the absence of the SHE but becomes much larger once the SHE
contribution is included. Meanwhile, the hierarchy between
$P_{z}(\phi),\overline{P}_{z}(\phi)$ will be reversed with and without the
SHE. Similar qualitative features can equally be seen in the “strange memory”
scenario, per the lessons we have just discussed when presenting Fig. 1.

In Fig. 2 (right), we present each of the four contributions, i.e., from the
SIP, the T-gradient and the vorticity in additional to the SHE, to $P_{y,z}$
at $\sqrt{s_{NN}}=7.7~{}$GeV. For longitudinal polarization $P_{z}(\phi)$, the
vorticity contribution is insignificant. Similar to results of hydrodynamic
calculations at top RHIC and LHC energies Fu  _et al._ (2021b); Becattini  _et
al._ (2021b), the sign of the second sine harmonics of the SIP and the
T-gradient contribution is opposite to each other at the lower beam energies.
The sign of the SHE contribution, as shown in red curve, is positive and is
the same as that from the SIP. Overall, the sign of second sine harmonics of
$P_{z}$ will be determined by the competition among the SHE, the SIP and the
T-gradient induced polarization. Turning to the polarization along
$y$-direction, the “global” polarization is non-zero mainly because of the
vorticity effect. Once again, the sign of the second cosine harmonics is
driven by the competition among contributions from different hydrodynamic
gradients. We note that while the sign of the SIP, T-gradient and vorticity
contribution at $\sqrt{s_{NN}}=7.7~{}$GeV is same as that at top RHIC and LHC
Fu  _et al._ (2021b); Becattini  _et al._ (2021b), the sign of the SHE
contribution to $P_{y}(\phi)$ is sensitive to the $\mu_{B}$ profile in the
longitudinal direction and hence can change from negative to positive with
decreasing beam energy, as we discussed earlier. The main insight provided by
Fig. 2 (right) is that the SHE contribution is indispensable for studying
differential spin polarization at lower beam energies. The net spin
polarization is more sensitive to the unique feature of the SHE than $P_{z,y}$
or $\overline{P}_{z,y}$ because the former largely isolates the effects of
chemical potential gradient from other mechanisms for polarization generation.

Besides results presented above, we have investigated the consequences of
changing the inputs in our hydrodynamic models. They include the initial
baryon density and flow profile, shear viscosity, bulk viscosity, baryon
diffusion constant, EoS, and freeze-out condition. We find that the SHE signal
is mostly sensitive to the initial profile but those variations within
realistic range don’t qualitatively change the conclusion in this letter; see
the forthcoming publication for more details Fu  _et al._ .

Summary— In this letter, we predict the spin Hall effect (SHE) signature for
the QCD matter created in relativistic heavy-ion collisions at RHIC-BES
energies. Such SHE arises from the gradient of baryon chemical potential and
is an essential mechanism generating spin polarization in baryon-rich QCD that
has not been fully explored before. Using the freeze-out profiles from MUSIC
hydrodynamics with AMPT initial conditions, we demonstrate that the SHE
contribution to the differential Lambda polarization is comparable to those
from the thermal vorticity and the shear-induced polarization (SIP) at lower
collision energies. We propose to use the second harmonics of net spin
polarization, $P_{2,z}^{\rm net}$ and $P_{2,y}^{\rm net}$ (defined in Eq.
(4)), to detect the SHE. We predict that these proposed SHE observables will
show the qualitative difference in both sign and collision energy dependence
in the absence and presence of the SHE. Future experimental observation of the
anticipated non-trivial signatures could provide the first evidence for the
SHE in hot and dense QCD matter.

###### Acknowledgements.

We gratefully acknowledge the contributions of Shuai Liu, who collaborated
with us on this research project during its early stages. We thank helpful
discussions with Francesco Becattini, Iurii Karpenko, Chun Shen and especially
thank Xiangyu Wu for the code verification on the part of the results
presented in this work. This work was supported in part by the NSFC under
grant No. 12075007 (B.F. and H.S.), No. 12147173 (B.F.) and No. 11861131009,
No. 12075098 (L.P.). Y.Y. thanks the support from the Strategic Priority
Research Program of Chinese Academy of Sciences, Grant No. XDB34000000. We
also acknowledge the extensive computing resources provided by the
Supercomputing Center of Chinese Academy of Science (SCCAS), Tianhe-1A from
the National Supercomputing Center in Tianjin, China and the High-performance
Computing Platform of Peking University.

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E\left|\frac{\left|\overline{y}_{i}\right|+\left|\overline{y}_{i,k}\right|}{\left(1+\overline{y}_{i}^{2}\right)\left(1+\overline{y}_{i,k}^{2}\right)}\left|\overline{y}_{i}-\overline{y}_{i,k}\right|\right|^{\kappa_{3}}$
$\displaystyle\leq$ $\displaystyle
E\left(c_{8}\left|\overline{y}_{i}-\overline{y}_{i,k}\right|I\left(\left|\overline{y}_{i}-\overline{y}_{i,k}\right|\leq
c_{9}^{k}\right)+c_{10}I\left(\left|\overline{y}_{i}-\overline{y}_{i,k}\right|\geq
c_{9}^{k}\right)\right)^{\kappa_{3}}$ $\displaystyle\leq$ $\displaystyle
c_{11}c_{9}^{k\kappa_{3}}+c_{12}P\left(\left|\overline{y}_{i}-\overline{y}_{i,k}\right|\geq
c_{9}^{k}\right)\leq
c_{11}c_{9}^{k\kappa_{3}}+c_{12}c_{9}^{-k}E\left|\overline{y}_{i}-\overline{y}_{i,k}\right|$
$\displaystyle\leq$ $\displaystyle c_{13}c_{14}^{k},$

for some $0<c_{9}<1$. Using the same arguments, for all
$\kappa_{4}<\kappa_{1}$

(C.34)
$E\left\|\frac{\mathbf{x}_{i}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}-\frac{\mathbf{x}_{i,k}\overline{y}_{i-1,k}}{1+\overline{y}_{i-1,k}^{2}}\right\|^{\kappa_{4}}\leq
c_{15}c_{16}^{k},$

and for all $\kappa_{5}<\kappa_{2}/2$

(C.35)
$E\left\|\frac{\mathbf{x}_{i}\mathbf{x}_{i}^{\intercal}}{1+\overline{y}_{i-1}^{2}}-\frac{\mathbf{x}_{i,k}\mathbf{x}_{i,k}^{\intercal}}{1+\overline{y}_{i-1,k}^{2}}\right\|^{\kappa_{5}}\leq
c_{17}c_{18}^{k},$

where $0<c_{16},c_{18}<1$. This entails that all the sequences studied above
are decomposable Benrnoulli shifts. Thus

$\left\|\mathbf{Q}_{m}^{\intercal}\mathbf{W}_{m}\mathbf{E}_{m}-\overline{\mathbf{E}}_{m}\right\|=O_{P}\left(1\right),$

where

$\overline{\mathbf{E}}_{m}=\sum_{i=2}^{m}\mathbf{\eta}_{i},$

and $\mathbf{\eta}_{i}$ is defined in (C.25).Repeating the arguments above, it
can also be shown that, for all $\kappa_{6}<\kappa_{1}$

$E\left\|\mathbf{\eta}_{i}-\mathbf{\eta}_{i,k}\right\|^{\kappa_{6}}\leq
c_{19}c_{20}^{k},$

with $\mathbf{\eta}_{i,k}$ defined as the other coupling constructions above
and $0<c_{20}<1$. The final results now follows from the strong approximation
derived in Aue et al. (2014).

######  Lemma C.7.

We assume that $E\log\left|\beta_{0}+\epsilon_{0,1}\right|<0$, and that
Assumptions 2.1, 4.1 and 4.2 are satisfied. Then it holds that

(C.36) $\max_{1\leq
k<\infty}k^{-1/2-\eta}\left\|\sum_{i=m+1}^{m+k}\left(y_{i-1},\mathbf{x}_{i}^{\intercal}\right)\frac{y_{i-1}}{1+y_{i-1}^{2}}-k\mathbf{a}\right\|=O_{P}\left(1\right),$

for all $\eta>0$, and

(C.37) $\max_{1\leq
k<\infty}\left|\sum_{i=m+1}^{m+k}\frac{\left(\epsilon_{i,1}y_{i-1}+\epsilon_{i,2}\right)y_{i-1}}{1+y_{i-1}^{2}}-\sum_{i=m+1}^{m+k}\frac{\left(\epsilon_{i,1}\overline{y}_{i-1}+\epsilon_{i,2}\right)\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right|=O_{P}\left(1\right).$

###### Proof.

The proof is based on the results derived in the proof of Lemma C.6. Equation
(C.36) follows from exactly the same argumens as the proof of (C.10), and
(C.37) follows from the proof of (C.11)-(C.12).

######  Lemma C.8.

We assume that $E\log\left|\beta_{0}+\epsilon_{0,1}\right|<0$, and that
Assumptions 2.1, 4.1 and 4.2 are satisfied. Then, on a suitably enlarged
probability space, it is possible to define two independent Wiener processes
$\left\\{W_{m,1}\left(k\right),1\leq k\leq m\right\\}$ and
$\left\\{W_{m,2}\left(k\right),1\leq k<\infty\right\\}$, whose distributions
do not depend on $m$, such that

$\left|\sum_{i=2}^{m}\mathbf{a}^{\intercal}\mathbf{Q\eta}_{i}-\mathscr{s}_{x,1}W_{m,1}\left(m\right)\right|=O_{P}\left(m^{1/2-\zeta}\right),$

and

$\max_{1\leq
k<\infty}\frac{1}{k^{1/2-\zeta}}\left|\sum_{i=m+1}^{m+k}\frac{\left(\epsilon_{i,1}\overline{y}_{i-1}+\epsilon_{i,2}\right)\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}-\mathscr{s}_{x,2}W_{m,2}\left(k\right)\right|=O_{P}\left(1\right),$

for some $0<\zeta<1/2$, where $\mathscr{s}_{x,1}$ and $\mathscr{s}_{x,2}$ are
defined in (4.7).

###### Proof.

In the proof of Lemma C.6 we have shown that $\mathbf{\eta}_{i}$ is a
decomposable Bernoulli shift; the desired result follows from Aue et al.
(2014).

######  Lemma C.9.

We assume that $E\log\left|\beta_{0}+\epsilon_{0,1}\right|>0$, and that
Assumptions 2.1, 4.1-4.3 are satisfied. Then it holds that

$\left\|\widehat{\mathbf{b}}_{m}-\mathbf{b}_{0}\right\|=O_{P}\left(m^{-1/2}\right).$

###### Proof.

Recall that

$\widehat{\mathbf{b}}_{m}-\mathbf{b}_{0}=\left(\mathbf{Q}_{m}^{\intercal}\mathbf{W}_{m}\mathbf{Q}_{m}\right)^{-1}\mathbf{Q}_{m}^{\intercal}\mathbf{W}_{m}\mathbf{E}_{m}.$

Repeating the proof of Theorem 3.1 in Berkes et al. (2009), it follows that
$\left|y_{i}\right|\rightarrow\infty$ a.s. exponentially fast; this
immediately yields

$\left\|\mathbf{Q}_{m}^{\intercal}\mathbf{W}_{m}\mathbf{Q}_{m}-\mathbf{B}_{m}\right\|=O_{P}\left(1\right),$

where $\mathbf{B}_{m}$ is a $\left(p+1\right)\times\left(p+1\right)$ symmetric
matrix with elements $\left\\{B_{i,j},1\leq i,j\leq p+1\right\\}$ defined as

$\displaystyle B_{1,1}=m,$
$\displaystyle\left(B_{1,2},...,B_{1,p+1}\right)=\sum_{i=2}^{\infty}\frac{\mathbf{x}_{i}^{\intercal}y_{i-1}}{1+y_{i-1}^{2}},$
$\displaystyle\left\\{B_{i,j},2\leq i,j\leq
p+1\right\\}=\sum_{i=2}^{\infty}\frac{\mathbf{x}_{i}\mathbf{x}_{i}^{\intercal}}{1+y_{i-1}^{2}}.$

Since we already know from the above that
$\left\|\mathbf{E}_{m}\right\|=O_{P}\left(m^{1/2}\right)$, the result follows
immediately.

##  D. Proofs

###### Proof of Theorem 3.1.

Recall (C.23). We begin by showing that

(D.1) $\max_{1\leq
k<\infty}\frac{\left|Z_{m}\left(k\right)-\Gamma_{m}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}=o_{P}\left(1\right).$

We begin by considering the case
$E\log\left|\beta_{0}+\epsilon_{0,1}\right|<0$; we write

$\displaystyle\left(\beta_{0}-\widehat{\beta}_{m}\right)\sum_{i=m+1}^{m+k}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}+\frac{k}{m}\sum_{i=2}^{m}\left(\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)$
$\displaystyle=$
$\displaystyle\left(\beta_{0}-\widehat{\beta}_{m}\right)\left(\sum_{i=m+1}^{m+k}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}-ka_{3}\right)$
$\displaystyle-
ka_{3}\left(\left(\sum_{i=2}^{m}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)^{-1}-\frac{1}{ma_{3}}\right)\left(\sum_{i=2}^{m}\left(\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}+\frac{\epsilon_{i,2}y_{i-1}}{1+y_{i-1}^{2}}\right)\right)$
$\displaystyle+\frac{k}{m}\sum_{i=2}^{m}\left(\epsilon_{i,1}\left(\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}-\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)+\epsilon_{i,2}\left(\frac{\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}-\frac{y_{i-1}}{1+y_{i-1}^{2}}\right)\right)$
$\displaystyle=$ $\displaystyle
R_{m,1}\left(k\right)+R_{m,2}\left(k\right)+R_{m,3}\left(k\right).$

Using Lemmas C.1 and C.2 with $\eta<1/2$

$\displaystyle\max_{1\leq
k<\infty}\frac{\left|R_{m,1}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(m^{-1}\right)\max_{1\leq k\leq
M}\frac{k^{1/2+\eta}}{\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle+O_{P}\left(m^{-1}\right)\max_{M<k<\infty}\frac{k^{1/2+\eta}}{\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\left(m^{-1/2}\max_{1\leq
k\leq M}k^{\eta}+\max_{M<k<\infty}k^{\eta-1/2}\right)=o_{P}\left(1\right).$

Using the same logic, it can also be shown that

$\displaystyle\max_{1\leq
k<\infty}\frac{\left|R_{m,2}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=o_{P}\left(1\right),$ $\displaystyle\max_{1\leq
k<\infty}\frac{\left|R_{m,3}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=o_{P}\left(1\right).$

Using Lemma C.3, it follows that

(D.2) $\displaystyle\max_{1\leq
k<\infty}\frac{\left|\displaystyle\frac{k}{m}\sum_{i=2}^{m}\left(\displaystyle\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\displaystyle\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)-\displaystyle\frac{k}{m}\left(a_{1}\sigma_{1}^{2}+a_{2}\sigma_{2}^{2}\right)^{1/2}W_{m,1}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\max_{1\leq
k<\infty}\frac{k}{m}\frac{m^{1/2-\zeta}}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(m^{-\zeta}\right)\max_{1\leq
k<\infty}\left(\frac{k}{m+k}\right)^{1-\psi}=o_{P}\left(1\right).$

By the same token, we can show that

(D.3) $\displaystyle\max_{1\leq
k<\infty}\frac{\left|\displaystyle\sum_{i=m+1}^{m+k}\left(\epsilon_{i,1}\left(\displaystyle\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}-\displaystyle\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\max_{1\leq
k<\infty}\frac{1}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(m^{-1/2}\right)\max_{1\leq
k<\infty}\frac{m}{m+k}\left(\frac{k}{m+k}\right)^{-\psi}=O_{P}\left(m^{-1/2+\psi}\right)=o_{P}\left(1\right),$

and

(D.4) $\displaystyle\max_{1\leq
k<\infty}\frac{\left|\displaystyle\sum_{i=m+1}^{m+k}\left(\epsilon_{i,2}\left(\displaystyle\frac{\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}-\displaystyle\frac{y_{i-1}}{1+y_{i-1}^{2}}\right)\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\max_{1\leq
k<\infty}\frac{1}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(m^{-1/2}\right)\max_{1\leq
k<\infty}\frac{m}{m+k}\left(\frac{k}{m+k}\right)^{-\psi}=O_{P}\left(m^{-1/2+\psi}\right)=o_{P}\left(1\right),$

having used Lemma C.2. Similarly, by Lemma C.3

(D.5) $\displaystyle\max_{1\leq
k<\infty}\frac{\left|\displaystyle\sum_{i=m+1}^{m+k}\left(\displaystyle\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}+\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)-\left(a_{1}\sigma_{1}^{2}+a_{2}\sigma_{2}^{2}\right)^{1/2}W_{m,2}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(1\right)\max_{1\leq
k<\infty}\frac{k^{1/2-\zeta}}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}=o_{P}\left(1\right).$

Putting all together, (D.1) has been shown for the stationary case.
Considering now the case $E\log\left|\beta_{0}+\epsilon_{0,1}\right|\geq 0$,
write

$\displaystyle\left(\beta_{0}-\widehat{\beta}_{m}\right)\sum_{i=m+1}^{m+k}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}+\frac{k}{m}\sum_{i=2}^{m}\epsilon_{i,1}$
$\displaystyle=$
$\displaystyle\left(\beta_{0}-\widehat{\beta}_{m}\right)\left(\sum_{i=m+1}^{m+k}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}-k\right)$
$\displaystyle-k\left(\left(\sum_{i=2}^{m}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)^{-1}-\frac{1}{m}\right)\left(\sum_{i=2}^{m}\left(\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}+\frac{\epsilon_{i,2}y_{i-1}}{1+y_{i-1}^{2}}\right)\right)$
$\displaystyle-\frac{k}{m}\left(\sum_{i=2}^{m}\frac{\epsilon_{i,1}}{1+y_{i-1}^{2}}+\sum_{i=2}^{m}\frac{\epsilon_{i,2}y_{i-1}}{1+y_{i-1}^{2}}\right)$
$\displaystyle=$ $\displaystyle
R_{m,4}\left(k\right)+R_{m,5}\left(k\right)+R_{m,6}\left(k\right).$

The arguments are similar to the above. Indeed, using Lemmas C.1-C.3

(D.6) $\displaystyle\max_{1\leq
k<\infty}\frac{\left|R_{m,4}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(m^{-1/2}\right)\max_{1\leq
k<\infty}\frac{km^{-\zeta}}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(m^{-\zeta}\right)=o_{P}\left(1\right).$

Similarly it can be shown that

(D.7) $\displaystyle\max_{1\leq
k<\infty}\frac{\left|R_{m,5}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}=O_{P}\left(m^{-\zeta}\right)=o_{P}\left(1\right),$
(D.8) $\displaystyle\max_{1\leq
k<\infty}\frac{\left|R_{m,6}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}=O_{P}\left(m^{-\zeta+\eta}\right)=o_{P}\left(1\right),$

having chosen $\eta<\zeta$ in the last equation. Lemma C.3 yields

(D.9) $\max_{1\leq
k<\infty}\frac{\left|\displaystyle\frac{k}{m}\sum_{i=2}^{m}\epsilon_{i,1}-\sigma_{1}\displaystyle\frac{k}{m}W_{m,1}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}=O_{P}\left(m^{\zeta-1/2}\right)=o_{P}\left(1\right),$

and

(D.10) $\max_{1\leq
k<\infty}\frac{\left|\displaystyle\sum_{i=m+1}^{m+k}\epsilon_{i,1}-\sigma_{1}\displaystyle
W_{m,2}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}=O_{P}\left(m^{\zeta-1/2}\right)=o_{P}\left(1\right).$

similarly, by Lemma C.2

(D.11) $\displaystyle\max_{1\leq
k<\infty}\frac{\left|\displaystyle\sum_{i=m+1}^{m+k}\displaystyle\frac{\epsilon_{i,1}}{1+y_{i-1}^{2}}\right|+\left|\displaystyle\sum_{i=m+1}^{m+k}\displaystyle\frac{\epsilon_{i,2}y_{i-1}}{1+y_{i-1}^{2}}\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(m^{-\zeta}\right)\max_{1\leq
k<\infty}\frac{k^{1/2+\eta}}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle=$ $\displaystyle O_{P}\left(m^{1/2-\zeta}\right)\max_{1\leq
k<\infty}\frac{k^{1/2+\eta-\psi}}{\left(m+k\right)^{1/2+\eta-\psi+1/2-\eta}}=O_{P}\left(m^{\eta-\zeta}\right)=o_{P}\left(1\right),$

having chosen $\eta<\zeta$. Putting all together, (D.1) now follows for the
case $E\log\left|\beta_{0}+\epsilon_{0,1}\right|\geq 0$.

Let now

(D.12)
$\Gamma\left(t\right)=\left\\{\begin{array}[]{ll}\mathscr{s}^{1/2}\left|W_{2}\left(t\right)-tW_{1}\left(1\right)\right|&\text{if
}E\log\left|\beta_{0}+\epsilon_{0,1}\right|<0\text{\ holds}\\\
\sigma_{1}\left|W_{2}\left(t\right)-tW_{1}\left(1\right)\right|&\text{if
}E\log\left|\beta_{0}+\epsilon_{0,1}\right|\geq 0\text{\
holds}\end{array}\right.,$

where $\left\\{W_{1}\left(k\right),1\leq k\leq m\right\\}$ and
$\left\\{W_{2}\left(k\right),1\leq k<\infty\right\\}$ are two independent
standard Wiener processes. Using the fact that the distribution of
$W_{m,1}\left(\cdot\right)$ and $W_{m,2}\left(\cdot\right)$ does not depend on
$m$, and exploiting the scale transformation and the continuity of the Wiener
process

$\displaystyle\max_{1\leq
k<\infty}\frac{\Gamma_{m}\left(k\right)}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle\overset{\mathcal{D}}{=}\max_{1\leq
k<\infty}\frac{\Gamma\left(\displaystyle\frac{k}{m}\right)}{\left(1+\displaystyle\frac{k}{m}\right)\left(\frac{\displaystyle
k/m}{\displaystyle 1\displaystyle+\displaystyle k/m}\right)^{\psi}}$
$\displaystyle\overset{\mathcal{D}}{=}\sup_{0<t<\infty}\frac{\Gamma\left(t\right)}{\left(1+t\right)\left(\displaystyle\frac{t}{1+t}\right)^{\psi}}.$

Computing the covariance kernel, it can be easily verified that

(D.13) $\left\\{\frac{W_{2}\left(t\right)-tW_{1}\left(1\right)}{1+t},t\geq
0\right\\}\overset{\mathcal{D}}{=}\left\\{W\left(\frac{t}{1+t}\right),t\geq
0\right\\},$

where $W\left(\cdot\right)$ denotes a standard Wiener. Hence

$\max_{1\leq
k<\infty}\frac{\Gamma_{m}\left(k\right)}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}\overset{\mathcal{D}}{=}\sup_{0<t<\infty}\frac{\mathscr{s}\left|W\left(\displaystyle\frac{t}{1+t}\right)\right|}{\left(\displaystyle\frac{t}{1+t}\right)^{\psi}},$

whence Theorem 3.1 follows.

###### Proof of Theorem 3.2.

Using exactly the same arguments as in the proof of Theorem 3.1, it follows
that

(D.14) $\max_{1\leq k\leq
m^{\ast}}\frac{\left|Z_{m}\left(k\right)-\Gamma_{m}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}=o_{P}\left(1\right).$

Further, it is immediate to see that

$\displaystyle\max_{1\leq k\leq
m^{\ast}}\frac{\Gamma_{m}\left(k\right)}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{\psi}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle\max_{1\leq k\leq
m^{\ast}}\frac{\Gamma\left(\displaystyle\frac{k}{m}\right)}{\left(1+\displaystyle\frac{k}{m}\right)\left(\frac{\displaystyle
k/m}{\displaystyle 1\displaystyle+\displaystyle k/m}\right)^{\psi}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\sup_{0<t<m_{0}}\frac{\Gamma\left(t\right)}{\left(1+t\right)\left(\displaystyle\frac{t}{1+t}\right)^{\psi}}.$

Using (D.13), it finally follows

$\sup_{0<t<m_{0}}\frac{\Gamma\left(t\right)}{\left(1+t\right)\left(\displaystyle\frac{t}{1+t}\right)^{\psi}}\overset{\mathcal{D}}{=}\sup_{0<t<m_{0}}\frac{\mathscr{s}\left|W\left(\displaystyle\frac{t}{1+t}\right)\right|}{\left(\displaystyle\frac{t}{1+t}\right)^{\psi}},$

whence the desired result.

###### Proof of Theorem 3.4.

We begin by noting that, repeating verbatim the proof of (D.1) with
$\psi=1/2$, it can be shown that

(D.15) $\max_{1\leq k\leq
m^{\ast}}\frac{\left|Z_{m}\left(k\right)-\Gamma_{m}\left(k\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{1/2}}=O_{P}\left(1\right).$

Recall also that the distribution of $\Gamma_{m}\left(k\right)$ does not
depend on $m$, so that

$\left\\{m^{-1/2}\Gamma_{m}\left(mt\right),t\geq
0\right\\}\overset{\mathcal{D}}{=}\left\\{\Gamma\left(t\right),t\geq
0\right\\},$

where $\Gamma\left(t\right)$ is defined in (D.12); further recall that, by
(D.13)

$\left\\{\frac{1}{\mathscr{s}}\frac{\Gamma\left(t\right)}{1+t},t\geq
0\right\\}\overset{\mathcal{D}}{=}\left\\{\left|W\left(\frac{t}{1+t}\right)\right|,t\geq
0\right\\}.$

We now report some results concerning $\Gamma_{m}\left(k\right)$. We begin by
studying

(D.16) $\displaystyle\frac{1}{\mathscr{s}}\max_{1\leq k\leq\log
m}\frac{\Gamma_{m}\left(k\right)}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\frac{1}{\mathscr{s}}\max_{1/m\leq t\leq\left(\log
m\right)/m}\frac{\Gamma\left(t\right)}{\left(1+t\right)\left(\displaystyle\frac{t}{1+t}\right)^{1/2}}\overset{\mathcal{D}}{=}\max_{1/m\leq
t\leq\left(\log
m\right)/m}\frac{\left|W\left(\displaystyle\frac{t}{1+t}\right)\right|}{\left(\displaystyle\frac{t}{1+t}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\max_{1/\left(m+1\right)\leq u\leq\left(\log
m\right)/\left(m+\log
m\right)}\frac{\left|W\left(u\right)\right|}{u^{1/2}}\overset{\mathcal{D}}{=}\max_{1\leq
v\leq\log m}\frac{\left|W\left(v\right)\right|}{v^{1/2}}$ $\displaystyle=$
$\displaystyle O_{P}\left(\sqrt{\log\log\log m}\right),$

where we have repeatedly used the scale transformation of the Wiener process,
and the Law of the Iterated Logarithm in the last passage. Similarly

(D.17) $\displaystyle\frac{1}{\mathscr{s}}\max_{m/\log m\leq k\leq
m^{\ast}}\frac{\Gamma_{m}\left(k\right)}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\frac{1}{\mathscr{s}}\max_{1/\log m\leq t\leq m^{\ast}/\log
m}\frac{\Gamma\left(t\right)}{\left(1+t\right)\left(\displaystyle\frac{t}{1+t}\right)^{1/2}}\overset{\mathcal{D}}{=}\max_{1/\log
m\leq t\leq m^{\ast}/\log
m}\frac{\left|W\left(\displaystyle\frac{t}{1+t}\right)\right|}{\left(\displaystyle\frac{t}{1+t}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle\max_{1/\left(\log
m+1\right)\leq u\leq
m^{\ast}/\left(m+m^{\ast}\right)}\frac{\left|W\left(u\right)\right|}{u^{1/2}}\overset{\mathcal{D}}{=}\max_{1\leq
v\leq m^{\ast}\left(\log
m+1\right)/\left(m+m^{\ast}\right)}\frac{\left|W\left(v\right)\right|}{v^{1/2}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(\sqrt{\log\log\frac{m^{\ast}\left(\log
m+1\right)}{m+m^{\ast}}}\right)=O_{P}\left(\sqrt{\log\log\log m}\right).$

Finally, by the same token, note that

$\displaystyle\frac{1}{\mathscr{s}}\max_{\log m\leq k\leq m/\log
m}\frac{\Gamma_{m}\left(k\right)}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\frac{1}{\mathscr{s}}\max_{\log m\leq k\leq m/\log
m}\frac{\Gamma_{m}\left(\displaystyle\frac{k}{m}\right)}{\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\frac{1}{\mathscr{s}}\max_{\left(\log m\right)/m\leq t\leq
1/\log
m}\frac{\Gamma\left(t\right)}{\left(1+t\right)\left(\displaystyle\frac{t}{1+t}\right)^{1/2}}\overset{\mathcal{D}}{=}\max_{\left(\log
m\right)/m\leq t\leq 1/\log
m}\frac{\left|W\left(\displaystyle\frac{t}{1+t}\right)\right|}{\left(\displaystyle\frac{t}{1+t}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle\max_{\left(\log
m\right)/\left(m+\log m\right)\leq t\leq 1/\left(\log
m+1\right)}\frac{\left|W\left(u\right)\right|}{u^{1/2}}\overset{\mathcal{D}}{=}\max_{1\leq
v\leq\left(m+\log m\right)/\left(\left(1+\log m\right)\log
m\right)}\frac{\left|W\left(v\right)\right|}{v^{1/2}},$

and using the Law of the Iterated Logarithm it follows that

(D.18)
$\lim_{m\rightarrow\infty}\frac{1}{\sqrt{2\log\log\displaystyle\frac{m+\log
m}{\left(1+\log m\right)\log m}}}\max_{1\leq v\leq\left(m+\log
m\right)/\left(\left(1+\log m\right)\log
m\right)}\frac{\left|W\left(v\right)\right|}{v^{1/2}}=1\text{ \ \ a.s.}$

Given that, as $m\rightarrow\infty$

$\frac{\log\log\displaystyle\frac{m+\log m}{\left(1+\log m\right)\log
m}}{\log\log m}=1,$

it follows that the term in (D.18) dominates, and therefore, putting all
together

$\lim_{m\rightarrow\infty}P\left\\{\max_{1\leq k\leq
m^{\ast}}\frac{Z_{m}\left(k\right)}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{1/2}}=\max_{1\leq
v\leq\left(m+\log m\right)/\left(\left(1+\log m\right)\log
m\right)}\frac{\left|W\left(v\right)\right|}{v^{1/2}}\right\\}=1.$

Using the Darling-Erdős theorem (Darling and Erdős, 1956), it follows that

$\displaystyle\lim_{m\rightarrow\infty}P\left\\{\gamma\left(\log\frac{m+\log
m}{\left(1+\log m\right)\log m}\right)\max_{1\leq v\leq\left(m+\log
m\right)/\left(\left(1+\log m\right)\log
m\right)}\frac{\left|W\left(v\right)\right|}{v^{1/2}}\right.$
$\displaystyle\leq$ $\displaystyle\left.x+\delta\left(\log\frac{m+\log
m}{\left(1+\log m\right)\log
m}\right)\right\\}=\exp\left(-\exp\left(-x\right)\right).$

The desired result follows upon noting that, by elementary arguments

$\displaystyle\left|\gamma\left(\frac{m+\log m}{\left(1+\log m\right)\log
m}\right)-\gamma\left(m\right)\right|$ $\displaystyle\rightarrow 0,$
$\displaystyle\left|\delta\left(\frac{m+\log m}{\left(1+\log m\right)\log
m}\right)-\delta\left(m\right)\right|$ $\displaystyle\rightarrow 0,$

as $m\rightarrow\infty$.

###### Proof of Theorem 3.3.

We start with the proof of (3.4), considering the case
$E\log\left|\beta_{0}+\epsilon_{0,1}\right|<0$. As in the proof of Theorem
3.1, it suffices to show that

$\max_{1\leq k\leq
m^{\ast}}\frac{Z_{m}^{\ast}\left(k\right)}{\overline{g}_{m,\psi}\left(k\right)}\overset{\mathcal{D}}{\rightarrow}\frac{1}{\overline{c}_{\alpha,\psi}}\sup_{0\leq
t\leq 1}\frac{\left|W\left(t\right)\right|}{t^{\psi}},$

where

$\displaystyle Z_{m}^{\ast}\left(k\right)=$
$\displaystyle\left|\left(\sum_{i=2}^{m}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}\right)^{-1}\left(\sum_{i=2}^{m}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=2}^{m}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)\left(\sum_{i=m+1}^{m+k}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}\right)\right.$
$\displaystyle+\left.\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right|,$

where recall that $\overline{y}_{i}$ is the stationary solution of (2.1).
Using the proofs of Lemmas C.1 and C.2, it follows by routine calculations
that

$\left|\left(\sum_{i=2}^{m}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}\right)^{-1}-\frac{1}{ma_{3}}\right|=O_{P}\left(m\right),$

and

$\left|\sum_{i=2}^{m}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=2}^{m}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right|=O_{P}\left(m^{1/2}\right),$

where $a_{3}$ is defined in (C.2). The approximations in Aue et al. (2014)
yield

$\left(m^{\ast}\right)^{-1/2+\psi}\max_{1\leq k\leq
m^{\ast}}\frac{1}{k^{\psi}}\left|\sum_{i=m+1}^{m+k}\left(\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}-a_{3}\right)\right|\overset{\mathcal{D}}{\rightarrow}a_{4}\sup_{0\leq
t\leq 1}\frac{\left|W\left(t\right)\right|}{t^{\psi}},$

where

$a_{4}=\sum_{h=-\infty}^{\infty}E\left[\left(\frac{\overline{y}_{0}^{2}}{1+\overline{y}_{0}^{2}}-a_{3}\right)\left(\frac{\overline{y}_{h}^{2}}{1+\overline{y}_{h}^{2}}-a_{3}\right)\right].$

Thus we conclude

$\displaystyle\left(m^{\ast}\right)^{-1/2+\psi}\left(\sum_{i=2}^{m}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}\right)^{-1}\left|\sum_{i=2}^{m}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=2}^{m}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right|\max_{1\leq
k\leq
m^{\ast}}\frac{1}{k^{\psi}}\sum_{i=m+1}^{m+k}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(m^{-1/2}\right)\left(m^{\ast}\right)^{-1/2+\psi}\max_{1\leq k\leq
m^{\ast}}\frac{1}{k^{\psi}}\sum_{i=m+1}^{m+k}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(m^{-1/2}\right)\left(m^{\ast}\right)^{-1/2+\psi}k^{1-\psi}$
$\displaystyle+O_{P}\left(m^{-1/2}\right)\left(m^{\ast}\right)^{-1/2+\psi}\max_{1\leq
k\leq
m^{\ast}}\frac{1}{k^{\psi}}\sum_{i=m+1}^{m+k}\left(\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}-a_{3}\right)$
$\displaystyle=$ $\displaystyle
O_{P}\left(\left(\frac{m^{\ast}}{m}\right)^{1/2}+m^{-1/2}\right)=o_{P}\left(1\right).$

Using Aue et al. (2014), we can define two independent Wiener processes
$\left\\{W_{m,1}\left(x\right),x\geq 0\right\\}$ and
$\left\\{W_{m,2}\left(x\right),x\geq 0\right\\}$ such that

$\displaystyle\max_{1\leq k\leq
m^{\ast}}\frac{1}{\left(m^{\ast}\right)^{1/2-\psi}k^{\psi}}\left|\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}-\left(a_{1}^{1/2}\sigma_{1}W_{m,1}\left(k\right)+a_{2}^{1/2}\sigma_{2}W_{m,2}\left(k\right)\right)\right|$
$\displaystyle=$ $\displaystyle o_{P}\left(1\right).$

By the scale transformation and the continuity of the Wiener process, it
follows that

$\displaystyle\max_{1\leq k\leq
m^{\ast}}\frac{1}{\left(m^{\ast}\right)^{1/2-\psi}k^{\psi}}\left|a_{1}^{1/2}\sigma_{1}W_{m,1}\left(k\right)+a_{2}^{1/2}\sigma_{2}W_{m,2}\left(k\right)\right|$
$\displaystyle\overset{\mathcal{D}}{=}\max_{1/m^{\ast}\leq k/m^{\ast}\leq
1}\left(\frac{k}{m^{\ast}}\right)^{-\psi}\left|a_{1}^{1/2}\sigma_{1}W_{m,1}\left(\frac{k}{m^{\ast}}\right)+a_{2}^{1/2}\sigma_{2}W_{m,2}\left(\frac{k}{m^{\ast}}\right)\right|$
$\displaystyle\overset{\mathcal{D}}{\rightarrow}\sup_{0\leq u\leq
1}\frac{1}{t^{\psi}}\left|a_{1}^{1/2}\sigma_{1}W_{1}\left(t\right)+a_{2}^{1/2}\sigma_{2}W_{2}\left(t\right)\right|,$

where $W_{1}$ and $W_{2}$ are independent Wiener processes. Since

$\left\\{a_{1}^{1/2}\sigma_{1}W_{1}\left(t\right)+a_{2}^{1/2}\sigma_{2}W_{2}\left(t\right),t\geq
0\right\\}\overset{\mathcal{D}}{=}\left\\{\mathscr{s}W\left(t\right),t\geq
0\right\\},$

where $\left\\{W\left(t\right),t\geq 0\right\\}$ is a Wiener process, the
result follows.

Next we consider the case $\psi=1/2$, and we establish a Darling-Erdős
limiting law for $Z_{m}^{\ast}\left(k\right)$. Firstly note that

$\displaystyle\max_{1\leq k\leq
m^{\ast}}\frac{1}{k^{1/2}}\left(\sum_{i=2}^{m}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}\right)^{-1}\left(\sum_{i=2}^{m}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=2}^{m}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)\sum_{i=m+1}^{m+k}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(\left(\frac{m^{\ast}}{m}\right)^{1/2}\right),$
$\frac{1}{\left(\log\log m^{\ast}\right)^{1/2}}\max_{1\leq k\leq
m^{\ast}}\frac{1}{k^{1/2}}\left|\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right|=O_{P}\left(1\right),$

and

$\frac{1}{\left(\log\log m^{\ast}\right)^{1/2}}\max_{\log m^{\ast}\leq k\leq
m^{\ast}/\log
m^{\ast}}\frac{1}{k^{1/2}}\left|\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right|\overset{\mathcal{P}}{\rightarrow}c>0.$

The above entails that we need to consider the maximum of
$Z_{m}^{\ast}\left(k\right)$ over $\log m^{\ast}\leq k\leq m^{\ast}/\log
m^{\ast}$. We have

$\displaystyle\max_{\log m^{\ast}\leq k\leq m^{\ast}/\log
m^{\ast}}\frac{1}{k^{1/2}}\left(\sum_{i=2}^{m}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}\right)^{-1}\left(\sum_{i=2}^{m}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=2}^{m}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)\sum_{i=m+1}^{m+k}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(\left(\frac{m^{\ast}}{m}\right)^{1/2}\left(\log
m^{\ast}\right)^{-1/2}\right).$

By the Law of the Iterated Logarithm, it follows that

$\max_{1\leq k\leq\log
m^{\ast}}\frac{1}{k^{1/2}}\left|\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right|=O_{P}\left(\left(\log\log\log
m^{\ast}\right)^{1/2}\right).$

Hence we need to show that, for all $-\infty<x<\infty$

$\displaystyle\lim_{m\rightarrow\infty}P\left\\{\gamma\left(\log
m^{\ast}\right)\frac{1}{\mathscr{s}}\max_{1\leq k\leq
m^{\ast}}\frac{1}{k^{1/2}}\left|\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right|\leq
x+\delta\left(\log m^{\ast}\right)\right\\}$ $\displaystyle=$
$\displaystyle\exp\left(-\exp\left(-x\right)\right),$

which follows from the same logic as the proof of Theorem 3.4.

We now turn to the case $E\log\left|\beta_{0}+\epsilon_{0,1}\right|\geq 0$.
Following the proof of Lemma C.1, it can be shown that

(D.19)
$\left|\left(\sum_{i=2}^{m}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)^{-1}-\frac{1}{m}\right|=O_{P}\left(m^{-3/2}\right),$
(D.20)
$\left|\left(\sum_{i=2}^{m}\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}+\sum_{i=2}^{m}\frac{\epsilon_{i,2}y_{i-1}}{1+y_{i-1}^{2}}\right)-\sum_{i=2}^{m}\epsilon_{i,1}\right|=O_{P}\left(1\right),$
(D.21) $\max_{1\leq
k<\infty}\left|\sum_{i=m+1}^{m+k}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}-k\right|=O_{P}\left(c^{m}\right),$

and

(D.22) $\max_{1\leq
k<\infty}\left|\left(\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}+\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,2}y_{i-1}}{1+y_{i-1}^{2}}\right)-\sum_{i=m+1}^{m+k}\epsilon_{i,1}\right|=O_{P}\left(c^{m}\right),$

with some $0<c<1$. Thus we get

$\displaystyle\left(\sum_{i=2}^{m}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}\right)^{-1}\left|\sum_{i=2}^{m}\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\sum_{i=2}^{m}\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right|\left(m^{\ast}\right)^{-1/2+\psi}\max_{1\leq
k\leq
m^{\ast}}\frac{1}{k^{\psi}}\sum_{i=m+1}^{m+k}\frac{\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}$
$\displaystyle=$ $\displaystyle
O_{P}\left(m^{-1/2}\right)\left(m^{\ast}\right)^{-1/2+\psi}\max_{1\leq k\leq
m^{\ast}}\frac{1}{k^{\psi}}\left(k+c^{m}\right)$ $\displaystyle=$
$\displaystyle
O_{P}\left(\left(\frac{m^{\ast}}{m}\right)^{1/2}\right)=o_{P}\left(1\right).$

Our assumptions entail that

$\left(m^{\ast}\right)^{-1/2+\psi}\max_{1\leq k\leq
m^{\ast}}\frac{1}{k^{\psi}}\left|\sum_{i=m+1}^{m+k}\epsilon_{i,1}\right|\overset{\mathcal{D}}{\rightarrow}\sigma_{1}\sup_{0\leq
t\leq 1}\frac{\left|W\left(t\right)\right|}{t^{\psi}},$

where $\left\\{W\left(t\right),t\geq 0\right\\}$ is a Wiener process; this
completes the proof of (3.4). Considering the case $\psi=1/2$, we can follows
the same arguments as above, using (D.19)-(D.22). We obtain

$\left|\max_{1\leq
k<m^{\ast}}\frac{\left|Z_{m}^{\ast}\left(k\right)\right|}{k^{1/2}}-\max_{\log
m^{\ast}\leq k<m^{\ast}/\log
m^{\ast}}\frac{1}{k^{1/2}}\left|\sum_{i=m+1}^{m+k}\epsilon_{i,1}\right|\right|=o_{P}\left(\left(\log\log
m^{\ast}\right)^{-1/2}\right).$

Hence we need to prove only that

$\lim_{m\rightarrow\infty}P\left\\{\gamma\left(\log
m^{\ast}\right)\frac{1}{\sigma_{1}}\max_{\log m^{\ast}\leq k<m^{\ast}/\log
m^{\ast}}\frac{1}{k^{1/2}}\left|\sum_{i=m+1}^{m+k}\epsilon_{i,1}\right|\leq
x+\delta\left(\log
m^{\ast}\right)\right\\}=\exp\left(-\exp\left(-x\right)\right),$

which is shown in Csörgő and Horváth (1997).

###### Proof of Theorem 3.5.

Since the distributions of $W_{m,1}\left(\cdot\right)$ and
$W_{m,2}\left(\cdot\right)$ do not depend on $m$, Corollary C.5 entails that,
for all $-\infty<x<\infty$

$P\left\\{\max_{h_{m^{\ast}}\leq k\leq
m^{\ast}}\frac{Z_{m}\left(k\right)}{g_{m,0.5}\left(k\right)}\geq
x\right\\}=P\left\\{\max_{h_{m^{\ast}}\leq k\leq
m^{\ast}}\frac{\left|W_{1}\left(k\right)-\displaystyle\frac{k}{m}W_{2}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{1/2}}\geq
x\right\\}+o\left(1\right),$

as $m^{\ast}\rightarrow\infty$. In turn, this entails that we can construct
critical values $\widehat{c}_{\alpha,0.5}$ based on

$P\left\\{\max_{h_{m^{\ast}}\leq k\leq
m^{\ast}}\frac{\left|W_{1}\left(k\right)-\displaystyle\frac{k}{m}W_{2}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{1/2}}\geq\widehat{c}_{\alpha,0.5}\right\\}=\alpha.$

Using the scale transformation for Wiener processes

$\displaystyle\max_{h_{m^{\ast}}\leq k\leq
m^{\ast}}\frac{\left|W_{1}\left(k\right)-\displaystyle\frac{k}{m}W_{2}\left(m\right)\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{m+k}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\max_{h_{m^{\ast}}/m^{\ast}\leq k/m^{\ast}\leq
1}\frac{\left(1+\displaystyle\frac{k}{m}\right)\left|W\left(\displaystyle\frac{k/m}{1+k/m}\right)\right|}{\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k/m}{1+k/m}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\max_{h_{m^{\ast}}/m^{\ast}\leq\tau\leq
1}\frac{\left|W\left(\displaystyle\frac{\tau}{1+\tau}\right)\right|}{\left(\displaystyle\frac{\tau}{1+\tau}\right)^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$
$\displaystyle\max_{h_{m^{\ast}}/\left(m^{\ast}+h_{m^{\ast}}\right)\leq u\leq
1/2}\frac{\left|W\left(u\right)\right|}{u^{1/2}}\overset{\mathcal{D}}{=}\max_{1\leq
v\leq\phi_{m}}\frac{\left|W\left(v\right)\right|}{v^{1/2}}$
$\displaystyle\overset{\mathcal{D}}{=}$ $\displaystyle\max_{1\leq
v\leq\exp\left(\log\phi_{m}\right)}\frac{\left|W\left(v\right)\right|}{v^{1/2}},$

where $W\left(\cdot\right)$ is a standard Wiener process and
$\phi_{m}=\left(m^{\ast}+h_{m^{\ast}}\right)/\left(2h_{m^{\ast}}\right)$.
Using equation (18) in Vostrikova (1981), we have

$P\left\\{\max_{1\leq
v\leq\exp\left(\log\phi_{m}\right)}\frac{\left|W\left(v\right)\right|}{v^{1/2}}\geq\widehat{c}_{\alpha,0.5}\right\\}\simeq\frac{\widehat{c}_{\alpha,0.5}\exp\left(-\frac{1}{2}\widehat{c}_{\alpha,0.5}^{2}\right)}{\left(2\pi\right)^{1/2}}\left(\log\phi_{m}+\frac{4-\log\phi_{m}}{\widehat{c}_{\alpha,0.5}^{2}}+O\left(\widehat{c}_{\alpha,0.5}^{-4}\right)\right),$

as $\widehat{c}_{\alpha,0.5}\rightarrow\infty$. This proves the claim.

###### Proof of Theorem 3.6.

The proof follows, with minor modifications, the proofs of the results above,
so we only outline its main passages. We begin with part (i), and consider the
case $E\log\left|\beta_{0}+\epsilon_{0,1}\right|<0$. Lemma C.1 entails that we
need to prove only that

(D.23) $\displaystyle\max_{1\leq
k<\infty}\frac{1}{g_{m,\psi}\left(k\right)}\max_{1\leq\ell\leq
k}\left|\sum_{i=m+\ell}^{m+k}\left(\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)\right.$
$\displaystyle\left.-\frac{k-\ell}{m}\sum_{i=2}^{m}\left(\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)\right|$
$\displaystyle\overset{\mathcal{D}}{\rightarrow}\sup_{0<x<\infty}\frac{\sup_{0\leq
t\leq
x}\left|\left(W_{2}\left(x\right)-W_{2}\left(t\right)\right)-\left(x-t\right)W_{1}\left(1\right)\right|}{\left(1+x\right)\left(x/\left(1+x\right)\right)^{\psi}},$

where recall that $\overline{y}_{i}$ is the stationary solution of (2.1). The
approximations in Lemma C.3 imply

$\displaystyle\max_{1\leq
k<\infty}\frac{1}{g_{m,\psi}\left(k\right)}\max_{1\leq\ell\leq
k}\left|\sum_{i=m+\ell}^{m+k}\left(\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)-\frac{k-\ell}{m}\sum_{i=2}^{m}\left(\frac{\epsilon_{i,1}\overline{y}_{i-1}^{2}}{1+\overline{y}_{i-1}^{2}}+\frac{\epsilon_{i,2}\overline{y}_{i-1}}{1+\overline{y}_{i-1}^{2}}\right)\right.$
$\displaystyle\left.-\mathscr{s}\left(W_{m,2}\left(k\right)-W_{m,2}\left(\ell\right)-\frac{k-\ell}{m}W_{m,1}\left(m\right)\right)\right|$
$\displaystyle=$ $\displaystyle o_{P}\left(1\right).$

By the scale transformation and the continuity of the Wiener process we have

$\displaystyle\max_{1\leq
k<\infty}\left(\frac{k+m}{k}\right)^{\psi}\frac{m}{m+k}\max_{1\leq\ell\leq
k}\left|\left(W_{m,2}\left(k\right)-W_{m,2}\left(\ell\right)-\frac{k-\ell}{m}W_{m,1}\left(m\right)\right)\right|$
$\displaystyle\overset{\mathcal{D}}{\rightarrow}\sup_{0<x<\infty}\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq
t\leq
x}\left|\left(W_{2}\left(x\right)-W_{2}\left(t\right)\right)-\left(x-t\right)W_{1}\left(1\right)\right|,$

whence the desired result.

Next we consider the nonstationary case. Applying Lemma C.2(ii), we can
replace (D.23) with the proof of

$\displaystyle\max_{1\leq
k<\infty}\frac{1}{g_{m,\psi}\left(k\right)}\max_{1\leq\ell\leq
k}\left|\sum_{i=m+\ell}^{m+k}\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}-\frac{k-\ell}{m}\sum_{i=2}^{m}\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}\right|$
$\displaystyle\overset{\mathcal{D}}{\rightarrow}\sup_{0<x<\infty}\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq
t\leq
x}\left|\left(W_{2}\left(x\right)-W_{2}\left(t\right)\right)-\left(x-t\right)W_{1}\left(1\right)\right|,$

which follows from the approximations in Lemma C.3(ii).

We conclude our proof by showing that

(D.24)
$P\left\\{\sup_{0<x<\infty}\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq
t\leq
x}\left|\left(W_{2}\left(x\right)-W_{2}\left(t\right)\right)-\left(x-t\right)W_{1}\left(1\right)\right|<\infty\right\\}=1.$

When $0<x\leq 1$, using Theorem 2.1 in Garsia et al. (1970),999See also Lemma
4.1 in Csörgő and Horváth, 1993. it follows that there exists a random
variable $\zeta$ such that $E\left|\zeta\right|^{p}<\infty$ for all $p\geq 1$
and

$\displaystyle\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq t\leq
x}\left|\left(W_{2}\left(x\right)-W_{2}\left(t\right)\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|\zeta\right|\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq
t\leq
x}\left|\left(\left(x-t\right)\log\left(\frac{1}{x-t}\right)\right)^{1/2}\right|$

a.s., whence

(D.25) $\displaystyle\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq
t\leq
x}\left|\left(W_{2}\left(x\right)-W_{2}\left(t\right)\right)-\left(x-t\right)W_{1}\left(1\right)\right|$
$\displaystyle\leq$
$\displaystyle\left|\zeta\right|\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq
t\leq
x}\left|\left(\left(x-t\right)\log\left(\frac{1}{x-t}\right)\right)^{1/2}\right|+W_{1}\left(1\right)\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq
t\leq x}\left|x-t\right|.$

It is now easy to see that

$\left|\zeta\right|\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq
t\leq
x}\left|\left(\left(x-t\right)\log\left(\frac{1}{x-t}\right)\right)^{1/2}\right|=O_{P}\left(1\right).$

As far as the second term in (D.25) is concerned, it immediately follows that

$\left|W_{1}\left(1\right)\right|\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq
t\leq
x}\left|x-t\right|=O_{P}\left(1\right)\left(\frac{1+x}{x}\right)^{\psi-1}=O_{P}\left(1\right).$

Also, by the Law of the Iterated Logarithm, it follows that

$\displaystyle\sup_{1\leq
x<\infty}\left(\frac{1+x}{x}\right)^{\psi}\frac{1}{1+x}\sup_{0\leq t\leq
x}\left|\left(W_{2}\left(x\right)-W_{2}\left(t\right)\right)-\left(x-t\right)W_{1}\left(1\right)\right|$
$\displaystyle\leq$ $\displaystyle\sup_{1\leq x<\infty}\frac{1}{x}2\sup_{0\leq
t\leq
x}\left|W_{2}\left(x\right)\right|+\left|W_{1}\left(1\right)\right|<\infty,$

a.s., whence (D.24) follows.

Finally, as far as parts (ii) and (iii) of the theorem are concerned, the
proofs are based on the observation that

$\max_{1\leq k\leq
m^{\ast}}\frac{1}{g_{m,\psi}^{\ast}\left(k\right)}\max_{1\leq\ell\leq
k}\frac{k-\ell}{m}\left|\sum_{i=2}^{m}\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}+\frac{\epsilon_{i,2}y_{i-1}}{1+y_{i-1}^{2}}\right|=o_{P}\left(1\right),$

and the same using $\overline{g}_{m,\psi}\left(k\right)$; hence, the arguments
above can be repeated with minor modifications.

###### Proof of Corollary 3.1.

The result is shown e.g. in the proof of Theorem 3.4 in Horváth and Trapani
(2016), or Corollary 3.1 in Horváth and Trapani (2023).

In order to prove the next set of results, we will make use of the following
decomposition, valid under (3.10)

(D.26) $\displaystyle Z_{m}\left(k\right)=$
$\displaystyle\left|\sum_{i=m+1}^{m+k^{\ast}}\frac{\epsilon_{i,1}y_{i-1}^{2}+\epsilon_{i,2}y_{i-1}}{1+y_{i-1}^{2}}-\left(\widehat{\beta}_{m}-\beta_{0}\right)\sum_{i=m+1}^{m+k^{\ast}}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right.$
$\displaystyle+\left(\beta_{A}-\beta_{0}\right)\sum_{i=m+k^{\ast}+1}^{m+k}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}+\sum_{i=m+k^{\ast}+1}^{m+k}\frac{\epsilon_{i,1}y_{i-1}^{2}+\epsilon_{i,2}y_{i-1}}{1+y_{i-1}^{2}}$
$\displaystyle-\left.\left(\widehat{\beta}_{m}-\beta_{0}\right)\sum_{i=m+k^{\ast}+1}^{m+k}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right|,$

with the convention that $\sum_{a}^{b}=0$ whenever $b<a$.

###### Proof of Theorem 3.7.

Based on the results above, the theorem follows from standard arguments - see
e.g. Theorem 4.1 in Horváth and Trapani (2023).

###### Proof of Theorem 4.1.

We begin by decomposing the detector $Z_{m}^{X}\left(k\right)$ as

(D.27)
$Z_{m}^{X}\left(k\right)=\left|\left(\mathbf{b}_{0}-\widehat{\mathbf{b}}_{m}\right)^{\intercal}\sum_{i=m+1}^{m+k}\left(y_{i-1},\mathbf{x}_{i}^{\intercal}\right)\frac{y_{i-1}}{1+y_{i-1}^{2}}+\sum_{i=m+1}^{m+k}\frac{\left(\epsilon_{i,1}y_{i-1}+\epsilon_{i,2}\right)y_{i-1}}{1+y_{i-1}^{2}}\right|.$

Henceforth, the proof is very similar to the proof of Theorem 3.1, and we only
report the main passages where the two proofs differ.

Consider first the case $E\log\left|\beta_{0}+\epsilon_{0,1}\right|<0$; using
Lemmas C.6-C.8 instead of Lemmas C.1-C.3, it can be shown along the same lines
as in the proof of Theorem 3.1 that

$\max_{1\leq
k<\infty}\frac{Z_{m}^{X}\left(k\right)}{m^{1/2}\left(1+\displaystyle\frac{k}{\mathscr{s}_{x}^{2}m}\right)\left(\displaystyle\frac{k}{k+\mathscr{s}_{x}^{2}m}\right)^{\psi}}\overset{\mathcal{D}}{\rightarrow}\sup_{0<t<\infty}\frac{\left|-\mathscr{s}_{x,1}tW_{1}\left(1\right)+\mathscr{s}_{x,2}W_{2}\left(t\right)\right|}{\left(1+\displaystyle\frac{t}{\mathscr{s}_{x}^{2}}\right)\left(\displaystyle\frac{t}{t+\mathscr{s}_{x}^{2}}\right)^{\psi}},$

where $\left\\{W_{1}\left(t\right),t\geq 0\right\\}$ and
$\left\\{W_{2}\left(t\right),t\geq 0\right\\}$ are two independent standard
Wiener processes. It is not hard to see that

$\sup_{0<t<\infty}\frac{\left|-\mathscr{s}_{x,1}tW_{1}\left(1\right)+\mathscr{s}_{x,2}W_{2}\left(t\right)\right|}{\left(1+\displaystyle\frac{t}{\mathscr{s}_{x}^{2}}\right)\left(\displaystyle\frac{t}{t+\mathscr{s}_{x}^{2}}\right)^{\psi}}\overset{\mathcal{D}}{=}\sup_{0<u<\infty}\frac{\left|-\mathscr{s}_{x,1}\mathscr{s}_{x}^{2}uW_{1}\left(1\right)+\mathscr{s}_{x,2}W_{2}\left(\mathscr{s}_{x}^{2}u\right)\right|}{\left(1+u\right)\left(\displaystyle\frac{u}{1+u}\right)^{\psi}},$

and that, for $u\leq v$

$\displaystyle
E\left(\left(-\mathscr{s}_{x,1}\mathscr{s}_{x}^{2}uW_{1}\left(1\right)+\mathscr{s}_{x,2}W_{2}\left(\mathscr{s}_{x}^{2}u\right)\right)\left(-\mathscr{s}_{x,1}\mathscr{s}_{x}^{2}vW_{1}\left(1\right)+\mathscr{s}_{x,2}W_{2}\left(\mathscr{s}_{x}^{2}v\right)\right)\right)$
$\displaystyle=$
$\displaystyle\mathscr{s}_{x,1}^{2}\left(\mathscr{s}_{x}^{2}\right)^{2}uv+\mathscr{s}_{x,2}^{2}\mathscr{s}_{x}^{2}u=\frac{\mathscr{s}_{x,2}^{4}}{\mathscr{s}_{x,1}^{2}}\left(uv+u\right).$

Hence it follows that

$\left\\{\frac{-\mathscr{s}_{x,1}\mathscr{s}_{x}^{2}uW_{1}\left(1\right)+\mathscr{s}_{x,2}W_{2}\left(\mathscr{s}_{x}^{2}u\right)}{1+u},u\geq
0\right\\}\overset{\mathcal{D}}{=}\left\\{\frac{\mathscr{s}_{x,2}^{2}}{\mathscr{s}_{x,1}}W\left(u\right),u\geq
0\right\\},$

where $\left\\{W\left(u\right),u\geq 0\right\\}$ is an independent standard
Wiener process. This completes the proof when
$E\log\left|\beta_{0}+\epsilon_{0,1}\right|<0$.

When $E\log\left|\beta_{0}+\epsilon_{0,1}\right|>0$, consider (D.27) again.
Seeing as

(D.28)
$\left\|\sum_{i=m+1}^{\infty}\frac{\mathbf{x}_{i}y_{i-1}}{1+y_{i-1}^{2}}\right\|=O_{P}\left(1\right),$

it is easy to see that

$\displaystyle\left|\left(\mathbf{b}_{0}-\widehat{\mathbf{b}}_{m}\right)^{\intercal}\left(\sum_{i=m+1}^{m+k}\left(y_{i-1},\mathbf{x}_{i}^{\intercal}\right)\frac{y_{i-1}}{1+y_{i-1}^{2}}-\sum_{i=m+1}^{m+k}\left(y_{i-1},\mathbf{0}_{p}^{\intercal}\right)\frac{y_{i-1}}{1+y_{i-1}^{2}}\right)\right|$
$\displaystyle\leq$
$\displaystyle\left\|\mathbf{b}_{0}-\widehat{\mathbf{b}}_{m}\right\|\left\|\sum_{i=m+1}^{m+k}\left(y_{i-1},\mathbf{x}_{i}^{\intercal}\right)\frac{y_{i-1}}{1+y_{i-1}^{2}}-\sum_{i=m+1}^{m+k}\left(y_{i-1},\mathbf{0}_{p}^{\intercal}\right)\frac{y_{i-1}}{1+y_{i-1}^{2}}\right\|$
$\displaystyle=$ $\displaystyle O_{P}\left(m^{-1/2}\right),$

where $\mathbf{0}_{p}$ is a $p$-dimensional vector of zeros. Similarly, using
(D.28) and standard algebra, it follows that

$\left\|\left(\mathbf{b}_{0}-\widehat{\mathbf{b}}_{m}\right)^{\intercal}-\left(\left(\sum_{i=2}^{m}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)^{-1}\left(\sum_{i=2}^{m}\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}\right),\mathbf{0}_{p}^{\intercal}\right)\right\|=O_{P}\left(m^{-1/2}\right),$

and by (C.13)

$\max_{1\leq
k<\infty}\frac{1}{k}\left|\left(\sum_{i=2}^{m}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)^{-1}\left(\sum_{i=2}^{m}\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)\left(\sum_{i=m+1}^{m+k}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}-k\right)\right|=O_{P}\left(m^{-1/2-\zeta}\right),$

for some $\zeta>0$, and similarly to (C.5)-(C.6)

$\left|\left(\sum_{i=2}^{m}\frac{y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)^{-1}\left(\sum_{i=2}^{m}\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}\right)-\frac{1}{m}\sum_{i=2}^{m}\epsilon_{i,1}\right|=O_{P}\left(m^{1/2-\zeta}\right).$

Thus, putting all together

$\max_{1\leq
k<\infty}\frac{1}{k}\left|\left(\mathbf{b}_{0}-\widehat{\mathbf{b}}_{m}\right)^{\intercal}\sum_{i=m+1}^{m+k}\left(y_{i-1},\mathbf{x}_{i}^{\intercal}\right)\frac{y_{i-1}}{1+y_{i-1}^{2}}-\frac{k}{m}\sum_{i=2}^{m}\epsilon_{i,1}\right|=O_{P}\left(m^{-1/2-\zeta}\right).$

Finally, similarly to (C.14) it can be shown

$\max_{1\leq
k<\infty}k^{-1/2-\eta}\left|\sum_{i=m+1}^{m+k}\frac{\epsilon_{i,1}y_{i-1}^{2}}{1+y_{i-1}^{2}}-\sum_{i=m+1}^{m+k}\epsilon_{i,1}\right|=O_{P}\left(m^{-\zeta}\right),$

for some $\zeta>0$ and for all $\eta>0$. Putting all together and recalling
that, when $E\log\left|\beta_{0}+\epsilon_{0,1}\right|>0$,
$\mathscr{s}_{x}^{2}=1$, it follows that

$\max_{1\leq
k<\infty}\frac{\left|Z_{m}^{X}\left(k\right)-\left|\displaystyle\sum_{i=m+1}^{m+k}\epsilon_{i,1}-\displaystyle\frac{k}{m}\displaystyle\sum_{i=2}^{m}\epsilon_{i,1}\right|\right|}{m^{1/2}\left(1+\displaystyle\frac{k}{m}\right)\left(\displaystyle\frac{k}{k+m}\right)^{\psi}}=o_{P}\left(1\right).$

Hereafter, the final result follows by making appeal to the proof of Theorem
3.1.

###### Proof of Theorem 4.2.

The proof follows immediately from the approximations derived above, and from
the proof of Theorem 3.4.

###### Proof of Theorem 4.10.

The proof follows by the same arguments as the proof of Theorem 3.6, and we
therefore omit it to save space.

###### Proof of Theorem 4.4.

The proof follows from the same arguments as in the case with no covariates.