r}|x|\tilde{\mu}_{n}(\\{s\\},dx)\geq\frac{c}{4M}\right\\}\cup\left\\{\sum_{s\leq
t}\int_{|x|\leq
r}|x|\tilde{\mu}(\\{s\\},dx)\geq\frac{c}{4M}\right\\}\right)<\epsilon,$

let $L(r)$, $A_{k}$ and $K$ be as defined earlier (for term four). Since
$\sum_{s\leq t}\left|\Delta(X^{n}-X)_{s}\right|1_{L(r)<|\Delta X^{n}_{s}|\leq
a}$ only contains a finite number of terms and since
$(\Delta(X^{n}-X))^{*}_{t}\xrightarrow{{\mathbb{P}}}0$ it follows that
$\sum_{s\leq t}\left|\Delta(X^{n}-X)_{s}\right|1_{L(r)<|\Delta X^{n}_{s}|\leq
a}\xrightarrow{{\mathbb{P}}}0$ as $n\to\infty$. Let

$\displaystyle E_{6}(n):=\left\\{\sup_{n}\sum_{s\leq t}\int_{|x|\leq
L(r)}|x|\tilde{\mu}_{n}(\\{s\\},dx)<\frac{c}{4M}\right\\}\cap\left\\{\sum_{s\leq
t}\int_{|x|\leq L(r)}|x|\tilde{\mu}(\\{s\\},dx)<\frac{c}{4M}\right\\}$
$\displaystyle\cap\left\\{\sum_{s\leq t}\left|1_{|\Delta X^{n}_{s}|\geq
L(r)}-1_{|\Delta X_{s}|\geq L(r)}\right|=0\right\\}\cap
A_{R}(n)\cap\left\\{(\Delta(X^{n}-X))^{*}_{t}<\frac{c}{4MK}\right\\}\cap\left(\bigcup_{k=1}^{K}A_{k}\right),$

then for large $n$ we can make $E_{6}(n)$ smaller than say $6\epsilon$. Now
for some $\theta(s,\omega)\in[0,1]$ we have on $E_{6}(n)$

$\displaystyle\left(\sum_{s\leq t}\left(\int_{|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right)^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\sum_{s\leq t}\left|\int_{|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right|$
$\displaystyle\leq$ $\displaystyle\sum_{s\leq t}\left|\int_{|x|\leq
L(r)}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right|$
$\displaystyle+$ $\displaystyle\sum_{s\leq t}\left|\int_{L(r)<|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right|\leq$
$\displaystyle 2M\sum_{s\leq t}\int_{|x|\leq
L(r)}|x|(\mu+\mu_{n})(\\{s\\},dx)+\sum_{s\leq
t,}\left|\left(\left(f_{n}(X^{n}_{s-}+\Delta
X_{s})-f_{n}(X^{n}_{s-}))\right)-\Delta
X_{s}f_{n}^{\prime}(X^{n}_{s-})\right)1_{L(r)<|\Delta X_{s}|\leq
a,s\in\mathcal{A}}\right.$
$\displaystyle\left.-\left(\left(f_{n}(X^{n}_{s-}+\Delta
X^{n}_{s})-f_{n}(X^{n}_{s-}))\right)-\Delta
X^{n}_{s}f_{n}^{\prime}(X^{n}_{s-})\right)1_{L(r)<|\Delta X^{n}_{s}|\leq
a,s\in\mathcal{A}_{n}}\right|$ $\displaystyle\leq$
$\displaystyle\frac{c}{2}+\sum_{s\leq t}\left|f_{n}(X^{n}_{s-}+\Delta
X_{s})-f_{n}(X^{n}_{s-}+\Delta
X^{n}_{s})-\Delta(X^{n}-X)_{s}f_{n}^{\prime}(X^{n}_{s-})\right|1_{L(r)<|\Delta
X_{s}|\leq a}=$ $\displaystyle\frac{c}{2}+\sum_{s\leq
t}\left|\left(f_{n}^{\prime}(X^{n}_{s-}+\Delta
X_{s}+\theta\Delta(X^{n}-X)_{s})-f_{n}^{\prime}(X^{n}_{s-})\right)\Delta(X^{n}-X)_{s}\right|1_{L(r)<|\Delta
X_{s}|\leq a}\leq$ $\displaystyle\frac{c}{2}+2M\sum_{s\leq
t}\left|\Delta(X^{n}-X)_{s}\right|1_{L(r)<|\Delta X_{s}|\leq
a}<\frac{c}{2}+2MK\frac{c}{4MK}=c,$

where we used the fact that $1_{L(r)<|\Delta X^{n}_{s}|\leq a}=1_{L(r)<|\Delta
X_{s}|\leq a}$ for $s\leq t$ on $E_{6}(n)$. We have now established that term
six does go to zero in probability. Assuming a)

$\displaystyle\left(\sum_{s\leq t}\left(\int_{|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right)^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle 2U\sum_{s\leq t}\int_{|x|\leq
L(r)}|x|(\mu+\mu_{n})(\\{s\\},dx),$

which is u.i. since $\sum_{s\leq t}\int_{|x|\leq
L(r)}|x|\mu_{n}(\\{s\\},dx)\xrightarrow{L^{1}}\sum_{s\leq t}\int_{|x|\leq
L(r)}|x|\mu(\\{s\\},dx)$. Assuming b)

$\displaystyle{\mathbb{E}}\left[\left(\sum_{s\leq t}\left(\int_{|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right)^{2}\right)^{\frac{1}{2}}1_{E}\right]$
$\displaystyle\leq$
$\displaystyle{\mathbb{E}}\left[2C\left(2|x|+2(X^{n})^{*}_{t}+2X^{*}_{t}\right)\sum_{s\leq
t}\int_{|x|\leq L(r)}|x|(\mu+\mu_{n})(\\{s\\},dx)1_{E}\right]$
$\displaystyle\leq$ $\displaystyle
2C\sqrt{{\mathbb{E}}\left[\left(2|x|+2(X^{n})^{*}_{t}+2X^{*}_{t}\right)^{2}1_{E}\right]}\sqrt{{\mathbb{E}}\left[\left(\sum_{s\leq
t}\int_{|x|\leq L(r)}|x|(\mu+\mu_{n})(\\{s\\},dx)\right)^{2}1_{E}\right]}.$

We now outline the proof in the case a) with $p=1$ and b) with $q=2$. In place
of the inequality (A.1.2) we instead estimate $[f_{n}(X^{n})-f(X)]_{t}$ by
using Lemma 2.3. Note that by doing so we get the constant $2^{6}$ in front of
each term (but this is obviously will not affect the $L^{1}$ convergence). Now
that we do not take the square root of each term this has the only effect that
it doubles the moment requirements and this will be the only difference in the
proof for this case. ∎

###  A.1.3 Proof of Theorem 3.15

###### Proof.

First we note that for any stopping time $T\leq t$,
$|\Delta(X^{n}-X)_{T}|\leq\sqrt{[X^{n}-X]_{T}}$ so
$X^{n}_{s-}\xrightarrow{a.s.}X_{s-}$. If we let
$B_{R}=\left\\{\sup_{n}(X^{n})^{*}_{t}\wedge X^{*}_{t}\leq R\right\\}$ then
${\mathbb{P}}\left(\cup_{R}B_{R}\geq
1\right)=\lim_{R\to\infty}{\mathbb{P}}(B_{R})=1$ so we may assume that
$X,\Delta X$, $\\{X^{n}\\}_{n}$ and $\\{\Delta X^{n}\\}_{n}$ are all uniformly
bounded by the constant $R$. We will let
$M=\sup_{n}\sup_{x\in[-R-a,R+a]}|f_{n}^{\prime}(x)|$. Recall from the proof of
Theorem 3.14 that

$\displaystyle[f_{n}(X^{n})-f(X)]^{\frac{1}{2}}_{t}\leq\left(\int_{0^{+}}^{t}f^{\prime}(X_{-})^{2}d\left[Z-Z^{n}\right]_{s}\right)^{\frac{1}{2}}+\left(\int_{0^{+}}^{t}\left(f_{n}^{\prime}(X^{n}_{s_{-}})-f^{\prime}(X_{s_{-}})\right)^{2}d[Z^{n}]_{s}\right)^{\frac{1}{2}}+$
$\displaystyle\left(\int_{0}^{t}\int_{|x|\leq
a}\left((f_{n}(X^{n}_{s-}+x)-f(X_{s-}+x)+f_{n}(X^{n}_{s-}))-f(X_{s-}))-x(f_{n}^{\prime}(X^{n}_{s-})-f^{\prime}(X_{s-}))\right)^{2}\tilde{\mu}(ds,dx)\right)^{\frac{1}{2}}$
$\displaystyle+\left[\int_{0}^{.}\int_{|x|\leq
a}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)((\tilde{\mu}-\nu_{c})(ds,dx)\right.$
$\displaystyle\left.\left.-\int_{0}^{.}\int_{|x|\leq
a}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)(\tilde{\mu}_{n}-(\nu_{n})_{c})(ds,dx)\right]_{t}^{\frac{1}{2}}\right.$
$\displaystyle+\left(\sum_{s\leq t}\left(\int_{|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f(X_{s-}+x)-f_{n}(X^{n}_{s-})+f(X_{s-})-x(f_{n}^{\prime}(X^{n}_{s-})-f^{\prime}(X_{s-}))\right)\mu(\\{s\\},dx)\right)^{2}\right)^{\frac{1}{2}}$
$\displaystyle+\left(\sum_{s\leq t}\left(\int_{|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right)^{2}\right)^{\frac{1}{2}}$
$\displaystyle+\left[\sum_{s\leq
t}\left(f_{n}(X^{n}_{s})-f_{n}(X^{n}_{s-})-\Delta
X^{n}_{s}f_{n}^{\prime}(X^{n}_{s-})\right)I_{|\Delta X^{n}_{s}|>a}-\sum_{s\leq
t}\left(f(X_{s})-f(X_{s-})-\Delta X_{s}f^{\prime}(X_{s-})\right)I_{|\Delta
X_{s}|>a}\right]_{t}^{\frac{1}{2}},$ (A.1.28)

so it will suffice to show that each term tends to zero a.s..If we choose
$a>R$ large then term seven in (A.1.3) is identically zero on $B_{R}$, so will
assume that term seven is identically zero.
For the first term in (A.1.3)

$\left(\int_{0^{+}}^{t}f^{\prime}(X_{-})^{2}d\left[Z-Z^{n}\right]_{s}\right)^{\frac{1}{2}}\leq
M[Z^{n}-Z]_{t}^{\frac{1}{2}}=M[X^{n}-X]_{t}^{\frac{1}{2}},$

which converges to zero a.s. by assumption.

For the second term in (A.1.3)

$\int_{0^{+}}^{t}\left(f_{n}^{\prime}(X^{n}_{s_{-}})-f^{\prime}(X_{s_{-}})\right)^{2}d[Z^{n}]_{s},$

note that since $[Z^{n}]_{s}\to[Z]_{s}$ a.s. and
$\left(f_{n}^{\prime}(X^{n}_{s_{-}})-f^{\prime}(X_{s_{-}})\right)^{2}\to 0$
a.s. for all $s\leq t$, as $n\to\infty$ it follows from the varying measure
variant of the dominated convergence theorem (see e.g. Serfozo (1982)),
applied pathwise a.s. that this term will vanish a.s..

For the third term

$\displaystyle\int_{0}^{t}\int_{|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f(X_{s-}+x)-f_{n}(X^{n}_{s-})+f(X_{s-})-x(f_{n}^{\prime}(X^{n}_{s-})-f^{\prime}(X_{s-}))\right)^{2}\tilde{\mu}(ds,dx)$
(A.1.29)

as in the proof of 3.14 we may dominate the integrand by

$8x^{2}\left(\left|f_{n}^{\prime}(X^{n}_{s-}+\theta_{1}x)\right|^{2}+\left|f^{\prime}(X_{s-}+\theta_{1}x)\right|^{2}+|f^{\prime}(X_{s-})|^{2}+|f_{n}^{\prime}(X^{n}_{s-})|^{2}\right),$

for some $\theta_{1}(s,\omega)$ and this can further be dominated by
$32M^{2}x^{2}$, which is $\tilde{\mu}$-integrable and since the integrand
converges pointwise to zero, (A.1.29) will vanish.

For term four we note that for any $r>0$ we may pick $L(r)\in[r/2,r]$ such
that ${\mathbb{P}}\left(\not\exists s\leq t:|\Delta X_{s}|=L(r)\right)=1$ and
it follows also that if we let $A_{\delta}(r)=\left\\{\not\exists s\leq
t:|\Delta X_{s}|\in[L(r)-\delta,L(r)+\delta]\right\\}$ then $\lim_{\delta\to
0^{+}}{\mathbb{P}}(A_{\delta}(r))=1$ for all $r>0$. On $A_{\delta}(r)$, if for
some stopping time $T\leq t$, $|\Delta X_{T}|<L(r)$ then $|\Delta
X_{T}|<L(r)-\delta$ and similarly $|\Delta X_{T}|>L(r)$ implies $|\Delta
X_{T}|>L(r)+\delta$. Since $|\Delta(X^{n}-X)_{T}|\xrightarrow{a.s.}0$ for any
stopping time $T\leq t$ it follows that if $T$ and $S$ are such that $|\Delta
X_{T}|\geq L(r)$ and $|\Delta X_{S}|<L(r)$ there exists some
$N_{r,\delta}(\omega)\in{\mathbb{N}}$ such that $n\geq N_{r,\delta}(\omega)$
implies that $|\Delta X^{n}_{T}|\geq L(r)$ and $|\Delta X^{n}_{S}|<L(r)$. On
$A_{\delta}(r)$,

$\displaystyle\sum_{s\leq t}(\Delta X^{n}_{s})^{2}I_{|\Delta X^{n}_{s}|\leq
L(r)}=\sum_{s\leq t}(\Delta(X^{n}-X)_{s}+\Delta X^{n}_{s})^{2}I_{|\Delta
X^{n}_{s}|\leq L(r)}$ $\displaystyle\leq$ $\displaystyle 2\sum_{s\leq
t}(\Delta(X^{n}-X)_{s})^{2}+2\sum_{s\leq t}(\Delta X_{s})^{2}I_{|\Delta
X^{n}_{s}|\leq L(r)}$ $\displaystyle\leq$ $\displaystyle 2\sum_{s\leq
t}(\Delta(X^{n}-X)_{s})^{2}+2\sum_{s\leq t}(\Delta X_{s})^{2}I_{|\Delta
X_{s}|\leq L(r)}\leq 2[X^{n}-X]_{t}+2\sum_{s\leq t}(\Delta
X_{s})^{2}I_{|\Delta X_{s}|\leq L(r)}.$ (A.1.30)

Recall that

$\displaystyle\left[\int_{0}^{.}\int_{|x|\leq
a}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)((\tilde{\mu}-(\nu)_{c})(ds,dx)\right.$
$\displaystyle\left.-\int_{0}^{.}\int_{|x|\leq
a}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)(\tilde{\mu}_{n}-(\nu_{n})_{c})(ds,dx)\right]_{t}^{\frac{1}{2}}$
$\displaystyle\leq\left[\int_{0}^{.}\int_{|x|\leq
L(r)}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)((\tilde{\mu}-\nu_{c})(ds,dx)\right]_{t}^{\frac{1}{2}}$
$\displaystyle+\left[\int_{0}^{.}\int_{|x|\leq
L(r)}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)(\tilde{\mu}_{n}-(\nu_{n})_{c})(ds,dx)\right]_{t}^{\frac{1}{2}}$
$\displaystyle+\left[\int_{0}^{.}\int_{L(r)<|x|\leq
a}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)(\tilde{\mu}-\tilde{\mu}_{n})(ds,dx)\right]_{t}^{\frac{1}{2}}$
$\displaystyle+\left[\int_{0}^{.}\int_{L(r)<|x|\leq
a}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)(\nu-\nu_{n})_{c}(ds,dx)\right]_{t}^{\frac{1}{2}},$
(A.1.31)

where the final term is zero since the process inside the quadratic variation
is continuous with finite variation. For the first term on the right-hand side
of (A.1.3) we have

$\displaystyle\left[\int_{0}^{.}\int_{|x|\leq
L(r)}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)((\tilde{\mu}-\nu_{c})(ds,dx)\right]_{t}^{\frac{1}{2}}=$
$\displaystyle\left(\int_{0}^{t}\int_{|x|\leq
L(r)}\left(\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-}))\right)-xf_{n}^{\prime}(X^{n}_{s-})\right)^{2}\tilde{\mu}(ds,dx)\right)^{\frac{1}{2}}\leq$
$\displaystyle\left(\int_{0}^{t}\int_{|x|\leq
L(r)}x^{2}\left(f_{n}^{\prime}(X^{n}_{s-}+\theta
x)-f_{n}^{\prime}(X^{n}_{s-})\right)^{2}\tilde{\mu}(ds,dx)\right)^{\frac{1}{2}}\leq
2M\left(\sum_{s\leq t}(\Delta X_{s})^{2}I_{|\Delta X_{s}|\leq
L(r)}\right)^{\frac{1}{2}}$

which can be arbitrarily small by choosing $r$ small enough.

The second term of (A.1.3) equals

$\displaystyle\left(\left[\int_{0}^{.}\int_{|x|\leq
L(r)}(((f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})))-xf_{n}^{\prime}(X^{n}_{s-}))(\tilde{\mu}_{n}-(\nu_{n})_{c})(ds,dx)\right]_{t}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\left(\int_{0}^{.}\int_{|x|\leq
L(r)}x^{2}\left(f_{n}^{\prime}(X^{n}_{s-}+\theta
x)-f_{n}^{\prime}(X^{n}_{s-})\right)^{2}\tilde{\mu}_{n}(ds,dx)\right)^{\frac{1}{2}}\leq
2M\left(\sum_{s\leq t}(\Delta X^{n}_{s})^{2}1_{|\Delta X^{n}_{s}|\leq
L(r)}\right)^{\frac{1}{2}}.$

which we know tends to zero uniformly over $n$ as $r\to 0^{+}$. Next, for the
third term in the right-hand side of (A.1.3)

$\displaystyle\left[\sum_{s\leq.}\left(f_{n}(X^{n}_{s-}+\Delta
X_{s})-f_{n}(X^{n}_{s-}+\Delta
X^{n}_{s})-\Delta(X^{n}-X)_{s}f_{n}^{\prime}(X^{n}_{s-})\right)1_{L(r)<|\Delta
X_{s}|\leq a}\right]_{t}$ $\displaystyle\leq\sum_{s\leq
t}\left(f_{n}(X^{n}_{s-}+\Delta X_{s})-f_{n}(X^{n}_{s-}+\Delta
X^{n}_{s})-\Delta(X^{n}-X)_{s}f_{n}^{\prime}(X^{n}_{s-})\right)^{2}1_{L(r)<|\Delta
X_{s}|\leq a},$

which contains finitely many terms, each of which converges to zero a.s..

For term five of (A.1.3) we recall

$\displaystyle\left(\sum_{s\leq t}\left(\int_{a<|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f(X_{s-}+x)-f_{n}(X^{n}_{s-})+f(X_{s-})-x(f_{n}^{\prime}(X^{n}_{s-})-f^{\prime}(X_{s-}))\right)\mu(\\{s\\},dx)\right)^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\sum_{s\leq t}\int_{|x|\leq
a}\left|f_{n}(X^{n}_{s-}+x)-f(X_{s-}+x)-f_{n}(X^{n}_{s-})+f(X_{s-})-x(f_{n}^{\prime}(X^{n}_{s-})-f^{\prime}(X_{s-}))\right|\mu(\\{s\\},dx).$
(A.1.32)

where we may argue as earlier and use dominated convergence applied to the
measure $\sum_{s\leq.}\mu(\\{s\\},dx)$ to show that this term converges to
zero.

For term six of (A.1.3), we first fix $r>0$ so small that $\sum_{s\leq
t,s\in\mathcal{A}}|\Delta X_{s}|1_{|\Delta X_{s}|>L(r)}$ is small enough for
our purposes and then fix some small $\delta>0$. Note that on $A_{\delta}(r)$
then $n\geq N_{r,\delta}(\omega)$ implies

$\left\\{s\in\mathcal{A}_{n}:|\Delta
X^{n}_{s}|>L(r),\right\\}=\left\\{s\in\mathcal{A}:|\Delta
X_{s}|>L(r)\right\\}.$

Therefore if $n\geq N_{r,\delta}(\omega)$,

$\sum_{s\leq t,s\in\mathcal{A}_{n}}|\Delta X^{n}_{s}|1_{|\Delta
X^{n}_{s}|>L(r)}=\sum_{s\leq t,s\in\mathcal{A}}|\Delta X^{n}_{s}|1_{|\Delta
X_{s}|>L(r)}\xrightarrow{a.s.}\sum_{s\leq t,s\in\mathcal{A}}|\Delta
X_{s}|1_{|\Delta X_{s}|>L(r)}$

and since $\sum_{s\leq t,s\in\mathcal{A}}|\Delta
X^{n}_{s}|\xrightarrow{a.s.}\sum_{s\leq t,s\in\mathcal{A}}|\Delta X_{s}|$,
this implies (for $n\geq N_{r,\delta}(\omega)$)

$\sum_{s\leq t,s\in\mathcal{A}_{n}}|\Delta X^{n}_{s}|1_{|\Delta X^{n}_{s}|\leq
L(r)}=\sum_{s\leq t,s\in\mathcal{A}}|\Delta X^{n}_{s}|1_{|\Delta X_{s}|\leq
L(r)}\xrightarrow{a.s.}\sum_{s\leq t,s\in\mathcal{A}}|\Delta X_{s}|1_{|\Delta
X_{s}|\leq L(r)}.$

This shows that $\sum_{s\leq t,s\in\mathcal{A}_{n}}|\Delta
X^{n}_{s}|1_{|\Delta X^{n}_{s}|\leq L(r)}$ can be made uniformly small in $n$
for small enough $r$.

$\displaystyle\left(\sum_{s\leq t}\left(\int_{|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right)^{2}\right)^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle\sum_{s\leq t}\left|\int_{|x|\leq
L(r)}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right|$
$\displaystyle+$ $\displaystyle\sum_{s\leq t}\left|\int_{L(r)<|x|\leq
a}\left(f_{n}(X^{n}_{s-}+x)-f_{n}(X^{n}_{s-})-xf_{n}^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right|\leq$
$\displaystyle 2M\sum_{s\leq t}\int_{|x|\leq
L(r)}|x|(\mu+\mu_{n})(\\{s\\},dx)+\sum_{s\leq
t,}\left|\left(\left(f_{n}(X^{n}_{s-}+\Delta
X_{s})-f_{n}(X^{n}_{s-}))\right)-\Delta
X_{s}f_{n}^{\prime}(X^{n}_{s-})\right)1_{L(r)<|\Delta X_{s}|\leq
a,s\in\mathcal{A}}\right.$
$\displaystyle\left.-\left(\left(f_{n}(X^{n}_{s-}+\Delta
X^{n}_{s})-f_{n}(X^{n}_{s-}))\right)-\Delta
X^{n}_{s}f_{n}^{\prime}(X^{n}_{s-})\right)1_{L(r)<|\Delta X^{n}_{s}|\leq
a,s\in\mathcal{A}_{n}}\right|$

the first term is uniformly small in $n$ as $r\to 0^{+}$, the second term
equals for $n\geq N_{r,\delta}(\omega)$ on $A_{\delta}(r)$

$\sum_{s\leq t,L(r)<|\Delta X_{s}|\leq
a,s\in\mathcal{A}}\left|\left(\left(f_{n}(X^{n}_{s-}+\Delta
X_{s})-f_{n}(X^{n}_{s-}))\right)-(\Delta X_{s}-\Delta
X^{n}_{s})f_{n}^{\prime}(X^{n}_{s-})-\left(f_{n}(X^{n}_{s-}+\Delta
X^{n}_{s})-f_{n}(X^{n}_{s-}))\right)\right)\right|$

which converges pathwise a.s. to zero. ∎

###  A.1.4 Proof of Theorem 3.16

###### Proof.

As in the proof of Theorem 3.15 we may assume that $X,\Delta X$,
$\\{X^{n}\\}_{n}$ and $\\{\Delta X^{n}\\}_{n}$ are all uniformly bounded by
some constant $R$. We will let $M=\sup_{x\in[-R-a,R+a]}|f^{\prime}(x)|$, which
must be finite since $f$ is locally Lipschitz. Similarly to the proof of
Theorem 3.14 the following inequality holds

$\displaystyle[f(X^{n})-f(X)]^{\frac{1}{2}}_{t}\leq\left(\int_{0^{+}}^{t}f^{\prime}(X^{n}_{-})^{2}d\left[Z-Z^{n}\right]_{s}\right)^{\frac{1}{2}}+\left(\int_{0^{+}}^{t}\left(f^{\prime}(X^{n}_{s_{-}})-f^{\prime}(X_{s_{-}})\right)^{2}d[Z]_{s}\right)^{\frac{1}{2}}+$
$\displaystyle\left(\int_{0}^{t}\int_{|x|\leq
a}\left((f(X^{n}_{s-}+x)-f(X_{s-}+x)+f(X^{n}_{s-}))-f(X_{s-}))-x(f^{\prime}(X^{n}_{s-})-f^{\prime}(X_{s-}))\right)^{2}\tilde{\mu}(ds,dx)\right)^{\frac{1}{2}}$
$\displaystyle+\left[\int_{0}^{.}\int_{|x|\leq
a}\left(\left(f(X^{n}_{s-}+x)-f(X^{n}_{s-}))\right)-xf^{\prime}(X^{n}_{s-})\right)((\tilde{\mu}-\nu_{c})(ds,dx)\right.$
$\displaystyle\left.\left.-\int_{0}^{.}\int_{|x|\leq
a}\left(\left(f(X^{n}_{s-}+x)-f(X^{n}_{s-}))\right)-xf^{\prime}(X^{n}_{s-})\right)(\tilde{\mu}_{n}-(\nu_{n})_{c})(ds,dx)\right]_{t}^{\frac{1}{2}}\right.$
$\displaystyle+\left(\sum_{s\leq t}\left(\int_{|x|\leq
a}\left(f(X^{n}_{s-}+x)-f(X_{s-}+x)-f(X^{n}_{s-})+f(X_{s-})-x(f^{\prime}(X^{n}_{s-})-f^{\prime}(X_{s-}))\right)\mu(\\{s\\},dx)\right)^{2}\right)^{\frac{1}{2}}$
$\displaystyle+\left(\sum_{s\leq t}\left(\int_{|x|\leq
a}\left(f(X^{n}_{s-}+x)-f(X^{n}_{s-})-xf^{\prime}(X_{s-})\right)(\mu-\mu_{n})(\\{s\\},dx)\right)^{2}\right)^{\frac{1}{2}}$
$\displaystyle+\left[\sum_{s\leq t}\left(f(X^{n}_{s})-f(X^{n}_{s-})-\Delta
X^{n}_{s}f^{\prime}(X^{n}_{s-})\right)I_{|\Delta X^{n}_{s}|>a}-\sum_{s\leq
t}\left(f(X_{s})-f(X_{s-})-\Delta X_{s}f^{\prime}(X_{s-})\right)I_{|\Delta
X_{s}|>a}\right]_{t}^{\frac{1}{2}},$ (A.1.33)

so it will suffice to show that each term tends to zero a.s.. Similarly to the
proof of Theorem 3.15 be choosing $a$ large we can assume that term seven in
(A.1.4) is identically zero on $B_{R}$. The first term in (A.1.4) can be dealt
with exactly like in the proof of Theorem 3.15 since we do not use any
continuity properties for $f^{\prime}$ here. Let $y\in{\mathbb{R}}$ then

$\displaystyle\int_{0}^{t}1_{\\{s:X_{s}=y\\}}d[X]^{c}_{s}=0.$

to see this note that if we consider the constant process $Y_{s}=y$ then,
since $X$ and $Y$ agree on the set $\\{s:X_{s}=y\\}$, so does $d[X]^{c}$ and
$d[Y]$, therefore

$\displaystyle\int_{0}^{t}1_{\\{s:X_{s}=y\\}}d[X]^{c}_{s}=\int_{0}^{t}1_{\\{s:X_{s}=y\\}}d[Y]_{s}\leq\int_{0}^{t}d[Y]_{s}=0.$

If we denote $\mathsf{disc}(f^{\prime})$ as the set of discontinuity points
for $f^{\prime}$ then, since $\mathsf{disc}(f^{\prime})$ is at most countable,
it follows from monotone convergence that

$\displaystyle\int_{0}^{t}1_{\\{s:X_{s}\in\mathsf{disc}(f^{\prime})\\}}d[X]^{c}_{s}=0.$

The above argument also trivially implies that $f$ in conjunction with $X$
(and $X^{n}$, for each $n$ with exactly the same type of argument) fulfils
Assumption 3.1.

Let $\\{T_{k}\\}_{k\geq 0}$ be the jump times for $X$ in $[0,t]$ (with the
convention that $T_{0}=0$). $X$ has no jumps in $[T_{k-1},T_{k})$ so it is
continuous and nowhere constant in this interval. Since
$X_{T_{k}-}=\left(X_{T_{k}-}-X_{T_{k-1}}\right)+X_{T_{k-1}}$ and
$X_{T_{k}-}-X_{T_{k-1}}$ is independent from $X_{T_{k-1}}$, it is continuously
distributed i.e., ${\mathbb{P}}\left(X_{T_{k}-}=y\right)=0$ for any
$y\in{\mathbb{R}}$. Therefore
${\mathbb{P}}\left(X_{T_{k}-}\in\mathsf{disc}(f^{\prime})\right)=0$. On the
other hand we have that since $X_{T_{k}}=X_{T_{k}-}+\Delta X_{T_{k}}$ and,
again, due to the independent increment property and the fact that
$X_{T_{k}-}$ does not charge the singletons,
${\mathbb{P}}\left(X_{T_{k}}\in\mathsf{disc}(f^{\prime})\right)=0$. In other
words the points $\\{X_{T_{k}-}\\}_{k\geq 0}$ as well as
$\\{X_{T_{k}}\\}_{k\geq 0}$ are a.s. continuity points for $f^{\prime}$. We
now re-write the expression inside the square root of the second term in
(A.1.4) as

$\displaystyle\int_{0^{+}}^{t}\left(f^{\prime}(X^{n}_{s_{-}})-f^{\prime}(X_{s_{-}})\right)^{2}d[Z]_{s}=\int_{0^{+}}^{t}\left(f^{\prime}(X^{n}_{s_{-}})-f^{\prime}(X_{s_{-}})\right)^{2}d[Z]^{c}_{s}+\sum_{k>0}(\Delta
X_{T_{k}})^{2}\left(f^{\prime}(X^{n}_{T_{k}})-f^{\prime}(X_{T_{k}})\right)^{2}$
$\displaystyle=\int_{0^{+}}^{t}\left(f^{\prime}(X^{n}_{s_{-}})-f^{\prime}(X_{s_{-}})\right)^{2}1_{\\{s:X_{s}\not\in\mathsf{disc}(f^{\prime})\\}}d[Z]^{c}_{s}+\sum_{k>0}(\Delta
X_{T_{k}})^{2}\left(f^{\prime}(X^{n}_{T_{k}})-f^{\prime}(X_{T_{k}})\right)^{2}$
(A.1.34)

The first term will vanish by the fact that on
$\\{s:X_{s}\not\in\mathsf{disc}(f^{\prime})\\}$,
$X^{n}_{s-}\xrightarrow{a.s.}X_{s-}$ implies
$f^{\prime}(X^{n}_{s-})\xrightarrow{a.s.}f^{\prime}(X_{s-})$ and we may
therefore apply dominated convergence as in the proof of Theorem 3.15. For the
second term we first note that the terms inside the sum all converge to zero
since $X_{T_{k}}$ is a.s. a continuity point for $f^{\prime}$. Also, $(\Delta
X_{T_{k}})^{2}\left(f^{\prime}(X^{n}_{T_{k}})-f^{\prime}(X_{T_{k}})\right)^{2}\leq
4M^{2}(\Delta X_{T_{k}})^{2}$ and thus

$\sum_{k>0}(\Delta
X_{T_{k}})^{2}\left(f^{\prime}(X^{n}_{T_{k}})-f^{\prime}(X_{T_{k}})\right)^{2}\leq
4M^{2}\sum_{k>0}(\Delta X_{T_{k}})^{2}\leq 4M^{2}[X]_{t}.$

It now follows from the dominated convergence theorem, applied pathwise to the
counting measure, that the sum in (A.1.4) vanishes.
For the third term we make the following estimate

$\displaystyle\int_{0}^{t}\int_{|x|\leq
a}\left(\Big{(}f(X^{n}_{s-}+x)-f(X_{s-}+x)+f(X^{n}_{s-}))-f(X_{s-})\Big{)}-x(f^{\prime}(X^{n}_{s-})-f^{\prime}(X_{s-}))\right)^{2}\tilde{\mu}(ds,dx)\leq$
$\displaystyle 2\int_{0}^{t}\int_{|x|\leq
a}\left(f(X^{n}_{s-}+x)-f(X_{s-}+x)+f(X^{n}_{s-}))-f(X_{s-})\right)^{2}\tilde{\mu}(ds,dx)+$
$\displaystyle 2\int_{0}^{t}\int_{|x|\leq
a}x^{2}\left(f^{\prime}(X^{n}_{s-})-f^{\prime}(X_{s-})\right)^{2}\tilde{\mu}(ds,dx).$

By continuity and dominated convergence similar to before the first term on
the right-hand side above is readily seen to vanish. For the second term we
note that $f^{\prime}(X_{s-})$ is only charged (since $\tilde{\mu}\leq mu$,
only charges a subset of $\\{(T_{k},\Delta X_{T_{k}})\\}_{k}$) at the points
$s\in\\{T_{k}\\}_{k}$ which we know are continuity points of $f^{\prime}$ a.s.
and therefore since $X^{n}_{s-}\xrightarrow{a.s.}X_{s-}$ it follows that the
integrand converges to zero under the measure $\tilde{\mu}$ a.s. and we can
then apply dominated convergence once again.
Term four of (A.1.4) can be dealt exactly as in the proof of Theorem 3.15
which does not use any continuity properties of $f^{\prime}$. Term five of
(A.1.4) is dealt with similarly to term three (compare with the proof of
Theorem 3.15). The sixth and final (recall that we eliminated term seven by
choice of $a$) can again be dealt with just as the same term in the proof of
Theorem 3.15 and this finishes the proof. ∎

###  A.1.5 Proof of Proposition 4.4

###### Proof.

The equivalence of 1) and 2) follows trivially from the hypothesis and the
linearity of ucp convergence. We will now establish that 2) and 3) are
equivalent. Let $\epsilon>0$, $c>0$ be arbitrary. Let $N$ denote the number of
jumps of $X$ with modulus larger or equal to $a$ on $[0,t]$ and denote
$A_{k}=\\{N=k\\}$ for $k\geq 0$. By the cadlag property of $X$ and continuity
of probability,
$1={\mathbb{P}}\left(\bigcup_{k=0}^{\infty}A_{k}\right)=\lim_{K\to\infty}{\mathbb{P}}\left(\bigcup_{k=0}^{K}A_{k}\right)$.
We may now choose $K_{0}$ such that
${\mathbb{P}}\left(\bigcup_{k=0}^{K_{0}}A_{k}\right)>1-\epsilon$. Moreover if
we let

$A_{t}(a,\delta)=\left\\{\exists s\leq t:|\Delta
X_{s}|\in[a,a+\delta)\right\\},$

then $\bigcap_{n\geq 1}A_{t}(a,\frac{1}{n})=\left\\{\exists s\leq t:|\Delta
X_{s}|=a\right\\}$ this event has probability zero for $a\in\mathbb{A}_{X}$
and so by continuity of probability we see that we may choose $\delta>0$ such
that for a given $a\not\in\mathbb{A}_{X}$,
${\mathbb{P}}\left(A_{t}(a,\delta)\right)<\epsilon/K_{0}$. Take such a
$\delta$ and assume we are given $a\not\in\mathbb{A}_{X}$. Define analogously
to $J_{u}$, $J^{n}_{u}=\sum_{s\leq u}\Delta X^{n}_{s}$. We will now show that
$X^{n}\xrightarrow{ucp}X$ implies $J^{n}(a)\xrightarrow{ucp}J(a)$ when
$a\not\in\mathbb{A}_{X}$. Let $N_{n}$ denote the number of jumps with modulus
larger or equal to $a$ for $X^{n}$ and denote $\\{T_{l}^{n}\\}_{l=1}^{N_{n}}$
the jump times of $X^{n}(a)$, let $\\{T_{l}\\}_{l=1}^{k}$ denote the jump
times of $X(a)$ on $A_{k}$. $X^{n}\xrightarrow{ucp}X$ implies that there
exists $n_{1}$ such that if $n\geq n_{1}$ then
${\mathbb{P}}\left(A_{k}\setminus\left\\{(X^{n}-X)^{*}_{t}\geq\min\left(\frac{c}{2K_{0}},\delta\right)\right\\}\right)<\epsilon/K_{0}$.
Let

$B_{(k,n)}=A_{k}\cap\left\\{(X^{n}-X)^{*}_{t}<\min\left(\frac{c}{2K_{0}},\delta\right)\right\\}\cap\left\\{|\Delta
X_{s}|\not\in[a,a+\delta]\quad\forall s\leq t\right\\},$

then ${\mathbb{P}}\left(A_{k}\setminus
B_{(k,n)}\right)<\frac{2\epsilon}{K_{0}}$. It follows that if $X^{n}$ has a
jump of size greater or equal to $a$ in $[0,t]$ on $B_{(k,n)}$ then so must
$X$ and vice versa since otherwise $|\Delta(X^{n}-X)|\geq\delta$ contradicting
the definition of $B_{(k,n)}$. i.e. the jump times of $J^{n}(a)$ and $J(a)$
coincide on $B_{(k,n)}$. Therefore we may write

$J(a)_{t}-J^{n}(a)_{t}=\sum_{l=1}^{k}\Delta(X^{n}-X)_{T_{l}},$

on $B_{(k,n)}$. We have for $1\leq k\leq K_{0}$ on $B_{(k,n)}$

$\displaystyle(J(a)-J^{n}(a))^{*}_{t}\leq\sum_{l=1}^{k}|\Delta(X^{n}-X)|_{T_{i}}\leq\sum_{l=1}^{k}\left((X^{n}-X)^{*}_{T_{i}}+(X^{n}-X)^{*}_{T_{i}-}\right)$
$\displaystyle\leq 2K_{0}(X^{n}-X)^{*}_{t}<c,$

and obviously for $k=0$, $J(a)-J^{n}(a)=0$. So

${\mathbb{P}}\left(A_{k}\cap\left\\{(J(a)-J^{n}(a))^{*}_{t}\geq
c\right\\}\right)\leq{\mathbb{P}}\left(B_{(k,n)}^{c}\cap
A_{k}\cap\left\\{(J(a)-J^{n}(a))^{*}_{t}\geq
c\right\\}\right)<\frac{\epsilon}{K_{0}}.$

This leads to

$\displaystyle{\mathbb{P}}\left((J(a)-J^{n}(a))^{*}_{t}\geq
c\right)\leq\sum_{k=1}^{K_{0}}{\mathbb{P}}\left(A_{k}\cap\left\\{(J(a)-J^{n}(a))^{*}_{t}\geq
c\right\\}\right)+{\mathbb{P}}\left(\bigcup_{k>K_{0}}A_{k}\right)$
$\displaystyle\leq K_{0}\frac{\epsilon}{K_{0}}+\epsilon=2\epsilon.$

This shows that if $a\not\in\mathbb{A}_{X}$ then
$J^{n}(a)\xrightarrow{ucp}J(a)$ and if we combine this with 2) (which is
equivalent to 1)) we see that this implies $X^{n}(a)\xrightarrow{ucp}X(a)$ and
it also trivially implies $\lim_{a\to
0^{+}}\lim_{n\to\infty}{\mathbb{P}}\left((J^{n}(a)-J(a))^{*}_{t}\geq
c\right)=0$ for all $c>0$. Assume that 3) is true. Take any $\epsilon>0$ and
take $a>0$ so small that
$\lim_{n\to\infty}{\mathbb{P}}\left(\left(\sum_{s\leq.}\Delta
X^{n}_{s}I_{|\Delta X^{n}_{s}|\leq a}\right)^{*}_{t}\geq c\right)<\epsilon$ as
well as ${\mathbb{P}}\left(\left(\sum_{s\leq.}\Delta X_{s}I_{|\Delta
X_{s}|\leq a}\right)^{*}_{t}\geq c\right)<\epsilon$, which we may since $J$ is
of finite variation and by 3). Therefore if $a\not\in\mathbb{A}_{X}$,

$\displaystyle\lim_{n\to\infty}{\mathbb{P}}\left((J-J^{n})^{*}_{t}\geq
3c\right)\leq\lim_{n\to\infty}{\mathbb{P}}\left((J-J(a))^{*}_{t}\geq
c\right)+\lim_{n\to\infty}{\mathbb{P}}\left((J(a)-J^{n}(a))^{*}_{t}\geq
c\right)+\lim_{n\to\infty}{\mathbb{P}}\left((J^{n}(a)-J^{n})^{*}_{t}\geq
c\right)$
$\displaystyle=\lim_{n\to\infty}{\mathbb{P}}\left(\left(\sum_{s\leq.}\Delta
X_{s}I_{|\Delta X_{s}|\leq a}\right)^{*}_{t}\geq
c\right)+0+\lim_{n\to\infty}{\mathbb{P}}\left(\left(\sum_{s\leq.}\Delta
X^{n}_{s}I_{|\Delta X^{n}_{s}|\leq a}\right)^{*}_{t}\geq c\right)<2\epsilon,$

letting $\epsilon\to 0^{+}$ shows that 2). If we instead assume 2) is true
then we have that if $a\not\in\mathbb{A}_{X}$

$\displaystyle\lim_{n\to\infty}{\mathbb{P}}\left(((J^{n}-J^{n}(a))-(J-J(a)))^{*}_{t}\geq
c\right)\leq\lim_{n\to\infty}{\mathbb{P}}\left((J^{n}-J)^{*}_{t}\geq
c\right)+$
$\displaystyle\lim_{n\to\infty}{\mathbb{P}}\left((J^{n}(a)-J(a))^{*}_{t}\geq
c\right)=\lim_{n\to\infty}{\mathbb{P}}\left((J^{n}(a)-J(a))^{*}_{t}\geq
c\right).$

Letting $a\to 0^{+}$ (outside $\mathbb{A}_{X}$) eliminates the remaining term
i.e.

$\lim_{a\to
0^{+},a\not\in\mathbb{A}_{X}}\lim_{n\to\infty}{\mathbb{P}}\left(\left(\sum_{s\leq.}\Delta
X^{n}_{s}I_{|\Delta X^{n}_{s}|\leq a}-\sum_{s\leq.}\Delta X_{s}I_{|\Delta
X_{s}|\leq a}\right)^{*}_{t}\geq c\right)=0,$

which implies

$\displaystyle\lim_{a\to
0^{+},a\not\in\mathbb{A}_{X}}\lim_{n\to\infty}{\mathbb{P}}\left(\left(\sum_{s\leq.}\Delta
X^{n}_{s}I_{|\Delta X^{n}_{s}|\leq a}\right)^{*}_{t}\geq
c\right)\leq\lim_{a\to
0^{+},a\not\in\mathbb{A}_{X}}\lim_{n\to\infty}{\mathbb{P}}\left(\left(\sum_{s\leq.}\Delta
X_{s}I_{|\Delta X_{s}|\leq a}\right)^{*}_{t}\geq c\right)$
$\displaystyle+\lim_{a\to
0^{+},a\not\in\mathbb{A}_{X}}\lim_{n\to\infty}{\mathbb{P}}\left(\left(\sum_{s\leq.}\Delta
X^{n}_{s}I_{|\Delta X^{n}_{s}|\leq a}-\sum_{s\leq.}\Delta X_{s}I_{|\Delta
X_{s}|\leq a}\right)^{*}_{t}\geq c\right)=0,$

since $X$ has jumps of finite variation. This shows 3). Suppose now that 4) is
true. Take $a^{\prime}\not\in\mathbb{A}_{X}$ so small that $a\leq a^{\prime}$
implies $\lim_{n\to\infty}{\mathbb{P}}\left(\sum_{s\leq t}|\Delta
X^{n}_{s}|I_{|\Delta X^{n}_{s}|\leq a}\geq c\right)<\epsilon$ as well as
$\lim_{n\to\infty}{\mathbb{P}}\left(\sum_{s\leq t}|\Delta X_{s}|I_{|\Delta
X_{s}|\leq a}\geq c\right)<\epsilon$ then

$\displaystyle{\mathbb{P}}\left((J^{n}(a)-J(a))^{*}_{t}\geq
c\right)\leq{\mathbb{P}}\left((J^{n}(a)-J^{n}(a^{\prime}))^{*}_{t}\geq
c\right)+{\mathbb{P}}\left((J^{n}(a^{\prime})-J(a^{\prime}))^{*}_{t}\geq
c\right)+$ $\displaystyle{\mathbb{P}}\left((J(a^{\prime})-J(a))^{*}_{t}\geq
c\right)\leq{\mathbb{P}}\left(\sum_{s\leq t}|\Delta X^{n}_{s}|I_{|\Delta
X^{n}_{s}|\leq a^{\prime}}\geq
c\right)+{\mathbb{P}}\left((J^{n}(a^{\prime})-J(a^{\prime}))^{*}_{t}\geq
c\right)+$ $\displaystyle{\mathbb{P}}\left(\sum_{s\leq t}|\Delta
X_{s}|I_{|\Delta X_{s}|\leq a^{\prime}}\geq c\right)\leq
2\epsilon+{\mathbb{P}}\left((J^{n}(a^{\prime})-J(a^{\prime}))^{*}_{t}\geq
c\right),$

since $a^{\prime}\not\in\mathbb{A}_{X}$ the second term goes to zero as
$n\to\infty$. This shows that
$\lim_{a\to\infty}\lim_{n\to\infty}{\mathbb{P}}\left((J^{n}(a)-J(a))^{*}_{t}\geq
c\right)=0$ but since 4) is stronger than 3) it implies 2) and combined with
this we have the final statement. ∎

## References

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  * Jacod and Shiryaev (2003) Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes. Springer-Verlag.
  * Liptser and Shiryaev (1986) Liptser, R. and Shiryaev, A. (1986) Theory of Martingales. Springer-Verlag.
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# Interplanetary Dust Emission as a Foreground for the _LiteBIRD_ CMB
Satellite Mission

K. Ganga  Université de Paris, CNRS, Astroparticule et Cosmologie, F-75013
Paris, France  M. Maris  INAF/Trieste Astronomical Observatory, Via G.B.
Tiepolo 11, ZIP I-34143 Trieste, Italy  Institute for Fundamental Physics of
the Universe, Via Beirut, 2–4 I–34151, Grignano, Trieste, Italy  M.
Remazeilles  Instituto de Fisica de Cantabria (CSIC-Universidad de Cantabria),
Avda. de los Castros s/n, E-39005 Santander, Spain  Jodrell Bank Centre for
Astrophysics, Department of Physics and Astronomy, The University of
Manchester, Manchester, M13 9PL, United Kingdom

for the _LiteBIRD_ collaboration

###### Abstract

As ever-more sensitive experiments are made in the quest for primordial CMB
$B$ Modes, the number of potentially significant astrophysical contaminants
becomes larger as well. Thermal emission from interplanetary dust, for
example, has been detected by the _Planck_ satellite. While the polarization
fraction of this Zodiacal, or interplanetary dust emission (IPDE) is expected
to be low, it is bright enough to be detected in total power. Here, estimates
of the magnitude of the effect as it might be seen by the _LiteBIRD_ satellite
are made. The _COBE_ IPDE model from [Kelsall et al. (1998)] is combined with
a model of the _LiteBIRD_ experiment’s scanning strategy to estimate potential
contamination of the CMB in both total power and in polarization power
spectra. _LiteBIRD_ should detect IPDE in temperature across all of its bands,
from 40 through 402 GHz, and should improve limits on the polarization
fraction of IPDE at the higher end of this frequency range. If the
polarization fraction of IPDE is of order 1%, the current limit from _ISO_
/CAM measurements in the mid-infrared, it may induce large-scale polarization
$B$ Modes comparable to cosmological models with an $r$ of order $10^{-3}$. In
this case, the polarized IPDE would also need to be modeled and removed. As a
CMB foreground, IPDE will always be subdominant to Galactic emissions, though
because it caused by emission from grains closer to us, it appears variable as
the Earth travels around the Sun, and may thereby complicate the data analysis
somewhat. But with an understanding of some of the symmetries of the emission
and some flexibility in the data processing, it should not be the primary
impediment to the CMB polarization measurement.

###### keywords:

Zodiacal dust - (Cosmology:) cosmic background radiation

##  1 Introduction

Measurements of the Cosmic Microwave Background, or CMB, are often cited as
helping transform cosmology into a precision science. Most of these
transformative measurements have been of the non-polarized brightness of the
CMB. The CMB has, in addition, a polarized component, measurements of which
are beginning to allow us to probe even more deeply into the history and
constituents of the Universe. This polarized signal, however, is fainter than
the non-polarized part. While building increasingly sensitive instruments to
detect this weaker signal, we will inevitably run into dim, new sources of
contamination which we could previously ignore, but which will not be
negligible in the future.

Zodiacal light, or Sunlight scattered from dust particles in our Solar System,
is often visible with the naked eye. These same interplanetary dust particles
also absorb some fraction of the incoming Sunlight, thereby rise in
temperature, and then radiate energy at longer wavelengths. With a spectral
maximum near 20 $\mu$m, this interplanetary dust emission (hereafter IPDE) is
best observed in the infrared. It began to be characterized in the 20th
century, and has now been imaged by a number of experiments, including being
mapped over the full sky by a few satellite missions. These latter include
_IRAS_ (Neugebauer et al., 1984), which mapped it at 12, 25, 60, and 100
$\mu$m, and _COBE_ /DIRBE, with a wider frequency coverage and better large-
scale stability but poorer angular resolution (Kelsall et al., 1998). More
recently, _WISE_ (Wright et al., 2010), _Astro-H_ (Pyo et al., 2010), and
_Planck_ (Planck Collaboration et al., 2011) have also detected IPDE (Planck
Collaboration et al., 2014) in full-sky maps. A recent review of observations
can be found in Lasue et al. (2020).

Though issued from larger grains, in many respects IPDE is similar to thermal
emission from grains in our Galaxy – it is, essentially, thermal emission from
dust particles heated by starlight. Given this, it should not be surprising
that the frequency dependence of the IPDE brightness is similar to that of
Galactic dust. Figure 1 shows the range of brightnesses, from _Planck_ ,
expected from both Galactic dust and from IPDE for wavelengths of interest
here.

Figure 1: Brightness spectra of IPDE, Galactic dust emission, and CMB
components. The total, monopole brightness of the CMB is shown in yellow. The
CMB dipole anisotropy is indicated in light green, while a rough idea of the
primordial CMB temperature anisotropies are shown in dark green. IPDE and
Galactic emissions from Planck are shown in red and blue, respectively. The
IPDE estimates cover brightnesses from the Ecliptic plane (the upper part of
the shaded red region) to the Ecliptic poles (the lower part of the shaded red
region) and are based on the _COBE_ /DIRBE model (Kelsall et al., 1998), with
extension to CMB frequencies based on fits to Planck data (Planck
Collaboration et al., 2014), as described in Section 2.2. Essentially, for
each frequency, the IPDE at the ecliptic equator and at the ecliptic pole are
calculated with the model, and the values between are filled in red. Galactic
emission is represented by an ensemble of light blue lines, each depicting the
spectrum of a sky location from the Beyond Planck project’s dust model
(https://beyondplanck.science/products/files/). To avoid saturating the
graphic, pixel spectra are shown after degrading the maps at each frequency to
HEALPix (Górski et al., 2005) resolution 64, with high transparency.

IPDE, however, is fainter than Galactic emission in total intensity, is in a
different magnetic environment, and is significantly less polarized than
Galactic dust emission. Though polarization has been detected in scattered
Zodiacal light (see, for example, Pitz et al., 1979; Weinberg & Hahn, 1980;
Leinert et al., 1982; Berriman et al., 1994), and though there are mechanisms
which might polarize the thermal emission (Onaka, 2000; Hoang & Lazarian,
2014, for example), none has been detected to this point. Siebenmorgen (1999)
assumed that IPDE was unpolarized in the mid-infrared to set limits of $1\pm
0.3$ % on instrumental polarization in the _ISO_ /CAM instrument. Reversing
this argument, one can infer that the mid-infrared polarization fraction of
IPDE is at most roughly one percent.

So while it will always be subdominant to Galactic Dust (Ade et al., 2018),
since _Planck_ has already detected IPDE at CMB wavelengths, _LiteBIRD_ should
detect it as well, with higher significance. While IPDE did not significantly
affect the _Planck_ results, its existence did complicated some aspects of the
_Planck_ /HFI analyses. We must recognize that IPDE models are still very
approximate, especially at long wavelengths (Dikarev & Schwarz, 2015), and
that the polarization properties of IPDE are still poorly constrained. But the
ambitious _LiteBIRD_ goals demand that we check that IPDE will not compromise
the measurements.

One difficult point in particular is that interplanetary dust is, relatively
speaking, close to the Earth. This complicates analyses, because as the Earth
orbits around the Sun, the column depth of interplanetary dust through which
one observes in a given direction on the distant sky changes depending upon
the time of year. As an extreme example, one can imagine observing some
distant object which happens to be at an angle close to $180^{\circ}$ from the
Sun (i.e., observing “away” from the Sun). If one observes this same point on
the sky six months later, the Solar angle will be small (i.e., we’ll be
observing “near” the Sun). The column depth of interplanetary dust through
which the observation is done will be larger in the latter case. Additionally,
the observation nearer to the Sun will be done though dust which includes some
grains that are closer to the Sun and thus hotter than in the first case. When
this is combined with changing polarization sensitivity angles, false
polarization signals might be inferred. In our toy model, if the first
observation were done with a detector sensitive to linear polarization
perpendicular to the plane of the ecliptic and the second with a detector
sensitive to linear polarization parallel to the plane of the ecliptic, one
would incorrectly infer polarization parallel to the plane of the ecliptic.

To investigate, this work is laid out as follows: the simulations that are
used to estimate the size of the effects are described in Section 2 and the
resultant maps and power spectra from these simulations are shown in Section
3. These are discussed in Section 4 and the paper concludes in Section 5.

##  2 Simulations

The simulated observations are based on the _LiteBIRD_ mission observation
strategy described in Hazumi et al. (2020), but comprise only a single
detector at each of 402, 119 and 40 GHz. The IPDE model used is the so-called
_COBE_ /DIRBE model (Kelsall et al., 1998), extended to the longer wavelengths
applicable to the CMB using measurements from _Planck_ (Planck Collaboration
et al., 2014). These elements are described below.

###  2.1 _Observation Strategy_

_LiteBIRD_ is a CMB mission which will observe in 15 bands between 40 and 402
GHz. It will observe from orbit around the second Earth-Sun Lagrange point,
which is always about 1% further from the Sun than the Earth, along the Sun-
Earth vector. The satellite will “spin” around an axis 50 degrees from the
focal plane “boresight” every 20 minutes, and this spin-axis itself will
precess at an angle of 45 degrees about the Sun-Earth vector with a period of
3.2058 hours (Hazumi et al., 2020). The simulation also includes the
modulation from a half-wave plate assumed to rotate at a rate of 46, 39, and
61 revolutions per minute at 40, 119111 _LiteBIRD_ ’s 119-GHz channel actually
comprises detectors from both its Low-Frequency Telescope (LFT) and its Mid-
Frequency Telescope (MFT). These two telescopes each have half-wave plates
which rotate at different frequencies; 46 rpm for the LFT and 39 rpm for the
MFT. This subtlety does not make a significant difference to the qualitative
results presented here, so is ignored., and 402 GHz, respectively.

As a first step, Figure 2 shows, for each pixel on the sky in ecliptic
coordinates, the average over all _LiteBIRD_ samples of the angle between the
Sun and the location of the pixel as it is observed from the spacecraft.

Figure 2: Average angle, in degrees, between a given observation direction and
the Sun, as seen from the spacecraft, for three years’ observations. The
minimum average value is about 87 degrees, the median and mean of the map are
both about 117 degrees, and the maximum is about 147 degrees. The grey lines
mark Galactic latitudes from -20 to 20 degrees.

For comparison, we note that _LiteBIRD_ ’s scanning strategy is better-adapted
to minimize the effects of IPDE than that of _Planck_. The density of
interplanetary dust is greater closer to the Sun than away from it, and
greater towards the ecliptic plane in general than further from it (see, for
example, Figure 4 of Kelsall et al., 1998). One therefore expects larger IPDE
signals as as one observes lines-of-sights at angles closer to the Sun. In the
ecliptic plane, where IPDE will be greatest, _LiteBIRD_ observes at larger
elongation angles than did _Planck_ , which observed the sky at angle between
about 87 and 103∘ from the Sun. That is, the observation strategy used here
looks through lower column depths of interplanetary dust than did _Planck_ ,
on average. As shown in Figure 2, in the ecliptic plane, _LiteBIRD_ observes
at an average angle of about $60^{\circ}$ from the anti-Solar direction (i.e.,
about $120^{\circ}$ from the Sun), while the equivalent angle for _Planck_ was
about $85^{\circ}$ (i.e., only about $95^{\circ}$ from the Sun), though with
smaller variations about this average.

_LiteBIRD_ will ultimately carry over 4500 detectors across its 15 frequency
channels. Using the emission model described below requires integrating over a
number of points along each line of sight for each sample of each detector in
the simulation. This represents a significant computing undertaking and is
beyond the scope of this initial estimation. Therefore, in order to reduce the
required computing resources, this work uses the frequencies 40, 119, and 402
GHz as representative bands. These are, respectively, _LiteBIRD_ ’s longest
wavelength band, most sensitive band to CMB, and shortest wavelength band (see
Table 3 of Hazumi et al., 2020). At each of these frequencies, a single
detector is simulated for the entire three-year mission. This will obviously
gloss over some of the details of how the data from different detectors are
combined, but this should not affect the rather broad-brush conclusions in
this work.

###  2.2 IPDE Model

The IPDE model used in this work was developed by the _COBE_ team in Kelsall
et al. (1998), which is in turn an extension of work done for _IRAS_ (Hauser
et al., 1985). The model supposes a Diffuse Cloud, three Infrared Dust Bands,
and a Circumsolar Ring and “Trailing Blob”. While Kelsall et al. (1998) used
DIRBE data and were thus limited to wavelengths shorter than 240 GHz, Fixsen &
Dwek (2002) used FIRAS data to extend large-scale observations to longer
wavelengths and show that the emissivity of the Diffuse Cloud falls off as
$\lambda^{-2}$ above about 150 $\mu$m.

With better angular resolution than FIRAS, Planck Collaboration et al. (2014)
was able to separate the smaller-scale features from the larger-scale Diffuse
Cloud and show that the Infrared Dust Bands appear to emit more efficiently at
longer wavelengths than does the Diffuse Could. We will see that the Infrared
Dust Bands may dominate the IPDE at the longer _LiteBIRD_ wavelengths.

The Ring and Blob were not well-detected by Planck, and are therefore
neglected here. This is done of necessity, since the long extrapolations done
here become untenable with very uncertain parameters. The Ring and Blob,
however, are points to keep in mind for _LiteBIRD_ in particular, since its
observation strategy includes observations with smaller Solar angles than
DIRBE and _Planck_ , and it may therefore observe these two structures more
directly than either DIRBE or _Planck_.

While more modern models can now be made (see, for example, Dikarev et al.,
2019, and references therein), this parameterized _COBE_ model should be
adequate for giving us an initial understanding of the order of magnitude of
the IPDE, and is in a form which is convenient to work with for full-sky CMB
maps. In particular, it has also been calibrated at CMB wavelengths by
_Planck_ (Planck Collaboration et al., 2014). But once again we underline the
approximate nature of the estimates here.

While the basic elements and geometry of the Kelsall et al. (1998) model are
used here, DIRBE did not observe wavelengths longer than 240 $\mu$m. Fixsen &
Dwek (2002) added FIRAS data to extend _COBE_ ’s emissivity measurement of the
Diffuse Cloud beyond 240 $\mu$m, but FIRAS’s $7^{\circ}$ beam prevented them
from discussing the Bands, Ring and Blob. In order to make predictions for the
longer wavelengths that _LiteBIRD_ will observe, therefore, linear fits to the
data from Table 3 of Planck Collaboration et al. (2016a) are done in frequency
and are interpolated and extrapolated to 402, 119 and 40 GHz to use for the
emissivities of the various IPDE components at _LiteBIRD_ ’s longer
wavelengths. The extrapolation of the Diffuse Cloud emissivity to 40 GHz,
_LiteBIRD_ ’s longest wavelength, is negative. In this special case, the value
is set to zero. Other extrapolation schemes can be used, but these makes no
appreciable difference to the results, essentially because the Bands, which
_Planck_ measured to have much flatter emissivities than the Diffuse Cloud,
are the predominate IPDE feature when this model is extrapolated to 40 GHz.

Table 1: 40, 119, and 402-GHz emissivities for the Diffuse Cloud and the three
IRAS Dust Bands used for the similations.

Component | 40 GHz | 119 GHz | 402 GHz
---|---|---|---
Cloud | –0.008 | 0.018 | 0.110
Band 1 | 1.040 | 1.142 | 1.508
Band 2 | 0.029 | 0.109 | 0.394
Band 3 | 0.694 | 0.959 | 1.907

Emissivity values as interpolated and extrapolated from fits to _Planck_ /HFI
data (Planck Collaboration et al., 2014). In principle, these unitless values
should be between 0 and 1. The slightly negative value at 40 GHz for the
Diffuse Cloud is assumedly the result of noise. The values greater than unity
arise from the complete degeneracy between the emissivity and the density of
particles used in Kelsall et al. (1998). One could repeat the work in Planck
Collaboration et al. (2014) with larger densities of particles to ensure these
values remain between 0 and 1, but for this work it was more straightforward
to simply keep the Kelsall et al. (1998) densities and live with the
inconsistency.

##  3 Results

With the prescription above, unpolarized time-lines of IPDE are simulated for
a single, fiducial detector at the center of the focal plane using equation 1
of Kelsall et al. (1998) to integrate along each line-of-sight for each
_LiteBIRD_ sample taken. These are projected into sky maps using the
elementary procedure used for making maps summarized in Section A. This is
repeated for each of 402, 119, and 40 GHz. The resultant simulated IPDE maps
in temperature obtained for three years of observations are shown in Figure 3.

Figure 3: Estimated 402 (top), 119 (middle), and 40-GHz (bottom) temperature
(Stokes I) IPDE contributions from a single detector for a mission lasting
three years. The maps are in ecliptic coordinates, with the basic structure of
the Galactic plane indicated by the series of grey lines, spanning Galactic
latitudes from -20 to 20 degrees.

While the model emission is assumed to be completely unpolarized, the
variability of the column depth of interplanetary dust through which
_LiteBIRD_ observes could mimic a polarized signal. Imagine a given point on
the distant celestial sphere observed once with a given satellite and half-
wave plate geometry, and then again, at a later time (usually around six
months), with a different set of orientations and angles. Because the column
depth of interplanetary dust through which the satellite observes may have
changed, the IPDE signal may be different for the two observations, and so a
naive observer might infer that the sky is polarized, even when it is not. The
apparent change in column depth when observing a given sky direction is
significant over timescales of weeks to months (as the satellite orbits the
Sun), so the polarization modulation from the fast-spinning half-wave plate
helps reduce these effects significantly in Q and U maps. Figure 4 shows plots
similar to 3, but for the Stokes Q values.

Figure 4: Estimated 402 (top), 119 (middle), and 40-GHz (bottom) Stokes Q IPDE
contributions from a single detector for a mission lasting one Solar year. The
maps are in ecliptic coordinates, with the basic structure of the Galactic
plane indicated by the series of grey lines, spanning Galactic latitudes from
-20 to 20 degrees. It should be emphasized that the noise seen here is not due
to intrinsic polarization in the IPDE, which is assumed to be zero for these
images. It is the result of multiple observations of the distant sky through
differing interplanetary dust column depths, the variations in which are
ultimately translated as small amounts of polarization by the map-making,
which cannot account for this variability.

The Stokes U maps are qualitatively similar to those of Q in Figure 4. In
these Figures, shown in ecliptic coordinates, one sees the IPDE signal
predominantly towards the ecliptic plane, with the thinner signals from the
so-called IRAS Bands (Sykes & Greenberg, 1986) becoming progressively more
prominent as one moves from higher to lower frequencies. This is a consequence
of the emissivity of the Diffuse Cloud falling more rapidly than that of the
IRAS Bands as one observes at longer wavelength (see Table 1), and is
presumably because the Dust Bands are the result of recent asteroid collisions
(Sykes & Greenberg, 1986) and thus have larger grains than the rest of the
cloud, having not yet been ground down with time as has the rest of the
Zodiacal Cloud.

The TT, EE and BB power spectra for the IPDE maps in Figures 3 and 4 (plus the
analogous Stokes U maps) are shown in Figure 5, along with models of expected
primordial CMB TT, $E$-Mode, and $B$-Mode power spectra for $r$=0.1, 0.01 and
0.001. These spectra are made simply using HEALPix tools (Górski et al., 2005)
without making any sky cuts, since most Galactic cuts don’t align with the
ecliptic plane and therefore don’t significantly affect the IPDE signal.

Figure 5: Single detector, noise-free, 402 (top), 119 (middle) and 40-GHz
(bottom) temperature (grey triangles), $E$-Mode (blue squares) and $B$-Mode
(red circles) power spectra of the maps shown in Figures 3 and 4 (plus the
analogous U maps). The grey, blue and red curves show the corresponding TT, EE
and primordial $B$-mode spectra from gravitational waves with $r=0.1,\ 0.01\
\&\ 0.001$, respectively, in progressively darker red, using parameters from
Planck Collaboration et al. (2016b). The shaded regions towards the left of
each solid curve indicate cosmic variance, the thin black lines represent an
estimate of the expected _LiteBIRD_ polarization noise levels (Hazumi et al.,
2020), and the violet line represents the spectrum of lensing. The map spectra
are made using HEALPix tools (Górski et al., 2005).

Figure 5 also shows the expected _polarization_ noise from Table 3 of Hazumi
et al. (2020) for each channel as a thin, black line. Note that these curves
are for entire _LiteBIRD_ channels, after co-addition of all the detectors at
the given channel frequency. There may be some subtleties in how the noise in
the actual instrument will integrate down. But, since these plots span many
orders of magnitude, and since instrumental noise in temperature maps should
be the same order of magnitude as instrumental noise in the polarization maps
in general, in the order-of-magnitude spirit of these calculations, this one
black curve will represent the noise, to be compared with both temperature and
polarization power spectra.

##  4 Discussion

Below, the implications for temperature and polarization measurements are
discussed independently.

###  4.1 Temperature

Figure 5 shows that the IPDE signal is generally smaller than that of the CMB
temperature anisotropies across the entire range of _LiteBIRD_ frequencies,
with the contamination only approaching the level of the CMB near the 402-GHz
channel, which will be used as a foreground monitor in any event. In the
119-GHz channel, _LiteBIRD_ ’s most sensitive to CMB anisotropies, the IPDE is
many orders of magnitude below the TT spectrum, and is still lower at 40 GHz,
the long-wavelength end of _LiteBIRD_ ’s frequency coverage. If one remembers
that Galactic foregrounds will need to be accounted for as well (see Figure 1,
for example), it is evident that the IPDE is not the primary foreground
obstacle for _LiteBIRD_. One might have guessed this, given the results of
_Planck_ and _WMAP_.

This does not mean, however, that _LiteBIRD_ will not be able to detect the
IPDE and that it need not be addressed. Table 2 of Hazumi et al. (2019) quotes
noise levels of $47.45,4.58,\ \mathrm{and}\ 37.42\,\mu$K$\cdot$arc-minute and
beam full-width-at-half-maximums of 17.9, 30222Note that this is a rough
average of the resolutions of the two different sets of detectors that
contribute to the 119 GHz channel – the Low-Frequency Telescope has a full-
with-at-half-maximum of 26.3 arc-minutes at 119 GHz, while the Mid-Frequency
Telescope’s FWHM is 33.6 arc-minutes., and 70.5 arc-minutes for _LiteBIRD_ ’s
402, 119, and 40-GHz channels, respectively. The thin, black lines in Figure 5
shows the expected sensitivity limits due to instrumental noise for each of
our reference channels. When compared with the power spectrum of IPDE in
Figure 5, it appears that, at least from the perspective of raw sensitivity,
_LiteBIRD_ will measure the total power IPDE over its entire frequency range.
Even at 40 GHz, _LiteBIRD_ ’s longest wavelength and least IPDE-sensitive
channel, a number of even-multipole TT power spectrum points, representing the
large-angular IPDE structure, are above the _LiteBIRD_ noise floor. This
indicates that the large-scale structure of IPDE should be detectable, if this
imperfect model and its extrapolation in frequency are close to reality.

To make this detection, one will have to remove the primary CMB anisotropies
themselves, as well as other more prominent foreground emissions such as
Galactic dust and synchrotron emission, as they will be larger than the IPDE
signature. But this can effectively be done by making judicious differences of
subsets of the data. For example, _LiteBIRD_ essentially makes two full
surveys of the sky per year. For most locations on the celestial sphere,
different column depths of interplanetary dust are seen during successive
surveys, even when observing the same direction on the distant sky. This means
that differences can be made of maps made of each half-year of data. In these
differences, the distant Galactic and cosmological signals will be removed,
but a significant fraction of the IPDE will remain in the difference, and can
therefore be modeled. This was suggested for _Planck_ in Maris et al. (2006)
and was one of the methods used by the Planck team (Planck Collaboration et
al., 2014) to address the effects of IPDE in temperature. It is illustrated
for _LiteBIRD_ in Figure 6.

Figure 6: An illustration of map differences that can be used to detect and
remove IPDE. The top map is made from six months of simulated 119-GHz
_LiteBIRD_ IPDE for data taken between July and December. The middle map is
made from six subsequent month of data. The bottom map shows the difference of
these first two maps. These maps include no noise or other systematics the
experiment will also see – only the IPDE and IPDE differences as seen between
successive surveys. Depending upon which frequencies and other detector
subsets used to make such differences, these estimated IPDE differences will
be hidden somewhat by the associated experimental noise.

While it will be difficult to see IPDE in the full temperature maps, once
alternating half-year maps are differenced, the IPDE will be clear and still
larger than the instrumental noise. This is illustrated in Figure 7, where the
power spectra of both the single-survey maps _and_ their difference are shown
to be significantly larger than the expected noise at 119 GHz over a wide
range of angular scales of interest.

Figure 7: Power spectra of 119-GHz, single-survey IPDE simulations and
differences of single-survey IPDE simulations. The power spectrum of a
temperature map made from an IPDE simulation of six months of data created
with observations beginning July 1, 2029 (i.e., the data shown in the top
panel of Figure 6) is shown with grey, left-pointing triangles. The power
spectrum of a temperature map made from an IPDE simulation of six months of
data created with observations beginning December 30, 2029 is shown with grey,
right-pointing triangles. The difference is show with the green points. The
black line again represents the experimental noise level expected from Hazumi
et al. (2020).

It can be modeled or otherwise characterized in order to return to the full
maps and remove it before subsequent analyses are done. If necessary, this
will need to be done before Galactic component separation and removal, since
IPDE will resemble Galactic Dust emission in many ways.

####  4.1.1 Difficulties

This apparent variability of the IPDE (which, it should be emphasized, is a
result in changes in lines-of-sight through the interplanetary dust cloud, not
intrinsic variations in the astrophysical source) may also create data
reduction confusion. As an example, it would be nice to use the half-year maps
in Figure 6 to try to track systematics such as calibration drifts on six-
month time scales. IPDE will, however, complicate this. In Table 2 the result
of fits of an offset and a dipole to the first two maps in Figure 6, along
with the combination of those data (which is quite similar to the middle panel
of Figure 3), are shown.

Table 2: Dipole fits to IPDE simulations presented in Figure 6. $\nu$ | Start | Dur. | Offset | Amp. | Lon. | Lat.
---|---|---|---|---|---|---
GHz333The frequency of the simulation in GHz. | Month444The date of the start of the simulation. More precisely, those simulations labeled ‘July’ began on July 1, 2029, while those labeled ‘Dec.’ started December 30, 2029. | Years555The length of the simulation in years |  $\mu$K666The amplitude of the fitted IPDE offset in $\mu$K. |  $\mu$K777The amplitude of the fitted IPDE dipole in $\mu$K. | deg.888The ecliptic longitude of the fitted IPDE dipole in degrees. | deg.999The ecliptic latitude of the fitted IPDE dipole in degrees.
402 | July | 0.5 | 75.0 | 3.757 | -4.5 | -87.9
402 | Dec. | 0.5 | 74.7 | 4.081 | 35.5 | 87.7
402 | July | 1.0 | 72.8 | 0.141 | 59.5 | -69.1
119 | July | 0.5 | 1.3 | 0.045 | 8.6 | -83.9
119 | Dec. | 0.5 | 1.2 | 0.041 | 6.7 | 82.9
119 | July | 1.0 | 1.2 | 0.004 | 16.7 | -52.0
40 | July | 0.5 | 0.4 | 0.006 | 14.0 | -59.3
40 | Dec. | 0.5 | 0.4 | 0.003 | -5.9 | -24.5
40 | July | 1.0 | 0.4 | 0.002 | 5.8 | -38.1

There are three lines for each of 402, 119, and 40 GHz in Table 2, with the
first two lines in each set being the fit to an offset and a dipole to six
months of data – the data shown in the first two images of Figure 6 for 119
GHz, for example. The third line in each of these three sets shows the same
fit to the combination of these data in a one-year map. It should be first
noted that while the Dipole amplitude found for the full-year data are quite
small, the intermediate fits on six-month subsets are much larger.

Delouis et al. (2021) quote a staggeringly small 40 nK uncertainty on the
amplitude of the 3.4 mK dipole amplitude at 100 and 143 GHz, and 710 nK at 353
GHz with _Planck_ , which is larger than the IPDE dipole amplitude found on
six-month timescales (though not yearly time scales). Therefore, if no
corrections for the IPDE are made, its apparent time variability will even
limit the measurement of the stability of the calibration on six-month time
scales with the Dipole.

While tracking calibration drift can be done on _yearly_ temperature map
differences, it is more difficult for some other systematic effects. Far
sidelobes, for example, are notoriously difficult to characterize. They too
appear as faint, slowly varying, large-scale features in the maps, they too
will show differences in half-year maps, but not in yearly maps, and they
therefore may need to be separated from IPDE with care. So, while smaller than
the CMB cosmic variance levels in those channels sensitive to the CMB, and
while large-scale, total power measurements are no longer the primary target
of future CMB experiments, this signal will be detected and must be fitted,
filtered, or otherwise removed or accounted for.

###  4.2 Polarization

Here, the IPDE is initially assumed to be unpolarized, but there will
sometimes be certain combinations of data taken at different times with
different polarization and half-wave plate orientations which might mimic a
polarization signal. Figure 5 also shows that the noise induced in _LiteBIRD_
polarization maps from IPDE observed through varying column depths of
interplanetary dust is negligible compared to instrumental noise if the IPDE
is inherently non-polarized. This is good news as far as _LiteBIRD_ ’s quest
for CMB $B$ Modes is concerned.

So, similar to the case of temperature, the largest effects from IPDE for
polarization may once again be in complications in data analysis pipeline. The
apparent dependence of the signal on time and on the orientation of the
detectors will make, for example, null tests based on the differences between
different detectors more delicate to design and interpret at the faintest
levels. Naive tests made by differencing the data from a detector with one
orientation from that with another will fail due to IPDE, and might lead one
to incorrectly assume that there is something wrong with the data when there
is not.

####  4.2.1 Intrinsic Polarization

There are important caveats to the assertion that the IPDE is unpolarized: in
preparing this work, the best limits found on the polarization fraction of
thermal IPDE (as opposed to scattered light) are about 1% (from _ISO_ /CAM;
Siebenmorgen, 1999). It should also be noted that these _ISO_ /CAM
observations were designed not to measure the IPDE, but simply to use IPDE to
set limits on the _ISO_ /CAM instrumental polarization. They consist of a few
hours of observations taken on a single day (February 8, 1997), observing
towards a single location on the sky (equatorial
lon./lat.=01h21m50.0s/+12d20m00s)101010See the _ISO_ Observation Log at
https://heasarc.gsfc.nasa.gov/W3Browse/iso/isolog.html. Since this 1% is also
roughly the polarization fraction that can, at least in principle, be created
via mechanisms such as those presented in Onaka (2000) and Hoang & Lazarian
(2014), it is a convenient target to test.

To this end, the simulations described previously have been redone with a toy
model, assuming that the IPDE is 1%-polarized. Lacking any other information,
the polarization direction is simply assumed to be perpendicular to the plane
defined by the Sun–satellite line, and the line-of-sight. This is what was
found in Berriman et al. (1994) for scattered polarization and is broadly
consistent with models such as that from Onaka (2000). While one would also
expect some dependence on the angle between these two lines, for example, this
is ignored here – the polarization amplitude is assumed to be 1% of the
unpolarized amplitude, regardless of observation geometry. The Stokes Q and U
maps obtained with these changes are shown in Figure 8.

Figure 8: Similar to Figure 4, but assuming that the IPDE is 1% polarized.
Estimated 402 (top), 119 (middle), and 40-GHz (bottom) Stokes Q (left) and U
(right) IPDE contributions from a single detector for a mission lasting one
Solar year. The maps are in ecliptic coordinates, with the basic structure of
the Galactic plane indicated by the series of grey lines, spanning Galactic
latitudes from -20 to 20 degrees.

While the high-spatial-frequency noise seen in Figure 4 remains, if the IPDE
is polarized, Figure 8 shows that there will be large-scale structure in the
Stokes Q maps, while the U map remains similar to that without IPDE
polarization. The relative lack of U versus Q signal arises from the geometry
of the observations and our fiat that the polarization will always be
perpendicular to the plane containing the Sun-Earth vector and the observation
line of sight. Near the ecliptic plane, which contains the bulk of the IPDE,
this constrains the polarization to be along the ecliptic z-axis – hence
mostly (negative) Q Stokes signal. The power spectra resulting from these maps
are shown in Figure 9.

Figure 9: Similar to Figure 5, but with 1% polarization for the emission,
assumed to be in the scan direction. Single detector, noise-free, 402 (top),
119 (middle) and 40-GHz (bottom) temperature (grey triangles), $E$-Mode (blue
squares) and $B$-Mode (red circles) power spectra of the maps shown in Figures
3 and 8. The grey, blue and red curves show the corresponding TT, EE and
primordial $B$-mode spectra from gravitational waves with $r$=0.1, 0.01 &
0.001, respectively, in progressively darker red, using parameters from Planck
Collaboration et al. (2016b). The shaded regions towards the left of each
solid curve indicate cosmic variance, the thin black lines represent an
estimate of the expected _LiteBIRD_ polarization noise levels (Hazumi et al.,
2020), and the violet line represents the spectrum of lensing. The spectra are
made using HEALPix tools (Górski et al., 2005).

A first take-away from Figure 9 is that _LiteBIRD_ should, in fact, be able to
improve upon current limits on the polarization fraction of IPDE. The top
panel shows, for example, that the low, even multipoles of the IPDE should be
detected with high significance at 402 GHz. In fact, if the model used here is
reasonable, the signal should be detectable up to multipoles of order 30 or
so. On a less positive note, while IPDE polarization will not be detectable at
the lowest _LiteBIRD_ frequencies, if the actual polarization fraction of IPDE
is actually close to the current 1% limit, IPDE could actually affect the CMB
$B$-Mode power spectrum in the 119-GHz and neighboring channels. In this case,
as was the case with temperature, the signal will need to be masked, modeled
and/or similarly removed. If one were to add a $15^{\circ}$ ecliptic plane
mask in addition to a fairly typical $30^{\circ}$ Galactic plane mask, the
usable sky fraction goes from 50% of the sky after the Galactic cut to 34%
after the Galactic plus Ecliptic rejections – that is, an additional Ecliptic
cut will remove over 30% of the remaining sky.

##  5 Conclusion

For analyses that need to approach _LiteBIRD_ ’s instrumental noise limits,
IPDE will need to be removed from the data, as was the case for the _Planck_
mission and as has been the case for many infrared measurements for some time.
In temperature, however, Figure 5 shows that CMB power spectrum contamination
will be at the level of a percent of the cosmic variance of the CMB or even
lower.

Given that _LiteBIRD_ observes at wavelengths a factor of 50 away from the
peak of IPDE, it is a testament to the exquisite polarization sensitivity
target that _LiteBIRD_ should be able to set new limits on the IPDE. On the
other hand, if the intrinsic polarization is close to the current limits, the
induced $B$-Mode signal will be roughly the same size as the _LiteBIRD_ CMB
$B$-Mode target at 119 GHz, and this will therefore present an additional
wrinkle added to Galactic foreground removal. This should be kept in mind as
_LiteBIRD_ data reduction and analysis projects progress.

Before finishing, it’s worth recalling that IPDE models are fairly uncertain
at these “long wavelengths” of interest to CMB experimenters. Dikarev &
Schwarz (2015) have compared the Kelsall et al. (1998) model with two other,
more recent models and found significant differences. While their Divine
(1993) and Dikarev et al. (2005) models have less “structure” in them, the
absolute level of the contamination is, in fact, estimated to be higher than
that in the Kelsall et al. (1998) model, and may, therefore, show more
contamination in CMB maps (see also Jorgensen et al., 2021). With this caveat,
a more thorough comparison is deferred to a later work.

Finally, we note that Figure 1 reinforces the notion that Galactic dust
emission will almost always dominate over IPDE. To first order, IPDE is less
bright than Galactic dust emission. But the IPDE brightness spectrum, as noted
previously, is similar to that of Galactic dust emission. In fact, it is
similar enough that this may make separation of the two difficult. While it
may be a coincidence that the brighter regions of IPDE are similar in
brightness to the dimmer regions of Galactic emission, it may also indicate
some confusion between the two. In addition, the interplanetary dust cloud’s
proximity also leads to an apparent brightness variation, and it is here that
IPDE may be at its most annoying. This apparent variability will limit
somewhat tests and analyses and make removing Galactic dust to below the level
of the IPDE more difficult. But given the differences in amplitude and
polarization fraction, IPDE should ultimately not be as difficult to address
as Galactic emissions.

###### Acknowledgements.

Nils Odegard and Janet Weiland signaled problems with earlier versions of the
code used here as part of their work on Odegard et al. (2019), and Martin
Bucher, Josquin Errard, and Marc Sauvage provided useful elements, discussions
and comments. The bulk of the computing for this work was done at the IN2P3’s
Computing Center (https://cc.in2p3.fr). Mathieu Remazeilles acknowledges
funding support from the ERC Consolidator Grant CMBSPEC (No. 725456) under the
European Union’s Horizon 2020 research and innovation program. This work is
supported in Japan by ISAS/JAXA for Pre-Phase A2 studies, by the acceleration
program of JAXA research and development directorate, by the World Premier
International Research Center Initiative (WPI) of MEXT, by the JSPS Core-to-
Core Program of A. Advanced Research Networks, and by JSPS KAKENHI Grant
Numbers JP15H05891, JP17H01115, and JP17H01125. The Italian _LiteBIRD_ phase A
contribution is supported by the Italian Space Agency (ASI Grants No.
2020-9-HH.0 and 2016-24-H.1-2018), the National Institute for Nuclear Physics
(INFN) and the National Institute for Astrophysics (INAF). The French
_LiteBIRD_ phase A contribution is supported by the Centre National d’Etudes
Spatiale (CNES), by the Centre National de la Recherche Scientifique (CNRS),
and by the Commissariat à l’Energie Atomique (CEA). The Canadian contribution
is supported by the Canadian Space Agency. The US contribution is supported by
NASA grant no. 80NSSC18K0132. Norwegian participation in _LiteBIRD_ is
supported by the Research Council of Norway (Grant No. 263011). The Spanish
_LiteBIRD_ phase A contribution is supported by the Spanish Agencia Estatal de
Investigación (AEI), project refs. PID2019-110610RB-C21 and AYA2017-84185-P.
Funds that support the Swedish contributions come from the Swedish National
Space Agency (SNSA/Rymdstyrelsen) and the Swedish Research Council (Reg. no.
2019-03959). The German participation in _LiteBIRD_ is supported in part by
the Excellence Cluster ORIGINS, which is funded by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s
Excellence Strategy (Grant No. EXC-2094 - 390783311). This research used
resources of the Central Computing System owned and operated by the Computing
Research Center at KEK, as well as resources of the National Energy Research
Scientific Computing Center, a DOE Office of Science User Facility supported
by the Office of Science of the U.S. Department of Energy.

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##  Appendix A Map-Making

Assuming white (or no) noise, we can make temperature and polarization maps
using the equation

$\mathbf{m}=\left(\mathbf{A}^{T}\mathbf{A}\right)^{-1}\mathbf{A}^{T}\mathbf{d}.$
(1)

Here, $\mathbf{d}$ is the vector of length $n_{\mathrm{samples}}$ with our
time-line of IPDE values for each 19 Hz sample, $i$. That is,

$\mathbf{d}=\left(\begin{array}[]{c}d_{1}\\\ d_{2}\\\ d_{3}\\\ \vdots\\\
d_{n_{\mathrm{samples}}}\end{array}\right).$

The compact notation in equation 1 hides the fact that $\mathbf{m}$ is
actually a set of three maps, one for each of the Stokes parameters T, Q, and
U (we avoid ‘I’, which is being used as a sample index in lower case). That
is,

$m_{s,p}=\left(\begin{array}[]{ccc}t_{1}&q_{1}&u_{1}\\\ t_{2}&q_{2}&u_{2}\\\
t_{3}&q_{3}&u_{3}\\\ \vdots&\vdots&\vdots\\\
t_{n_{\mathrm{pixels}}}&q_{n_{\mathrm{pixels}}}&u_{n_{\mathrm{pixels}}}\\\
\end{array}\right),$

where $s$ is an index over the three Stokes parameters and $p$ is an index
over all map pixels. The so-called “pointing matrix”, $\mathbf{A}$, is defined
such that

$A_{i,1,p}=\left\\{\begin{array}[]{cl}1&\mathrm{if\ sample}\ i\ \mathrm{lands\
in\ pixel}\ p\\\ 0&\mathrm{otherwise}\end{array}\right.,$

$A_{i,2,p}=\left\\{\begin{array}[]{cl}\cos\left(2\theta_{i}\right)&\mathrm{if\
sample}\ i\ \mathrm{lands\ in\ pixel}\ p\\\
0&\mathrm{otherwise}\end{array}\right.,$

and

$A_{i,3,p}=\left\\{\begin{array}[]{cl}\sin\left(2\theta_{i}\right)&\mathrm{if\
sample}\ i\ \mathrm{lands\ in\ pixel}\ p\\\
0&\mathrm{otherwise}\end{array}\right..$

This means that for pixel $p$,

$\displaystyle\mathbf{A}^{T}\mathbf{A}$ $\displaystyle=$
$\displaystyle\sum_{s_{p}}\left(\begin{array}[]{ccc}1&\cos
2\theta_{s_{p}}&\sin 2\theta_{s_{p}}\\\ \cos
2\theta_{s_{p}}&\cos^{2}2\theta_{s_{p}}&\cos 2\theta_{s_{p}}\sin
2\theta_{s_{p}}\\\ \sin 2\theta_{s_{p}}&\cos 2\theta_{s_{p}}\sin
2\theta_{s_{p}}&\sin^{2}2\theta_{s_{p}}\end{array}\right),$ (5)

where $s_{p}$ is the set of samples $s$ which fall in pixel $p$. This depends
only on the scanning and sampling patterns.