Asymptotically independent increments random elements: Billingsley Ch:$4$

Question

Let $X_n$ be random elements of $D$ (space of cad lag functions on $[0,1]$ as domain). $X_n$ has asymptotically independents if $0\leq s_1 \leq t_1 \leq s_2 \leq \ldots < s_r \leq t_r \leq 1$, then for all linear Borel sets $H_1,\ldots,H_r$ we have

$P\{X_n(t_i)-X_n(s_i)\in H_i, i = 1,\ldots,r \}-\prod_{i=1}^{r}P\{X_n(t_i)-X_n(s_i)\in H_i\} \star$

converges to zero as $n\to\infty$. Now Theorem $19.2$ in Billingsley's book on convergence of probability measures reads

Let $X_n$ have the following properties,

1.) asymptotically independent increments,

2.) $\{X_n^{2}(t)\}$ is uniformly integrable for each $t$,

3.) $E\{X_n(t)\}\to 0$, $E\{X_n^{2}(t)\}\to t$ as $n\to\infty$,

4.) Also for each positive $\epsilon$ and $\eta$, there exists a positive $\delta$ such that $P\{w(X_n,\delta)\geq \epsilon\}\leq \eta$ for all sufficiently large $n$.

Then $X_n \to W$ (Wiener measure/process).

Its proof seems pretty standard, I had one doubt in it. Tightness of $\{X_n\}$ follows from condition $4$, which also implies that if $X$ is a limit point, then $P(X\in C) = 1$. It remains to show that $X$ has distribution of $W$.

Conditions $2,3$ ensure that $E\{X(t)\} = 0$ and $E\{X^{2}(t)\} = t$. Next is written that Condition $1$ implies that increments of $X(t)$ are independent. I had a doubt here. How does weak convergence and the condition $\star$ imply that the $X(t)$ has independent increments? I mean the probabilities need not converge, except for the continuity sets of $\mathcal{D}$.

NEW DOUBT

The following doubt in relaxing the condition $t_i <s_{i+1}$ using the fact $\mathbb{P}(X \in C) = 1$. Let us consider the case of only $3$ points in time, i.e. $0,s,t$, where assume that $0<s<t$. My intuition is that this case will entail the main idea. We would like to show that

$\mathbb{P}(X_s \in H_1, X_t-X_s \in H_2) = \mathbb{P}(X_s \in H_1)\mathbb{P}(X_t-X_s \in H_2) \star$.

Also we are given that if $s_1 <s$, then

$\mathbb{P}(X_{s_{1}} \in A, X_t-X_s \in B) = \mathbb{P}(X_{s_{1}} \in A)\mathbb{P}(X_t-X_s \in B) \star\star$.

Using the fact that $\mathbb{P}(X\in C) = 1$, we can write

$\{X_s \in H_1, X_t-X_s \in H_2 \}\equiv \{X_{s^{-}} \in H_1, X_t-X_s \in H_2 \}\equiv \cap_{k=1}^{\infty}\cup_{m=1}^{\infty}E_{k,m}$, where $E_{k,m} := \{X_l \in B_{\frac{1}{k},H_1} 1-\frac{1}{m}\leq l<s \}$, where $B_{\frac{1}{k},H_1}$ denotes an open ball of radius $\frac{1}{k}$ around the set $H_1$. Now I know that I have to use $\star \star$ somehow, but cannot proceed.

Answer 1

Assume that we have proved that if $0\leqslant s_1\leqslant t_1\lt s_2\leqslant t_2\lt\dots\lt s_r\leqslant t_r\leqslant 1$, we have that $(X_{t_i}-X_{s_i})_{i=1}^r$ is independent. Then using $\mathbb P\{X\in C\}=1$, we will deduce the independence of $(X_{t_i}-X_{t_{i-1}})_{i=1}^r$ where $0\leqslant t_0\lt t_1\lt \dots\lt t_{t-1}\lt t_r$ (in the definition of asympotically independent increments, we consider disjoint $[s_i,t_i]$).

So let $r\geqslant 1$ and $0\leqslant s_1\leqslant t_1\lt s_2\leqslant t_2\lt\dots\lt s_r\leqslant t_r\leqslant 1$ be fixed. We have for each $H_i$, $1\leqslant i\leqslant r$ such that $\mathbb P\{X_{t_i}-X_{s_i}\in \partial H_i\}=0$ that $$\tag{*}\mathbb P\left(\bigcap_{i=1}^r\{X_{t_i}-X_{s_i}\in H_i\}\right)=\prod_{i=1}^r\mathbb P\left(\{X_{t_i}-X_{s_i}\in H_i\}\right).$$ We now have to extend this to each Borel sets $H_i$. If the $H_i$ are finite union of disjoint intervals whose endpoint $c_{i,j},d_{i,j}$ are such that $\mathbb P\{X_{t_i}-X_{s_i}=c_{i,j}\}=0=\mathbb P\{X_{t_i}-X_{s_i}=d_{i,j}\}$, then we are done. Approximation by below a $H_i$ which is an arbitrary finite union of disjoint intervals, we get that $(*)$ holds for such $H_i$.