A **NEW DOUBT** arose in a previous question, so I am posting it again since the previous edits did not draw views.


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.


**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


$\{X\_s \in H\_1, X\_t-X\_s \in H\_2 \}\equiv \{X\_{s^{-}} \in H\_1, X\_t-X\_s \in H\_2 \}\equiv \cap\_{k=1}^{\infty}\cup\_{m=1}^{\infty}E\_{k,m}\cap\{X\_t-X\_s \in H\_2\}$


where $E\_{k,m} := \{X\_l \in B\_{\frac{1}{k},H\_1} 1-\frac{1}{m}\leq l<s \}$,and $B\_{\frac{1}{k},H\_1}$ 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. 



Billingsley Ch: $4$ Asymptotically Independent random elements
Take $s\_n:=s-1/n$: then $(X\_{s\_n},X\_t-X\_s)\to (X\_s,X\_t-X\_s)$ almost surely (by the property $\mathbb P\{X\in C\}=1$). Now we can check, uisng characteristic functions, that $X\_s$ and $X\_{t}-X\_s$ are independent. 


A similar argument applies for more than two increments.


