Suppose that $X_1, X_2, \dots$ be a seq. of r.v.s such that
$(a) ~ X_{n+m} - X_n$ is independent of $\mathcal{F}(X_1, \cdots, X_n), $
$(b)$ the distribution of $X_{n+m} -X_n$ does not depend on $n$
for every $n,m = 1,2,\dots$.
I tried to prove that $these ~ statements ~ hold ~ even ~ if ~ we ~ replace$ $n$ $by ~ a ~ stopping ~ time$ $N$.
But I succeed in proving the above claim only when I assumed that $P(N<\infty) = 1$.
Meanwhile, how can I define $X_N$ or $\mathcal{F}(X_1, \cdots , X_N)$ on the set $\{\omega:N(\omega)=\infty\}$?
To prove the above one, I thought it was natural to regard $N$ as a finite stopping time.
Q1. Can we prove that the statements (a) and (b) hold even if we replace $n$ by $N$ without the assumption that $P(N<\infty)=1$?
Q2. How can I define $X_N$ or $\mathcal{F}(X_1, \cdots , X_N)$ on the set $\{N=\infty\}$?
Any help or comment would be appreciated!