Let $ε_n, n > 1$, and $V_n, n > 0$, be independent random variables, with $P(ε_n = 1) = P(ε_n = −1) = 1/2$, $P(V_n = 1) = p_n, P(V_n = 0) = 1 − p_n$, for all n. Define $X_n$ inductively by $X_0 = 1$ and, for n > 0, $X_{n+1} = X_n + V_nε_{n+1}$ if X_n > 0$ $ and $X_{n+1} = 0$ if $X_n = 0$.
(i) Show that $(X_n)_{n>0}$ is a martingale with respect to a filtration that you should define.
(ii) Suppose that $p_n = 1$ for all n. Show that $X_n → 0$ almost surely.
(iii) Now let $p_n = 1/(n+ 1)$ for all n. Does $X_n → 0$ almost surely? What if p_n = $1/(n+1)^2$ for all n?
[You may assume that for real numbers $0 < x_n < 1$, $\prod_1^∞(1−x_n) = 0$ if and only if $\sum_1^∞ x_n =\infty$]
Part (i) fine to show under the filtration $\mathcal{F}_n=\sigma((X_m)_{m \leq n},(V_m)_{m \leq n},(ε_m)_{m \leq n})$. For part (ii) We can use Doob's Forward convergence thm to get that $X_n → X_\infty$ a.s. for some a.s. finite r.v. $X_\infty$. However how can I show that $X_\infty=0$ a.s. here? I have been trying to show that the hitting time to 0 is a.s. finite (which would do the trick here), but have not been successful.
Have not tried (iii) yet as I am yet to do (ii). Thanks.
For (i): as it stands, the sequence $\left(\mathcal F_n\right)_{n\geqslant 1}$ is not nondecreasing, hence it is not a filtration.
For (ii) If $\omega$ is such that $X_n(\omega)>0$ for each $n$, then we have $1+\sum_{j=1}^n\xi_j(\omega)=:1+S_n(\omega)\gt 0$ for each $n$. But the set of such $\omega$'s has a null probability. Indeed, by the Kolmogorov's 0-1 law, the random variables $\limsup_n S_n$ and $\liminf_n S_n$ are almost surely constant (these one may be finite or not). Since $\xi_1$ is symmetric and non-degenerated, we necessarily have $\limsup_n S_n=+\infty$ and $\liminf_n S_n=-\infty$ almost surely.