I am having some problems with the following exercise:
Let $(Y_n ,n ≥ 1)$ be a sequence of independent random variables such that:
$P(Y_n = e^n − 1) = e^{−n}$, $P(Y_n = −1) = 1 − e^{−n}$, $∀n ≥ 1$.
Then we define $S_n ,X_n$ by:
$S_0 = 0$, $S_n = Y_1 + ··· + Y_n$ , $n ≥ 1$.
$X_0 = 0$, $X_n = \frac{Y_1 + ··· + Y_n}n$, $n ≥ 1$.
(i) Prove that $P(\limsup\limits_{n\rightarrow\infty}{\left\{Y_n\ne-1\right\}})=0$
(ii) Prove that $\liminf\limits_{n\rightarrow\infty} {\left\{Y_n = −1\right\}}$ ⊂ $\left\{\lim_{n \to \infty}X_n=-1\right\}$ ⊂ $\left\{\lim_{n\to\infty}S_n=\infty\right\}$
(iii) Prove that $X_n \to −1$ a.s. Compare this result with law of large numbers and comment.
(iv) Show that $S_n$ is a martingale with $S_n \to -\infty$ a.s.
(v) Are $X_n$ and $S_n$ uniformly integrable?
So far, I have proved (i) with the Borel-Cantelli lemma. I have also tried to use Borel-Cantelli to prove (ii) and (iii), but I was not very successful. I would be grateful for some help there.
For (iv), I have tried to prove that $S_n$ is a martingale by induction, but I am struggling with the inductive step.
For (v), I think that we could use the following theorem:
Let $(X_n)$ be a martingale. Then $(X_n)$ is U.I. $\iff$ $X_n \to X_\infty$ a.s. and in $L^1(P)$
However, I don't know how to show if $X_n$, $S_n$ converge in $L^1$.
Thank you for your help!
(ii) Let $\omega$ such that $\omega\in\liminf_n\{Y_n=-1\}$. There $N=N(\omega)$ such that $Y_n(\omega)=-1$ for $n\geqslant N$. In particular, for these $n$, we have $X_n(\omega)=\frac 1n\sum_{j=1}^NY_j(\omega)-\frac{n-N+1}n$, hence $X_n(\omega)\to -1$. This implies that $nX_n(\omega)\to -\infty$.
(iii) We have to prove that $\mu(\liminf_n\{Y_n=-1\})=1$, but it's a consequence of (i).
(iv) For each $n$, $S_n$ is bounded hence integrable. In order to check $\mathbb E[S_{n+1}\mid\mathcal F_n]$, use linearity of conditional expectation and the fact that $X_{n+1}$ is independent of $\mathcal F_n$ (provided we consider natural filtration).
(v) $\{S_n,n\geqslant 1\}$ cannot be uniformly integrable. Otherwise, $$\mathbb E|S_n|\leqslant \sup_k\mathbb E[|S_k|\chi_{\{|S_k|\gt R\}}]+\mathbb E[|S_n|\chi_{\{|S_n|\leqslant R\}}]$$ so for a fixed $\varepsilon$, take $R$ such that $\sup_k\mathbb E[|S_k|\chi_{\{|S_k|\gt R\}}]\lt \varepsilon$: by dominated convergence, $\lim_{n\to \infty}\mathbb E|S_n|=0$, which is not possible.
The expectation of $X_n$ is $0$ and $X_n\to -1$ almost surely: in case of uniform integrability we would have $\mathbb E|X_n-(-1)|\to 0$.