0
$\begingroup$

Suppose we have a sequence of non decreasing random variables that converges to 0 in distribution. How can we prove that this sequence converges to 0 almost surely?

$\endgroup$
0
$\begingroup$

I will use two standard theorems :

(1) Convergence in Distribution to a constant r.v. $\Rightarrow$ Convergence in Probability

(2) If $X_n$ converges in probability to $X$ then there exists a subsequence $X_{n_k}$ such that $X_{n_k}$ converges almost surely to $X$ as $k \rightarrow \infty$.

For the given problem, $X_n(w)$ is a bounded non-decreasing sequence, therefore, $\lim_{n\rightarrow \infty}X_n(w)$ exists for all $w$. Let $n_k$ be the subsequence from (2) above. Let $T = \mathrm{P}[w : X_{n_k}(w) \rightarrow 0]$. Then $X_n(w) \rightarrow 0$ for all $w \in T$. From (2) we know $P(T) = 1$ and we are done.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.