$$(X_n)$$ is a sequence of $$L^2$$ random variables with $$EX_n=0$$ for all $$n$$ and suppose there is a constant $$c$$ s.t. $$\operatorname{Var}(X_{n+k}−X_n)\leq ck$$, for all $$n,k\geq0$$. Show that $$X_n/n$$ converges to $$0$$ a.s. (Hint: First prove along a suitable subsequence).

I can see that we are trying to make the probabilities summable along a subsequence but may I know how to choose a subsequence so that the upper bound we use for variance makes the probabilities summable?

$$P(|X_{n_k}|>nϵ)≤\operatorname{Var}(X_{n_k})/{n_k}^2ϵ^2\leq?$$

• Are you allowed to use the Borel Cantelli lemma ? – P. Quinton Dec 25 '19 at 19:21
• Yes, I was trying to use that with summability – manifolded Dec 25 '19 at 19:22
• You first need to complete your inequality with an upper bound on the variance as given in the problem. – Michael Dec 25 '19 at 21:16
• If it helps you can use $(a+b)^2\leq 2a^2+2b^2$. – Michael Dec 25 '19 at 21:54

Let $$\| X \|_2$$ denote the $$L^2$$ norm of $$X$$. Note that if $$EX = 0$$, then $$\| X \|_2^2 = \operatorname{Var}(X)$$.
Consider the subsequence given by $$n_k = 2^k$$. Then, $$|\| X_{2^{k + 1}} \|_2 - \| X_{2^k} \|_2| \leq \| X_{2^{k + 1}} - X_{2^k} \|_2$$ by the reverse triangle inequality. Squaring and using the given condition yields $$|\| X_{2^{k + 1}} \|_2 - \| X_{2^k} \|_2|^2 \leq c2^k$$ Since $$\| X_{2^k} \|_2 \geq 0$$, it follows that $$\| X_{2^{k + 1}} \|_2 \leq \| X_{2^k} \|_2 + \sqrt{c}2^{k / 2}$$ By induction, it follows that $$\| X_{2^{k + 1}} \|_2 \leq \| X_1 \|_2 + \sqrt{2c}\frac{2^{k / 2} - 1}{\sqrt{2} - 1}$$ Finally, $$\operatorname{Var}(X_{n_k}) / n_k^2 \leq \left( \| X_1 \|_2 2^{-k} + \sqrt{2c}2^{-k}\frac{2^{(k - 1) / 2} - 1}{\sqrt{2} - 1} \right)^2 = O(2^{-k})$$ so $$\operatorname{Var}(X_{n_k}) / n_k^2$$ is summable.