Consider a collection of 0-1 random variables $Y_{n,N}$, for all $n$ and $N$. The random variable $Y_{n,N}$ is a deterministic function of the collection of random variables in $\mathcal{F}_n = \{(U_k)_{k=1,\ldots,n}, (V_k)_{k=1,\ldots,n} \}$ where the $U_k$'s and the $V_k$'s are all independent among them and uniformly distributed over $[0,1]$. Therefore, $Y_{n,N}=\mathbb{E}[Y_{n,N} | \mathcal{F}_n ]$.

Let me denote by $\mathcal{F}_n\setminus V_n$ the set $\mathcal{F}_n$ where $V_n$ has been removed.

My question is the following. Assuming that $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N Y_{n,N} $$ exists almost surely and that $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N \mathbb{E}[Y_{n,N} | \mathcal{F}_n\setminus V_n] = Z $$ almost surely (where $Z$ is some other random variable), is it true that necessarily $$ \lim_{N\to\infty} \frac{1}{N} \sum_{n=1}^N Y_{n,N} = Z $$ almost surely? Why?

Though different (because I am assuming the existence of the first limit), this question is related to the following questions: Strong law of large numbers for function of random vector: can we apply it for a component only? and Law of large numbers with one dependency


With the notations of the opening post,let $\mathcal G_n:=\mathcal F_{n+1}\setminus V_{n+1}$ and $D_{n,N}:=Y_{n,N}-\mathbb{E}\left[Y_{n,N} | \mathcal{F}_n\setminus V_n\right]$. Then $D_{n,N}$ is $\mathcal G_n$-measurable, $\mathbb{E}\left[D_{n,N} | \mathcal{G}_{n-1} \right]=0$ and $\left\lvert D_{n,N}\right\vert\leqslant 2$ hence by Hoeffding's inequality, we derive that $$ \mathbb P\left\{\frac 1N\left\lvert \sum_{n=1}^ND_{n,N}\right\rvert \gt \varepsilon\right\}\leqslant\exp\left(-\frac{N^2\varepsilon^2}{4N}\right) $$ hence an application of the Borel-Cantelli lemma shows that $\frac 1N \sum_{n=1}^ND_{n,N}\to 0$ almost surely.

  • $\begingroup$ The $Y_{n,N}$ are not necessarily independent, so the $D_{n,N}$'s are not necessarily independent as well, right? Isn't the Hoeffding's inequality valid for i.i.d random variables only? $\endgroup$ – user52227 Apr 29 '18 at 14:09
  • $\begingroup$ Usually the version of Hoeffding inequality for martingale differences is called Azuma, but is contained in the paper by Hoeffding. $\endgroup$ – Davide Giraudo Apr 29 '18 at 14:18
  • $\begingroup$ Oh, thank you!!! $\endgroup$ – user52227 Apr 29 '18 at 14:22

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.