# Strong law of large numbers for the conditional expectation of functions of random vectors

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

## 1 Answer

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.

• 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? – user52227 Apr 29 '18 at 14:09
• Usually the version of Hoeffding inequality for martingale differences is called Azuma, but is contained in the paper by Hoeffding. – Davide Giraudo Apr 29 '18 at 14:18
• Oh, thank you!!! – user52227 Apr 29 '18 at 14:22