SLLN with dependent Bernoullis implies convergence of sum with conditional means?

Question

Suppose I have $n$ bernoulli (values zero or one), possibly dependent and nonidentically distributed, random variables (like the generalized binomial model), where a law of large numbers holds. Let $S_n = \sum_{i=1}^n X_i$ and

$$\lim_{n\rightarrow\infty}\frac{S_n-ES_n}{n}=0\text{ a.s.}$$

My question is, does this imply a function like $$F(n)=\frac{1}{n}\sum_{i=0}^{\lfloor ES_n \rfloor}(ES_n-i)\Pr(S_n=i)+\frac{1}{n}\sum_{i=\lfloor ES_n \rfloor+1}^n(i-ES_n)\Pr(S_n=i) $$ also converges to zero almost surely?

My work: $$nF(n) = \Pr(S_n\leq ES_n)(ES_n - E(S_n \mid S_n\leq ES_n)) + (1-\Pr(S_n \leq ES_n))(E(S_n\mid S_n >ES_n )-ES_n) $$

Replacing $\Pr(S_n\leq ES_n)$ with $p_n$,

$$nF(n) = 2p_n ES_n - ES_n - p_nE(S_n \mid S_n\leq ES_n) + (1-p_n)E(S_n \mid S_n> ES_n) $$

$$nF(n) = 2p_n ES_n - 2p_nE(S_n \mid S_n\leq ES_n).$$

Here, I'm getting stuck. I think I can use the tower property to show $EF(n)=0$, but I'd like to show $F(n)\rightarrow 0$ as $n\rightarrow \infty$ a.s.

Answer 1

Yes. In fact, you just need to assume $(S_n-E[S_n])/n$ converges to $0$ in probability (almost sure convergence is not needed). It is helpful that, in your case, $0 \leq S_n \leq n$ for all $n$.

You are trying to show that $\lim_{n\rightarrow\infty} E\left[\left|\frac{S_n-E[S_n]}{n}\right|\right]=0$. Define $A_n = |S_n-E[S_n]|/n$. A hint is to fix $\epsilon>0$ and consider $E[A_n] = E[A_n|A_n\leq \epsilon]Pr[A_n\leq \epsilon] + E[A_n|A_n>\epsilon]Pr[A_n>\epsilon]$.

More generally, if $\{Z_n\}_{n=1}^{\infty}$ is a sequence of random variables that satisfies $Z_n\rightarrow 0$ in probability, and if the $Z_n$ values are deterministically bounded by some constant $C$ (so that $|Z_n|\leq C$ for all $n$), then $\lim_{n\rightarrow\infty}E[Z_n]=0$. This is a special case of the "Bounded Convergence Theorem" for random variables (see section 13.6 of Probability with Martingales by David Williams).

An even more general statement is the "bounded moment convergence theorem," which removes the requirement of deterministic bounds and only requires $E[|Z_n|^{1+\delta}] \leq C$ for all $n$ (for some real numbers $C\geq 0$, $\delta>0$). I needed such a statement once but couldn't find any ref, so I gave a quick proof in Appendix A here: http://ee.usc.edu/stochastic-nets/docs/low-power-computing-chapter.pdf