Assume that we have a sequence of i.i.d. random variables that are Gaussian mean $\mu$, and variance $\sigma^2$. I am trying to study the converge of the following sequence of random variables:
\begin{equation} S_n = \frac{1}{n}\sum^n_{i=2} \bigg\{X_i \frac{\sum^{i-1}_{j=1}X_j}{i} \bigg\} \end{equation}
I suspect this might converge to $\mu^2$ but I am not sure how to apply the law of large numbers here, since quantities in the sum are not independent. I would really appreciate if someone could provide a hint. Thanks!
This can indeed be treated using the strong law of large numbers. Let $$\Omega':=\left\{\omega\in\Omega,\frac 1i\sum_{l=1}^iX_l\to \mu \right\}.$$ Then $\Omega'$ has probability one. Let $\omega\in\Omega'$ and fix $\varepsilon\gt 0$. Let $i_0$ be such that for all $i\geqslant i_0$, $\left\lvert \frac 1i\sum_{l=1}^{i-1} X_l(\omega)-\mu\right\rvert\lt\varepsilon $. Then $$S_n(\omega) =\frac 1n \sum_{i=1}^{i_0-1} X_i(\omega)\frac{\sum_{l=1}^{i-1} X_l(\omega)}l +\frac 1n\sum_{i=i_0}^{n} X_i(\omega)\left( \frac{\sum_{l=1}^{i-1} X_l(\omega)}l -\mu\right)+\frac\mu n\sum_{i=i_0}^{n} X_i(\omega).$$ The first term goes to zero, the second one do not exceed $\varepsilon n^{-1}\sum_{i=1}^n\left\lvert X_i\left(\omega\right)\right\rvert$, which goes to $\varepsilon \mathbb E\left\lvert X_1\right\rvert $ by the strong law of large numbers. By the strong law of large numbers, the third term goes to $\mu^2$.