Math 6040 • Probability

Suppose \(\{r_{k}\}_{k \in \mathbb {Z}_{+}}\) is a sequence of real numbers such that \(n^{-1}\sum _{k=1}^{n}|r_{k}| \to 0\) as \(k \to \infty \). Let \(\{X_{k}\}_{k \in \mathbb {N}}\) be a sequence of \(L^{2}\) random variables such that \(\mathbb {E}[X_{k}] = 0\) and \(\mathbb {E}[X_{k}X_{\ell }] \le \) \(r_{\ell -k}\) for all \(k \le \ell \). Let \(S_{n} = X_{1} + \cdots + X_{n}\). Show that \(n^{-1}S_{n}\) converges in \(L^{2}\) as \(n \to \infty \) and identify the limit.