Math 6010 • Linear Models

Let \(M_n\) be a martingale with \(M_0=0\) and increments \(X_n=M_n-M_{n-1}\) that satisfy \(\mathbb {E}[X_n^2]<\infty \) for all \(n\). Assume that \((b_n)_{n=1}^{\infty }\) is an increasing sequence of positive real numbers such that \(b_n\to \infty \) and \[ \sum _{n=1}^{\infty }\frac {\mathbb {E}[X_n^2]}{b_n^2}<\infty . \] Show that \(M_n/b_n\to 0\) almost surely as \(n\to \infty \). Hint: Since the sum of the \(X_k\) is a martingale so too should be a weighted sum; prove this idea and use it to produce a new martingale out of the \(X_k\) and \(b_k\). Derive convergence properties of this new martingale, and then use Kronecker’s Lemma: if \(a_n\) is a sequence of numbers such that \(\sum _{n=1}^{\infty }a_n\) exists and is finite and \(b_n\) is a sequence of the type above, then \[ \frac {1}{b_n}\sum _{k=1}^{n}a_kb_k\to 0 \qquad \text {as } n\to \infty . \]