Problem: Partial sums of independent stochastic variables

(a) Let \(X_1, X_2,\ldots\) be independent stochastic variables with \(E[X_n] = 0\) for \(n \geq 1\). Let \(M_0 = 0\) and \(M_n = X_1 + \ldots + X_n\) for \(n \geq 1\). Show that \(M_n\) is a (mean-zero) martingale.

(b) Let \(X_1, X_2,\ldots\) be independent stochastic variables with \(E[X_n] = \mu\) for \(n \geq 1\). Let \(S_0 = 0\) and \(S_n = X_1 + \ldots + X_n\) for \(n \geq 1\). Find \(E[S_n|\mathcal{F}_{n-1}]\), where \(\mathcal{F}_m\) contains all information about the \(X\) process up to and including time \(m\). Is \(S_n\) a (mean-zero) martingale? Can you find a simple transformation of \(S_n\) which is a zero-mean martingale?

(c) Let \(X_1, X_2,\ldots\) be independent stochastic variables with \(E[X_n] = 1\) for \(n \geq 1\). Let \(M_0 = 1\) and \(M_n = X_1 \cdot \ldots \cdot X_n\) for \(n \geq 1\). Show that \(M_n\) is a martingale. What is \(E[M_n]\)?

Hints: To get some hints for how to solve the problem see the link at the bottom of this page.