Question

Let $$X_1, X_2,...$$ be independent, identically distributed random variables with $${P(X_j=2)} = {1\over3}$, ${P({X_j}={1\over2})} = {2\over3}$$ . Let $$M_0 = 1$$ and for $$n \geq 1$$, $$M_n = X_1*X_2*...*X_n$$ .

(a) Show that $$M_n$$ is a martingale.
(b) Explain why $$M_n$$ satisfies the conditions of the martingale convergence theorem

(c)  Let $$M_\infty = \lim_{n\to\infty}M_n$$ . Explain why $$M_{\infty} = 0$$

(Hint: there are at least two ways to show this. One is to consider $$\log{M_n}$$ and use the law of large numbers. Another is to note that with probability one $\frac{M_{n+1}}{M_n}$ does not converge)
(d) Use the optional sampling theorem to determine the probability that $$M_n$$ ever attains a value as large as 64.
(e) Does there exist a $C < \infty$ such that $$E[M_n^2] \leq C$$ for all $$n$$?

Answer 1

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