Example of a Non-negative Martingale Satisfying Certain Conditions

Question

Question

The question is to find an example of a non-negative martingale $(X_n)$ with $EX_n=1$ for all $n$ such that $X_n$ converges almost surely to a random variable $X$ where $EX\neq 1$ and $\text{Var}(X)>0$.

My attempt

An example of a martingale that I thought could fit the bill was the product martingale with $X_n=\prod_{i=1}^n Y_i$ where $(Y_{i})$ are i.i.d non-negative random variables with mean $1$ and $P(Y_i=1)<1$. Unfortunately $X_n\to 0$ a.s and hence the limit is degenerate. Other examples, I tried to cook up (e.g. branching process with one individual) all had degenerate limits.

I am having trouble coming up with an example that does not have a degenerate limit.

Answer 1

Modifying your example a little, by letting $Y_0$ be any non-negative random variable independent of $\{Y_n\}_{n\ge 1}$ with $E(Y_0)=1$, $$ M_0=\frac12Y_0+\frac 12,\ \ M_n=\frac12\left(Y_0+\prod_{i=1}^nY_i\right),\ \ n\ge 1 $$ is a martingale converging to $\frac 12 Y_0$.

Answer 2

Hint

Convex combinations of martingales are martingale. Your example converges to $0$. modify if to get an example converging to something else, mix the two.

Answer

Let $X_n$ be the martingale you described; $X_n=\prod_{i=1}^n Y_i$, where $Y_i\stackrel{\text{iid}}\sim \operatorname{Unif}\{1/2,3/2\}$.

Let $(X_n')_{n\ge 1}$ be an independent copy of $(X_n)_{n\ge 1}$.

Finally, let $\xi$ be Bernoulli$(1/3)$, independent of the previous variables. Then $$ X_n\cdot{\bf 1} ({\xi=1})+(2-X_n')\cdot {\bf 1}(\xi=0) $$ is a martingale whose expectation is always one, and in the limit equals $0$ with probability $1/3$ and equals $2$ with probability $2/3$.