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.
Best Answer
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 $\frac 12 Y_0$.