# Example of a Non-negative Martingale Satisfying Certain Conditions

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.

## 2 Answers

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 $$\frac 12 Y_0$$.

### 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 $$2/3$$.