# Showing that a martingale $Y_k$ does not converge almost surely.

Let $$X_i$$ be iid with

$$\mathbb{P}(X_i=1)= \mathbb{P}(X_i= -1) = \frac{1}{2i}, \mathbb{P}(X_i=0)=1-\frac{1}{i},$$

where $$i=1,2,...$$

And define $$Y_1=X_1$$ and for $$k\geq2$$

$$Y_k= \begin{cases} X_k, \text{ if } Y_{k-1}=0\\ kY_{k-1}|X_k|, \text { if } Y_{k-1} \neq 0 \end{cases}$$

I have shown that $$Y_k$$ is a martingale wrt to the natural filtration $$\sigma(X_1,...,X_k)$$. And also that it converges in probability to zero (by conditioning on the events $$(Y_{k-1} = 0 )$$ and $$(Y_{k-1} \neq0)$$). I am however stuck on showing that it doesn't converge almost surely. I guess one has to use the Borell Cantelli Lemma but I don't see how to apply that in this situation.

• What is $M_n$ ?
– Surb
Nov 6, 2021 at 11:42
• Yes sorry, that should be $Y_k$ Nov 6, 2021 at 11:52

One can show that $$\Bbb P(Y_k\neq0\mid X_0,\ldots,X_{k-1})=\frac1k$$ for all $$k\ge1$$. Thus $$\sum_{k=1}^\infty\Bbb P(Y_k\neq0\mid X_0,\ldots,X_{k-1})=\infty.$$ By the conditional Borel-Cantelli lemma, this means that the event $$\limsup\:\{Y_k\neq0\}=\limsup\:\{|Y_k|\ge1\}$$ is almost sure.