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I am reading a paper with the following set up. Let $(\Omega, \mathcal{F},P)$ be a probability space, $B \in \mathcal{F}$, and $\{\mathcal{F}_n : n \in \mathbb{N} \}$ a filtration with $\mathcal{F}_n \uparrow \mathcal{F}$.

Let $q_n = P(B \mid \mathcal{F}_{n-1})$. The sequence $\{q_n \}$ is a martingale with values in $[0,1]$ almost surely. Now, the paper asserts $$E\Big(\sum_{n=1}^\infty(q_{n+1} - q_n)^2 \Big) \leq 1.$$

I can't see why this is true. Of course, for all $n$, $E(q_{n+1}-q_n)^2 \leq 1$, but I don't see why the inequality should hold after summing over all $n$. Could someone please point out what I'm missing?

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Increments of a martingale are orthogonal hence $$\mathbb E\left[\sum_{n=1}^{+\infty}\left(q_{n+1}-q_n\right)^2 \right] =\mathbb E\left[\left(\sum_{n=1}^{+\infty}q_{n+1}-q_n\right)^2  \right].$$ By the martingale convergence theorem, $$\sum_{n=1}^{+\infty}q_{n+1}-q_n=\mathbb P\left(B\mid\mathcal F\right)-\mathbb P\left(B\mid\mathcal{F} _0\right),$$ which gives the result.

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  • $\begingroup$ Thanks for the quick and helpful answer! $\endgroup$
    – aduh
    May 6, 2017 at 15:52

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