Convergence of conditional expectations in $L_p$ for non-negative adapted processes

Question

Given an adapted processes $(X_n, \mathscr{F}_n)$ that satisfies $X_n \geq 0$ for all $n > 0$ and $\sum\limits_{n=1}^{+\infty} X_n = 1$, can we conclude that $\sum\limits_{n=1}^{+\infty} \mathbb{E}(X_n | \mathscr{F}_{n-1})$ converges in any $L_p$?

Answer 1

Yes. In fact, we even have convergence a.s. The process $Y_n := \sum_{i=1}^n \mathbb E[X_i|\mathcal F_{i-1}]$ is a uniformly integrable submartingale (to see this, use the fact that $X_i$ are uniformly integrable, and the tower property of conditional expectation). So it converges a.s. and in every $L^p$ by the submartingale convergence theorem.

Answer 2

Convergence holds in any $L^p$ for $p\geqslant 1$. By Theorem III.4.3 page 106 in

Garsia, Adriano M. Martingale inequalities: Seminar notes on recent progress. Math. Lecture Note Ser. W. A. Benjamin, Inc., Reading, Mass.-London-Amsterdam, 1973. viii+184 pp.

the inequality $$ \mathbb E\left[\left(\sum_{i=1}^n\mathbb E\left[Z_i\mid\mathcal F_{i-1}\right]\right)^p\right]\leqslant p^{p+1}\mathbb E\left[\left(\sum_{i=1}^n Z_i\right)^p\right] $$ holds for any $p\geqslant 1$, any non-negative random variables $Z_i$ and any increasing collection of $\sigma$-algebra $(\mathcal F_i)$.

Applying this for fixed $M<N$ to $Z_i=0$ if $i\leqslant M-1$ and $Z_i=X_i$ otherwise gives $$ \mathbb E\left[\left(\sum_{i=M}^N\mathbb E\left[X_i\mid\mathcal F_{i-1}\right]\right)^p\right]\leqslant p^{p+1}\mathbb E\left[\left(\sum_{i=M}^N X_i\right)^p\right] $$ and using the assumption $\sum_i X_i\leqslant 1$ gives $$ \mathbb E\left[\left(\sum_{i=M}^N\mathbb E\left[X_i\mid\mathcal F_{i-1}\right]\right)^p\right]\leqslant p^{p+1}\mathbb E\left[ \sum_{i=M}^N X_i \right]. $$ By monotone convergence theorem, $\sum_{i=1}^\infty \mathbb E[X_i]$ converges hence the sequence $\left(\sum_{i=1}^n \mathbb E\left[X_i\mid\mathcal F_{i-1}\right]\right)$ is Cauchy in $L^p$.