$L^p$ submartingale convergence theorem

Question

In class today, we learned about the familiar $L^p$ martingale convergence theorem: For $p >1,$ if $X_n$ is a martingale with $\sup \mathbb{E}|X_n|^p <\infty$, then $X_n \rightarrow X$ a.s. and in $L^p$.

Comparing this result with many other convergence theorems, we find that is requires X to be a martingale instead of just a submartingale. I assume this must have some reason, i.e. this result does not hold for submartingales.

Intuitively, I would assume that this is caused by something related to a restriction to the nonnegativity of the submartingale, i.e. the result will not hold when $X_n$ is a negative submartingale. Is this true? Or does the theorem also hold for submartingales?

If it does not hold, could you give me a counterexample?

Answer 1

The $L^p$ martingale convergence theorem holds also true for non-negative submartingales. The proof relies on Doob's maximal inequality:

Let $(X_j)_{j \in \mathbb{N}}$ be a non-negative submartingale (or a martingale). Then $X_n^* := \sup_{j \leq n} |X_j|$ satisfies $$\|X_n^*\|_p \leq \frac{p}{p-1} \|X_n\|_p$$ for any $p>1$. Moreover, for $X_{\infty}^* := \sup_{j \geq 1} |X_j|$ we have $$\|X_{\infty}^*\|_p \leq \frac{p}{p-1} \sup_{j \geq 1} \|X_j\|_p. \tag{1}$$

For a proof see e.g. René Schilling: Measures, Integrals and Martingales, Theorem 19.12.

So let's prove the convergence theorem for non-negative submartingales:

Let $(X_j)_{j \in \mathbb{N}}$ be a non-negative submartingale which is bounded in $L^p$ for some $p>1$, i.e.

$$\sup_{j \geq 1} \|X_j\|_p <\infty \tag{2}.$$

Then $(X_j)_{j \in \mathbb{N}}$ is in particular bounded in $L^1$ and therefore, by a standard convergence theorem for submartingales, we have $X_j \to X$ almost surely for some random variable $X$. As $|X| \leq X_{\infty}^*$, we find by $(1)$ and $(2)$

$$|X-X_j| \leq 2 X_{\infty}^* \in L^p.$$

Consequently, the dominated convergence theorem proves

$$\|X-X_j\|_p \to 0 \qquad \text{as $j \to \infty$.}$$