Strong Law of Large Numbers for martingale difference sequence

Question

Could anyone provide a reference for the following theorem?

Theorem. Suppose that $\{X_k\}$ is a martingale difference sequence, that is to say $\mathbb{E}(|X_k|) < \infty$ and $\mathbb{E}(X_k \mathrel{|} X_1, ..., X_{k-1}) = 0$ for each $k \ge 1$. If $\sum_{k=1}^\infty \mathbb{E}(X_k^2)/k^2 < \infty$, then $n^{-1}\sum_{k=1}^{n} X_k \to 0$ almost surely.

N.B.

1. Instead of $\sum_{k=1}^\infty \mathbb{E}(X_k^2)/k^2 < \infty$, the following stronger assumption is sufficient for my application: $\sup_{k\ge 1}\|X_k\|_\infty < \infty$.

2. This theorem can be found (with proofs) at, e.g.,

   Law of Large Numbers for Martingales

   but no reference is given. I would like to apply this theorem in a paper (without repeating its proof) and hence a reference is needed.

3. The following post

   Law of Large Numbers for Martingale Difference Sequences (in probability)

   suggests Chow 1971, which establishes the convergence under the (weaker) assumption that $\{X_k\}$ is uniformly integrable. This is nice. However, a reference presenting the above-mentioned theorem would be better.

Answer 1

You can directly apply the theorem in Chow 1967, which states the following.

Let $Y_n = X_1 + \dots + X_n$ be a martingale such that there exists $\alpha \ge 1$ such that $$ \sum_{k=1}^\infty \mathbb{E}(|X_k|^{2\alpha}) / k^{1+\alpha} \;<\; \infty, $$ then we have $$\lim_{n\to\infty} Y_n / n = 0 \quad \text{a.s.}$$

Answer 2

Define $T_n:=\sum_{k=1}^n {X_k\over k}$, $n=1,2,\ldots$. This is a martingale, and $$ \Bbb E[T_n^2]=\sum_{k=1}^n{\Bbb E[X_k^2]\over k^2}\le \sum_{k=1}^\infty{\Bbb E[X_k^2]\over k^2}<\infty. $$ That is, $(T_n)$ is an $L^2$-bounded martingale, hence a.s. convergent. Now apply Kronecker's lemma to conclude that $\lim_n{1\over n}\sum_{k=1}^n X_k=0$ a.s.

Answer 3

This is an application of Theorem 2.18 page 35 in Martingale Limit Theory and Its Application Hall, P.; Heyde, C. C.. Indeed, this result says that if $(X_i)$ is a martingale difference sequence and $(U_n)$ a nondecreasing sequence of positive random variables such that $U_n$ is $\mathcal F_{n-1}$-measurable, then $U_n^{-1}\sum_{i=1}^n X_i$ converges almost surely to $0$ on the set $\{U_n\to\infty\}\cap \left\{\sum_{i=1}^\infty U_i^{-p}\mathbb E\left[\lvert X_i \rvert^p\mid\mathcal F_{i-1}\right]<\infty\right\}$.

Apply this with $p=2$ and $U_n=n$. Note that the non-negative random variable $\sum_{i=1}^\infty U_i^{-p}\mathbb E\left[\lvert X_i \rvert^p\mid\mathcal F_{i-1}\right]= \sum_{i=1}^\infty i^{-2}\mathbb E\left[ X_i ^2\mid\mathcal F_{i-1}\right]$ is almost surely finite, since by assumption and the monotone convergence theorem, its expectation is finite.