I am concerned with CLTs for m.d.a. A famous result is due to McLeish (https://projecteuclid.org/download/pdf_1/euclid.aop/1176996608). The notes from Sethuraman (http://math.arizona.edu/~sethuram/notes/wi_mart1.pdf) are very helpful to understand the proof. I have two questions:
- I have a problem with their conditions on the m.d.a.
Let $(X_{ni})_{i=1}^{k_n}$ be a m.d.a.
McLeish's conditions: (i) $\max_{i\le k_n} |X_{ni}|$ converges in probability to $0$ (as $n\to\infty$) (ii) $\max_{i\le k_n} |X_{ni}|$ is uniformly bounded in $L^2$ and (iii) $\sum_{i\le k_n}X_{ni}^2$ converges is probability to 1.
Sethuramans conditions: (i) $\max_{i\le k_n} |X_{ni}|$ converges in $L^1$ to $0$ and (iii) from McLeish.
I don't see why (i) and (ii) from McLeish is equivalent(?) to (i) from Sethuraman, although the proof is basically the same and Sethurman is always refering to McLeish's paper.
- In Thm3 Sethuraman looks at a special case (ergodic and stationary sequence) and concludes that condition (ii) (=(iii) from McLeish) follows from the ergodicity of the sequence. By the ergodic theorem I know (using his notations) that $\frac{1}{n}\sum_{i=1}^{n}Z_i\to E(Z_1)$ a.s., but can I therefore conclude $\frac{1}{n}\sum_{i=1}^{n}Z_i^2\to E(Z_1^2)$ a.s.?
1.
$$\require{cancel} \left. \begin{array}{l} \text{pr-}\lim\limits_{n\to\infty}\max_{i\le k_n}|X_{ni}|=0,\\ \sup\limits_{n}\sup\limits_{i\le k_n}\mathsf{E}[X_{ni}|^2\le C<\infty.\end{array} \right\} \stackrel{\implies}{\cancel{\impliedby}} \lim_{n\to\infty}\mathsf{E}[\max_{i\le k_n}|X_{ni}|]=0,$$ since $\mathsf{E}[X^2_{ni}]$ deals with the second order moment.
2. Let $\{Z_n,n\in\mathbb{N}_+\}$ be a ergodic stationary sequence, then for any nonegative Borel-measurable function $f$, $$\lim_{n\to\infty}\frac1n\sum_{i=1}^nf(Z_i)=\mathsf{E}[f(Z_1)]. \qquad\text{a.s.}$$ cf. O. Kallenberg, Foundations of Modern Probability, 2ed. Springer Verlag, 2001. Th.10.6, p.181. Taking $f(x)=x^2$ give what you want.