Martingale approximations for sums of stationary processes. (Q1879842)

Central limit theorems for sums of stationary and ergodic processes may be obtained by either one of two major methods. Under conditions of weak dependence such as mixing conditions, blocking techniques may to exploited effectively which usually reduce the problem to the behavior of sums of independent random variables and thus to classical limit theory. The second method originates from \textit{M. I. Gordin} [Sov. Math., Dokl. 10, 1174--1176 (1969); translation from Dokl. Akad. Nauk SSSR 188, 739--741 (1969; Zbl 0212.50005)] and uses martingale approximations in order to apply martingale central limit theory. The paper under review is a major contribution to the latter approach because it reveals the fact that the existence of a martingale approximation satisfying a Lindeberg-Feller condition is necessary and sufficient for a certain conditional form of asymptotic normality to hold. To be specific, let the sums \(S_n=\sum_{i=1}^ng(X_i)\) be additive functionals of a stationary and ergodic Markov chain \((X_n)_{n\in Z}\) with values in the state space \(\mathcal X\), where \(g(X_0)\) for the measurable function \(g:{\mathcal X}\rightarrow R\) has mean zero and finite variance. This framework covers the partial sums of any stationary and ergodic process \((\xi_n)_{n\in Z}\) by letting \(X_n=(\ldots,\xi_{n-1},\xi_n)\) and \(g(X_n)=\xi_n\). Set \(\sigma_n^2=E(S_n^2)\), and let \(P^x\) denote the regular conditional probability for \({\mathcal F}_\infty=\sigma(X_n:n\in Z)\) given \(X_0=x\). Moreover, let \(F_n(x;z)=P^x(S_n/\sigma_n\leq z)\) denote the conditional distribution functions of the normalized partial sums, whereas \(\Phi\) is the standard normal distribution function and \(\Delta\) the Lévy distance between two distribution functions. Then, asymptotic normality of \(S_n/\sigma_n\) given \(X_0\) is defined to mean \[ \lim_{n\to\infty}\int_{\mathcal X}\Delta(\Phi,F_n(x;\cdot))\,\pi(dx)=0, \tag{1} \] where \(\pi\) is the marginal distribution of \(X_0\). In one of the main results of the paper it is shown that \((1)\) holds if and only if there exists a doubly indexed family \(D_{nj}\) of random variables with the following properties: \begin{align*} &D_{nj},j\geq2,\text{is a martingale difference sequence with respect to the filtration }\tag{2}\\ &{\mathcal F}_j=\sigma(\ldots,X_{j-1},X_j)\text{ for every }n\geq1;\\ &\max_{k\leq n}E[(S_k-M_{nk})^2]=o(\sigma_n^2)\text{ as }n\to\infty, \text{ where }M_{nk}=D_{n1}+\ldots+D_{nk};\tag{3}\\ &\sigma_n^{-2}\sum_{k=1}^nE(D_{nk}^2\mid {\mathcal F}_{k-1})\rightarrow1 \text{ in probability as }n\to\infty\tag{4}\\ &\text{and}\\ &\sigma_n^{-2}\sum_{k=1}^nE(D_{nk}^21_{\{| D_{nk}| \geq\varepsilon\sigma_n\}}\mid {\mathcal F}_{k-1})\rightarrow0\text{ in probability as }n\to\infty \text{ for each }\varepsilon>0.\tag{5} \end{align*} Moreover, there is always a martingale approximation \((D_{nj})\) to \((S_n)\) such that the sequence \(D_{nj},j\geq1\), is stationary for every \(n\geq1\), and a nontrivial one, i.e. one which is independent of \(n\). Clearly, \((1)\) implies that \(S_n/\sigma_n\) is asymptotically standard normal, but it is stronger in general. This is demonstrated by an example from the theory of linear processes. A different example shows that \((1)\) or, equivalently, \((2)\)--\((5)\) are not strong enough to imply a functional central limit theorem, and slightly stronger sufficient conditions for the functional central limit theorem are given.