Contributions to Central Limit Theory for Dependent Variables

Abstract

Considerations on stochastic models frequently involve sums of dependent random variables (rv's). In many such cases, it is worthwhile to know if asymptotic normality holds. If so, inference might be put on a nonparametric basis, or the asymptotic properties of a test might become more easily evaluated for certain alternatives. Of particular interest, for example, is the question of when a weakly stationary sequence of rv's possesses the central limit property, by which is meant that the sum $\sum^n_1 X_i$, suitably normed, is asymptotically normal in distribution. The feeling of many experimenters that the normal approximation is valid in situations "where a stationary process has been observed during a time interval long compared to time lags for which correlation is appreciable" has been discussed by Grenander and Rosenblatt ([10]; 181). (See Section 5 for definitions of stationarity.) The general class of sequences $\{X_i\}_{-\infty}^\infty$ considered in this paper is that whose members satisfy the variance condition \begin{equation*}\tag{1.1}\operatorname{Var} (\sum^{a+n}_{a+1} X_i) \sim nA^2\text{uniformly in} a (n \rightarrow \infty) (A^2 > 0).\end{equation*} Included in this class are the weakly stationary sequences for which the covariances $r_j$ have convergent sum $\sum_1^\infty r_j$. A familiar example is a sequence of mutually orthogonal rv's having common mean and common variance. As a mathematical convenience, it shall be assumed (without loss of generality) that the sequences $\{X_i\}$ under consideration satisfy $E(X_i) \equiv 0$, for the sequences $\{X_i\}$ and $\{X_i - E(X_i)\}$ are interchangeable as far as concerns the question of asymptotic normality under the assumption (1.1). As a practical convenience, it shall be assumed for each sequence $\{X_i\}$ that the absolute central moments $E|X_i - E(X_i)|^\nu$ are bounded uniformly in $i$ for some $\nu > 2$ ($\nu$ may depend upon the sequence). When (1.1) holds, this is a mild additional restriction and a typical criterion for verifying a Lindeberg restriction ([15]; 295). We shall therefore confine attention to sequences $\{X_i\}$ which satisfy the following basic assumptions (A): \begin{equation*}\tag{A1}E(X_i) \equiv 0,\end{equation*}\begin{equation*}\tag{A2}E(T_a^2) \sim A^2 \text{uniformly in} a (n \rightarrow \infty) (A^2 > 0),\end{equation*}\begin{equation*}\tag{A3}E|X_i|^{2+\delta} \leqq M (\text{for some} \delta > 0 \text{and} M < \infty),\end{equation*} where $T_a$ denotes the normed sum $n^{-\frac{1}{2}} \sum^{a+n}_{a+1} X_i$. Note that the formulations of (A2) and (A3) presuppose (A1). We shall say, under assumptions (A), that a sequence $\{ X_i\}$ has the central limit property (clp), or that $T_1$ is asymptotically normal (with mean zero and variance $A^2$), if \begin{equation*}\tag{1.2}P\{(nA^2)^{-\frac{1}{2}}\sum^n_1 X_i \leqq z\} \rightarrow (2\pi)^{-\frac{1}{2}} \int^z_{-\infty} e^{-\frac{1}{2}t{}^2}dt\quad (n \rightarrow \infty).\end{equation*} The assumptions (A) do not in general suffice for (1.2) to hold. (The reader is referred to Grenander and Rosenblatt ([10]; 180) for examples in which (1.2) does not hold under assumptions (A), one case being a certain strictly stationary sequence of uncorrelated rv's, another case being a certain bounded sequence of uncorrelated rv's.) It is well known, however, that in the case of independent $X_i$'s the assumptions (A) suffice for (1.2) to hold. It is desirable to know in what ways the assumption of independence may be relaxed, retaining assumptions (A), without sacrificing (1.2). Investigators have weakened considerably the moment requirements (A2) and (A3) while retaining strong restrictions on the dependence. However, in many situations of practical interest, assumptions (A) hold but neither strong dependence restrictions nor strong stationarity restrictions seem to apply. Thus it is important to have theorems which take advantage of assumptions (A) when they hold, in order to utilize conclusion (1.2) without recourse to severe additional assumptions. A basic theorem in this regard is offered in Section 4. It is unfortunate that the additional assumptions required, while relatively mild, are not particularly amenable to verification, with present theory. This difficulty is alleviated somewhat by the strong intuitive appeal of the conditions. The variety of ways in which the assumption of independence may be relaxed in itself poses a problem. It is difficult to compare the results of sundry investigations in central limit theory because of the ad hoc nature of the suppositions made in each instance. In Section 2 we explore the relationships among certain alternative dependence restrictions, some introduced in the present paper and some already in the literature. Conditions involving the moments of sums $\sum^{a+n}_{a+1} X_i$ are treated in detail in Section 3. The central limit theorems available for sums of dependent rv's embrace diverse areas of application. The results of Bernstein [2] and Loeve [14], [15] have limited applicability within the class of sequences satisfying assumptions (A). A result that is apropos is one of Hoeffding and Robbins [11] for $m$-dependent sequences (defined in Section 2). In addition to assumptions (A1) and (A3), their theorem requires that, defining $A_a^2 = E(X^2_{a+m}) + 2 \sum^m_1E(X_{a+m-j}X_{a+m},$ \begin{equation*}\tag{H}\lim_{n\rightarrow\infty} n^{-1}\sum^n_{i = 1} A^2_{a+i} = A^2 \text{exists uniformly in} a (n \rightarrow \infty).\end{equation*} Now it can be shown easily that conditions (A2) and (H) are equivalent in the case of an $m$-dependent sequence satisfying (A1) and (A3). Therefore, a formulation relevant to assumptions (A) is THEOREM 1.1 (Hoeffding-Robbins). If $\{ X_i\}$ is an $m$-dependent sequence satisfying assumptions (A), then it has the central limit property. In the case of a weakly stationary (with mean zero, say) $m$-dependent sequence, the assumptions of the theorem are satisfied except for (A3), which then is a mild additional restriction. For applications in which the existence of moments is not presupposed, e.g., strictly stationary sequences, Theorem 1.1 has been extended by Diananda [6], [7], [8] and Orey [16] in a series of results reducing the moment requirements while retaining the assumption of $m$-dependence. In the present paper the interest is in extensions relaxing the $m$-dependence assumption. A result of Ibragimov [12] in this regard implies THEOREM 1.2 (Ibragimov). If $\{ X_i\}$ is a strictly stationary sequence satisfying assumptions (A) and regularity condition (I), then it has the central limit property. (Condition (I) is defined in Section 2.) Other extensions under condition (I) but not involving stationarity assumptions are Corollary 4.1.3 and Theorem 7.2 below. See also Rosenblatt [17]. Other extensions for strictly stationary sequences, further reducing the dependence restrictions, appear in [12] and [13] and Sections 5 and 6 below. Section 2 is devoted to dependence restrictions. The restrictions (2.1), (2.2) and (2.3), later utilized in Theorem 4.1, are introduced and shown to be closely related to assumptions (A). Although conditional expectations are involved in (2.2) and (2.3), the restrictions are easily interpreted. It is found, under assumptions (A), that if (2.3) is sufficiently stringent, then (2.1) holds in a stringent form (Theorem 2.1). A link between regularity assumptions formulated in terms of joint probability distributions and those involving conditional expectations is established by Theorem 2.2 and corollaries. Implications of condition (I) are given in Theorem 2.3. Section 3 is devoted to the particular dependence restriction (2.1). Theorem 3.1 gives, under assumptions (A), a condition necessary and sufficient for (2.1) to hold in the most stringent form, (3.1). The remaining sections deal largely with central limit theorems. Section 4 obtains the basic result and its general implications. Sections 5, 6 and 7 exhibit particular results for weakly stationary sequences, sequences of martingale differences and bounded sequences. NOTATION AND CONVENTIONS. We shall denote by $\{ X_i\}^\infty_{-\infty}$ a sequence of rv's defined on a probability space. Let $\mathscr{M}_a ^b$ denote the $\sigma$-algebra generated by events of the form $\{(X_{i_1},\cdots, X_{i_k}) \varepsilon E\}$, where $a - 1 < i_1 < \cdots < i_k < b + 1$ and $E$ is a $k$-dimensional Borel set. We shall denote by $\mathscr{P}_a$ the $\sigma$-algebra $\mathscr{M}^a_{-\infty}$ of "past" events, i.e., generated by the rv's $\{ X_a, X_{a-1},\cdots\}$. Conditional expectation given a subfield $\mathscr{B}$ will be represented by $E(\cdot\mid\mathscr{B}),$ which is to be regarded as a function measurable ($\mathscr{B}$). All expectations will be assumed finite whenever expressed.