Rates of Weak Convergence and Asymptotic Expansions for Classical Central Limit Theorems

Abstract

Let $Q_n(n = 1,2, \cdots), Q$ be probability measures on the Borel $\sigma$-field of $R^k$. The sequence $\{Q_n\}$ converges weakly to $Q$ if for every real-valued, bounded, almost surely $(Q)$ continuous function $g$ on $R^k$ the convergence \begin{equation*}\tag{0.1} \lim_n \inf g dQ_n = \int g dQ\end{equation*} holds (cf. [9], Chapter 1). Such functions $g$ are called $Q$-continuous. If the indicator function $I_A$ of the set $A$ is $Q$-continuous, then $A$ is also called $Q$-continuous. If $\mathscr{F}$ is a class of $Q$-continuous functions $g$ over which the convergence (0.1) is uniform for every sequence $\{Q_n\}$ converging weakly to $Q$, then $\mathscr{F}$ is called a $Q$-uniformity class. A class of sets is called a $Q$-uniformity class if the indicator functions of the sets of this class form a $Q$-uniformity class of functions. A systematic study of $Q$-uniformity (in separable metric spaces) was initiated by Ranga Rao [21], who obtained a number of nice results. His studies were carried further in a very useful manner by Billingsley and Topsoe [10]. In this article the error of normal approximation $|\int g d(Q_n - \Phi)|$ is estimated for arbitrary $\Phi$-continuous $g, \Phi$ being the $k$-dimensional standard normal distribution and $Q_n$ the distribution of the appropriately normalised $n$th partial sum of a sequence of independent $k$-dimensional random vectors $\{X^{(r)}; r = 1,2,\cdots\}$. The classical central limit theorems assert weak convergence of $\{Q_n\}$ to $\Phi$ under certain moment conditions. It is shown here (Theorem 1, Section 3) that for an arbitrary real-valued, bounded, measurable $g$ on $R^k$ one has $(0.2) \quad |\int g d(Q_n - \Phi)| \leqq c(k, \delta)\omega_g(R^k) \rho^{3(1 + \delta)/(3 + \delta)}_{3 + \delta,n} n^{-\frac{1}{2}} + \int \omega_g(S(x, \varepsilon_n)) d\Phi(x)$, where $\delta$ is any positive number; $\rho_{3+\delta, n}$ is defined by (1.4), and \begin{equation*}\tag{0.3}\begin{align*} \omega_g(A) &= \sup \{|g(x) - g(y)|; x, y \epsilon A\}, S(x, \varepsilon) = \{y; |x - y| < \varepsilon\}, \\ \varepsilon_n &= c(k) \rho^{3/(3+\delta)}_{3 + \delta, n} n^{-\frac{1}{2}} \log n,\end{align*}\end{equation} $c(k), c(k, \delta)$ being positive constants depending only on their respective arguments. If $\varepsilon_n$ goes to zero as $n$ goes to infinity, then the right side of (0.2) goes to zero for every $\Phi$-continuous $g$. For the rest of this section let us assume that $\{\rho_{3+\delta,n\}}$ is bounded. By (0.2), if $\int \omega_g(S(x, \varepsilon)) d\Phi(x) = O(\varepsilon)$ as $\varepsilon$ goes to zero, then the error of approximation is $O(n^{-\frac{1}{2}} \log n)$. One may also use (0.2) to obtain uniform upper bounds for errors of approximation over arbitrary $\Phi$-uniformity classes. In particular, $(0.4) \quad \sup\{|\int g d(Q_n - \Phi)|; g \epsilon \mathscr{F}_1(\Phi; c, d, \varepsilon_0)\} = O(n^{-\frac{1}{2}}\log n)$, where $(0.5) \quad \mathscr{F}_1(\Phi; c, d, \varepsilon_0) = \{g; \omega_g(R^k) \leqq c, \int \omega_g(S(\cdot, \varepsilon)) d\Phi \leqq \varepsilon\text{for} 0 < \varepsilon \leqq \varepsilon_0\}$, $c, d, \varepsilon_0$ being arbitrary positive constants. For the largest translation-invariant subclass of this class a sharper bound $O(n^{-\frac{1}{2}})$, which is best possible, was obtained in [3], [5], (cf. [4]). A similar result (in the i.i.d. case) has been independently obtained by Von Bahr [22]. As applications one obtains precise bounds for many interesting classes of sets and functions. However, there are $\Phi$-continuous functions and $\Phi$-uniformity classes of functions for which the technique used in [5] or [22] is not effective. An example in Section 3 shows that there are Borel sets $A$ such that the upper bound for $|Q_n(A) - \Phi(A)|$ as provided by [5] (Theorem 1) is $O(1)$, while (0.2) provides the bound $O(n^{-\frac{1}{2}}\log n)$. A modification of the bound (0.2) when applied to $g = I_A$ for an arbitrary Borel set $A$ enables one to show that the Prokhorov distance between $Q_n$ and $\Phi$ is $O(n^{-\frac{1}{2}}\log n)$. It is not known whether the factor $\log n$ in the expression for $\varepsilon_n$ in (0.3) (and, hence, in (0.4) and in the estimate of Prokhorov's distance) may be dispensed with or not. However, under Cramer's condition (3.42) $\log n$ may be replaced by one. The remaining theorems are proved for the i.i.d. case, partly for the sake of simplicity and partly because of the non-availability in the existing literature of complete proofs for some of the expansions related to the characteristic function of $Q_n$ in the non-identically distributed case. Theorem 2 provides an asymptotic expansion for $\int g dQ_n$ with a remainder term which is $o(n^{-(s-2)/2})$ uniformly over all $g$ in $\mathscr{F}_1^\ast(\Phi; c, d, \varepsilon_0)$, the largest translation-invariant subclass of $\mathscr{F}_1(\Phi; c, d, \varepsilon_0)$, when $E|X^{(1)}|^s < \infty$ for some integer $s$ not smaller than three and the characteristic function of $X^{(1)}$ obeys Cramer's condition (3.42)'. Applications to the class $\mathscr{L}$ of all measurable convex sets, the class $L(c, d)$ (see (3.54)) of bounded Lipschitzian functions, etc., are immediate. Theorem 3 gives an asymptotic expansion for $\int g dQ_n$ for a very special class of functions $g$ when no restriction like (3.42)' is imposed. Theorem 4 provides some classes of functions $g$ (under varying restrictions on the distribution of $X^{(1)}$) for which the error of approximation $|\int g d(Q_n - \Phi)|$ is of the order $O(n^{-1})$. Section 1 introduces notation to be used throughout the article. Section 2 provides basic lemmas for proving the results (outlined above) of Section 3.