Approximate Central Limit Theorems

Abstract

We refine the classical Lindeberg–Feller central limit theorem by obtaining asymptotic bounds on the Kolmogorov distance, the Wasserstein distance, and the parameterized Prokhorov distances in terms of a Lindeberg index. We thus obtain more general approximate central limit theorems, which roughly state that the row-wise sums of a triangular array are approximately asymptotically normal if the array approximately satisfies Lindeberg’s condition. This allows us to continue to provide information in nonstandard settings in which the classical central limit theorem fails to hold. Stein’s method plays a key role in the development of this theory.

Appendix: Proof of Theorem 3.1

We follow [4], Sect. 2. We keep a continuously differentiable $h : \mathbb {R} \rightarrow [0,1]$, with bounded derivative, fixed, and let $f_h$ be its Stein transform defined by (6). Also, we put

$$\begin{aligned} \sigma _{n,k}^2 = \mathbb {E}[\xi _{n,k}^2]. \end{aligned}$$

The following lemma is easily verified. It can be found in, e.g., [1] (pp. 10–11).

Lemma 1

$f_h$ is twice continuously differentiable, has bounded first and second derivatives, and

$$\begin{aligned} \mathbb {E}\left[ h(\xi )\right] - h(x)= x f_h(x) - f_h^\prime (x). \end{aligned}$$

(31)

The following lemma can be found in [4] (Lemma 2.4). We give the proof for completeness.

Lemma 2

Put

$$\begin{aligned} \delta _{n,k} = f_h\left( \sum _{i \ne k} \xi _{n,i} + \xi _{n,k}\right) - f_{h}\left( \sum _{i \ne k} \xi _{n,i}\right) - \xi _{n,k} f^\prime _h\left( \sum _{i \ne k} \xi _{n,i}\right) \end{aligned}$$

and

$$\begin{aligned} \epsilon _{n,k} = f^\prime _h\left( \sum _{i \ne k} \xi _{n,i} + \xi _{n,k}\right) - f^\prime _{h}\left( \sum _{i \ne k} \xi _{n,i}\right) - \xi _{n,k} f^{\prime \prime }_h\left( \sum _{i \ne k}\xi _{n,i}\right) . \end{aligned}$$

Then

$$\begin{aligned}&\mathbb {E}\left[ \left( \sum _{k=1}^n \xi _{n,k}\right) f_h\left( \sum _{k=1}^n \xi _{n,k}\right) - f_h^\prime \left( \sum _{k=1}^n \xi _{n,k}\right) \right] \nonumber \\&\quad = \sum _{k=1}^n \mathbb {E}\left[ \xi _{n,k}\delta _{n,k}\right] - \sum _{k=1}^n \sigma _{n,k}^2 \mathbb {E}\left[ \epsilon _{n,k}\right] . \end{aligned}$$

(32)

Proof

Recalling that $\xi _{n,k}$ and $\sum _{i \ne k} \xi _{n,i}$ are independent, $\mathbb {E}\left[ \xi _{n,k}\right] = 0$, and $\sum _{k=1}^n \sigma _{n,k}^2 = 1$, we get

$$\begin{aligned}&\sum _{k=1}^n \mathbb {E}\left[ \xi _{n,k} \delta _{n,k}\right] - \sum _{k=1}^n \sigma _{n,k}^2 \mathbb {E}\left[ \epsilon _{n,k}\right] \\&\quad = \sum _{k=1}^n \mathbb {E}\left[ \xi _{n,k} f_{h}\left( \sum _{k = 1}^n \xi _{n,k}\right) \right] - \mathbb {E}\left[ \xi _{n,k} f_{h}\left( \sum _{i \ne k} \xi _{n,i}\right) \right] \\&\qquad - \sum _{k=1}^n \mathbb {E}\left[ \xi _{n,k}^2 f^\prime _h\left( \sum _{i \ne k} \xi _{n,i}\right) \right] - \sum _{k=1}^n \sigma _{n,k}^2 \mathbb {E}\left[ f_{h}^{\prime }\left( \sum _{k=1}^n \xi _{n,k}\right) \right] \\&\qquad + \sum _{k=1}^n \mathbb {E}\left[ \xi _{n,k}^2\right] \mathbb {E}\left[ f^\prime _{h}\left( \sum _{i \ne k} \xi _{n,i}\right) \right] + \sum _{k=1}^n \sigma _{n,k}^2 \mathbb {E}\left[ \xi _{n,k} f^{\prime \prime }_h\left( \sum _{i \ne k}\xi _{n,i}\right) \right] . \end{aligned}$$

The last expression further reduces to

$$\begin{aligned}&\mathbb {E}\left[ \left( \sum _{k=1}^n\xi _{n,k}\right) f_{h}\left( \sum _{k = 1}^n \xi _{n,k}\right) \right] - \mathbb {E}\left[ \xi _{n,k}\right] \mathbb {E}\left[ f_{h}\left( \sum _{i \ne k} \xi _{n,i}\right) \right] \\&\qquad - \sum _{k=1}^n \mathbb {E}\left[ \xi _{n,k}^2 f^\prime _h\left( \sum _{i \ne k} \xi _{n,i}\right) \right] - \mathbb {E}\left[ f_{h}^{\prime }\left( \sum _{k=1}^n \xi _{n,k}\right) \right] \\&\qquad + \sum _{k=1}^n \mathbb {E}\left[ \xi _{n,k}^2 f^\prime _{h}\left( \sum _{i \ne k} \xi _{n,i}\right) \right] + \sum _{k=1}^n \sigma _{n,k}^2 \mathbb {E}\left[ \xi _{n,k}\right] \mathbb {E}\left[ f^{\prime \prime }_h\left( \sum _{i \ne k}\xi _{n,i}\right) \right] , \end{aligned}$$

which is easily seen to equal

$$\begin{aligned} \mathbb {E}\left[ \left( \sum _{k=1}^n \xi _{n,k}\right) f_h\left( \sum _{k=1}^n \xi _{n,k}\right) - f_h^\prime \left( \sum _{k=1}^n \xi _{n,k}\right) \right] . \end{aligned}$$

This finishes the proof. $\square$

The following lemma is an application of Taylor’s theorem.

Lemma 3

For any $a, x \in \mathbb {R}$,

$$\begin{aligned}&\left| f_h(a + x) - f_h(a) - f_h^\prime (a) x \right| \nonumber \\&\quad \le \min \left\{ \left( \sup _{x_1,x_2 \in \mathbb {R}}\left| f_h^\prime (x_1) - f^{\prime }(x_2)\right| \right) \left| x\right| ,\frac{1}{2} \left\| f_h^{\prime \prime }\right\| _\infty x^2\right\} . \end{aligned}$$

(33)

We are now in a position to present a proof of Theorem 3.1.

Proof of Theorem 3.1

For n and $\epsilon > 0$, we have, by (31), (32), and (33),

$$\begin{aligned} \left| \mathbb {E}\left[ h\left( \xi \right) - h\left( \sum _{k=1}^{n}\xi _{n,k}\right) \right] \right|= & {} \left| \mathbb {E}\left[ \left( \sum _{k=1}^n \xi _{n,k}\right) f_h\left( \sum _{k=1}^n \xi _{n,k}\right) - f_h^\prime \left( \sum _{k=1}^n \xi _{n,k}\right) \right] \right| \\\le & {} \sum _{k=1}^n \mathbb {E}\left[ \left| \xi _{n,k}\delta _{n,k}\right| \right] + \sum _{k=1}^n \sigma _{n,k}^2 \mathbb {E}\left[ \left| \epsilon _{n,k}\right| \right] \\\le & {} \frac{1}{2}\left\| f_h^{\prime \prime }\right\| _\infty \sum _{k=1}^n \mathbb {E}\left[ \left| \xi _{n,k}\right| ^3 ; \left| \xi _{n,k}\right| <\epsilon \right] \\&+ \left( \sup _{x_1,x_2 \in \mathbb {R}} \left| f_h^\prime (x_1) - f_{h}^\prime (x_2)\right| \right) \sum _{k=1}^n \mathbb {E}\left[ \left| \xi _{n,k}\right| ^2;\left| \xi _{n,k}\right| \ge \epsilon \right] \\&+ \left( \sup _{x_1, x_2 \in \mathbb {R}} \left| f_h^{\prime \prime }(x_1) - f_h^{\prime \prime } (x_2)\right| \right) \sum _{k=1}^n\sigma _{n,k}^2 \mathbb {E}\left[ \left| \xi _{n,k}\right| \right] , \end{aligned}$$

which proves the desired result since $\sum _{k=1}^n \sigma _{n,k}^2 = 1$. $\square$