MATHEMATICS

I. A. Ibragimov

On the Rate of Convergence to the Normal Distribution

(Presented by Academician Yu. V. Linnik, 13 XI 1964)

1. Let

\[ \xi_1,\ \xi_2,\ldots,\xi_n,\ldots \]

be a sequence of independent identically distributed random variables with common distribution function (d.f.) \(F(x)\) and characteristic function (c.f.) \(f(t)\). We shall assume that the d.f. \(F(x)\) belongs to the domain of attraction of the normal law

\[ \Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-z^2/2}\,dz. \]

In this case the variables \(\xi_j\) have a finite first moment ((\(^{1}\), p. 192), and we shall assume that \(E\xi_j=0\). By \(F_n(x)\) we denote the d.f. of the sum

\[ S_n=\frac{\xi_1+\cdots+\xi_n}{B_n}-A_n, \]

normalized so that \(F_n(x)\to \Phi(x)\).

There are simple sufficient conditions guaranteeing that

\[ \sup_x |F_n(x)-\Phi(x)|=O(n^{-\delta/2}), \tag{1} \]

where \(0<\delta\le 1\) ((\(^{1}\), p. 215).

In Theorem 1 of the present note we give necessary and sufficient conditions for relation (1) to hold.

2. Theorem 1. In order that

\[ |F_n(x)-\Phi(x)|=O(n^{-\delta/2}), \qquad 0<\delta\le 1, \]

it is necessary and sufficient that the following conditions hold:

\[ \text{1) }\quad \sigma^2=\int x^2\,dF(x)<\infty; \]

\[ \text{2) }\quad \int_{|x|>z} x^2\,dF(x)=O(z^{-\delta}), \qquad z\to\infty. \]

In this case the normalizing constants necessarily have the form \(A_n=0\), \(B_n=\sigma\sqrt n\).

Remark. From condition 2) it follows that all moments of order less than \(2+\delta\) are finite.

It is easy to see that the d.f. \(F_n(x)\) for lattice distributions \(F(x)\) cannot be approximated by the continuous function \(\Phi(x)\) with an accuracy exceeding \(1/\sqrt n\). Therefore, in wishing to extend the results of Theorem 1 to exponents \(\delta>1\), we must impose some additional requirement on the d.f. \(F(x)\). As such a requirement we choose Cramér’s condition (C).

We shall say that the d.f. \(F(x)\) satisfies condition (C) if

\[ \overline{\lim}_{t\to\infty}|f(t)|<1. \tag{C} \]

Theorem 2. In order that

\[ |F_n(x)-\Phi(x)|=O(n^{-\delta/2}),\qquad 2k<\delta\le 2(k+1),\qquad k=0,1,\ldots, \]

it is necessary, and for distributions satisfying condition (C) sufficient, that the following conditions hold:

1) all integral moments of the d.f. \(F(x)\) of order less than \(\delta+2\) are finite and coincide with the corresponding moments of the normal distribution;

2)

\[ \int_{|x|>z} x^{2(k+1)}\,dF(x)=O(z^{2k-\delta}),\qquad \text{if } \delta<2(k+1), \]

\[ \int x^{2(k+2)}\,dF(x)<\infty,\qquad \text{if } \delta=2(k+1). \]

Corollary. If for all \(\delta>0\)

\[ |F_n(x)-\Phi(x)|=O(n^{-\delta/2}), \tag{2} \]

then the d.f. \(F(x)\) is normal.

Remark. For d.f.’s \(F(x)\) belonging to the domains of attraction of stable laws with exponent \(\alpha<2\), a superfast convergence to the limiting distribution similar to (2) is quite possible. Whatever stable d.f. \(G_\alpha(x)\) with exponent \(\alpha<2\) is given, there exists a d.f. \(F(x)\), not stable, for which

\[ |F_n(x)-G_\alpha(x)|=O(e^{-cn}),\qquad c>0. \]

1. We now give analogues of Theorem 1 corresponding to local limit theorems ((1), pp. 238–258). Let first the random variables \(\xi_j\) be lattice-valued with span \(h\). Without loss of generality one may suppose that the possible values of \(\xi_j\) are \(lh\), \(l=0,\pm1,\ldots\).

It follows from Theorem 1 that one may restrict oneself to the case \(\mathbf E\xi_j=0\), \(\mathbf D\xi_j=1\). Then \(A_n=0\), \(B_n=\sqrt n\). Put

\[ P_n(l)=\mathbf P\{\sqrt n S_n=lh\},\qquad l=0,\pm1,\ldots \]

By \(\varphi(x)=\dfrac{1}{\sqrt{2\pi}}e^{-x^2/2}\) we denote the normal density.

Theorem 3. In order that

\[ \sup_{-\infty<l<\infty}\left|\frac{\sqrt n}{h}P_n(l)-\varphi\left(\frac{lh}{\sqrt n}\right)\right| =O(n^{-\delta/2}), \]

it is necessary and sufficient that:

1) the d.f. \(F(x)\) satisfy conditions 1), 2) of Theorem 2;

2) the span of the distribution \(h\) be maximal.

Put \(p_n(x)=F_n'(x)\).

Theorem 4. In order that

\[ \sup_{-\infty<x<\infty}|p_n(x)-\varphi(x)|=O(n^{-\delta/2}),\qquad \delta>0, \]

it is necessary and sufficient that:

1) the d.f. \(F(x)\) satisfy conditions 1), 2) of Theorem 2;

2) there exist a number \(N\) such that the d.f. \(F_N(x)\) is absolutely continuous and

\[ \sup_x p_N(x)<\infty. \]

1. The proofs of Theorems 1—4 are based on similar ideas. We briefly outline the proof of Theorem 1, restricting ourselves to symmetric distributions \(F(x)\) and assuming, for simplicity, that already \(\sigma^2<\infty\). Then one may take \(\sigma=1\), \(B_n=\sqrt n\).

In a neighborhood of zero the characteristic function is
\(f(t)=\exp\{-\tfrac12 t^2(1+\gamma(t))\}\), where \(\gamma(t)=o(1)\), \(t\to0\).

Lemma. Under the assumptions made above, equality (1) holds if and only if

\[ \int_0^x t^2|\gamma(t)|\,dt=O(x^{3+\delta}),\qquad x\to0. \tag{3} \]

Proof. Necessity. In a neighborhood of zero the difference \(f(t)-e^{-t^2/2}\) can vanish only a finite number of times (if \(f(t)\not\equiv e^{-t^2/2}\)), and consequently, in some neighborhood of zero the function \(\gamma(t)\) has constant sign. Therefore, using Parseval’s equality, we have

\[ \frac{1}{\sqrt{2\pi}}\left|\int_{-\infty}^{\infty} [F_n(x)-\Phi(x)]\,x e^{-x^2/2}\,dx\right| = \left|\int_{-\infty}^{\infty} [f_n(t)-e^{-t^2/2}]e^{-t^2/2}\,dt\right| \ge \]

\[ \ge \frac{1}{e}\int_0^1 \left|1-e^{\frac12 t^2\gamma(t/\sqrt n)}\right|\,dt +O(e^{-\ln^2 n}), \]

\[ \int_0^{1/\sqrt n} t^2|\gamma(t)|\,dt = O\!\left(n^{-(3+\delta)/2}\right), \]

which is equivalent to (3).

The proof of the sufficiency of (3) is obtained by the usual methods connected with the application of Esseen’s theorem ((1), pp. 211—218).

Subsequently the proof of the theorem is continued as follows. It is proved that (3) is equivalent to the equality

\[ \int\left(\frac{\sin xu}{xu}-1+\frac{x^2u^2}{6}\right)\,dF(u) = O(x^{2+\delta}),\qquad x\to0, \]

which, as is not difficult to verify, is already equivalent to condition 2) of Theorem 1.

Leningrad State University
named after A. A. Zhdanov

Received
5 XI 1964

REFERENCES

1. B. V. Gnedenko, A. I. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Moscow—Leningrad, 1949.