UDC 519.21

MATHEMATICS

L. V. OSIPOV

ON THE ACCURACY OF THE APPROXIMATION OF THE DISTRIBUTION OF A SUM OF INDEPENDENT RANDOM VARIABLES TO THE NORMAL DISTRIBUTION

(Presented by Academician Yu. V. Linnik on 13 IV 1967)

1. Consider a sequence of independent identically distributed random variables \(X_1,\ldots,X_n,\ldots\) with common distribution function \(V(x)\), characteristic function \(v(t)\), and positive variance \(\sigma^2=DX_n<\infty\). Without loss of generality one may assume that \(\mathbf{E}X_n=0\). Denote by \(F_n(x)\) the distribution function of the normalized sum

\[ \frac{1}{\sigma\sqrt n}\sum_{j=1}^{n}X_j, \]

and by \(\Phi(x)\) the normal distribution function with zero mean and unit variance.

It is known that \(\sup_x |F_n(x)-\Phi(x)|\to 0\) as \(n\to\infty\). The asymptotic behavior of \(F_n(x)-\Phi(x)\) as \(n\to\infty\) has been studied by many authors. Under the assumption of the existence of the third moment \(a_3=\mathbf{E}X_n^3\), it follows from the known asymptotic expansions of the function \(F_n(x)\) \((^1)\) that

\[ \sup_x |F_n(x)-\Phi(x)|=n^{-1/2}(A+o(1))\qquad (n\to\infty), \tag{1} \]

where \(A=|a_3|/6\sqrt{2\pi}\sigma^3\) if \(X_n\) has a nonlattice distribution, and
\(A=|a_3|/6\sqrt{2\pi}\sigma^3+h/2\sqrt{2\pi}\sigma\) if \(X_n\) has a lattice distribution with maximal span \(h\).

We shall consider the case where \(\mathbf{E}|X_n|^3=\infty\). If, in addition, the condition \(\limsup_{|t|\to\infty}|v(t)|<1\) (Cramér’s condition (C)) is satisfied, then we obtain the relation
\(\sup_x |F_n(x)-\Phi(x)|\asymp \psi_{n,2}\), where the quantity \(\psi_{n,2}\) is a certain functional of the function \(nV(x\sigma\sqrt n)\) (Theorem 1)*. In the case of lattice distributions the relation

\[ \sup_x |F_n(x)-\Phi(x)|\asymp \left(\psi_{n,2}+\frac{1}{\sqrt n}\right) \]

holds (Theorem 2). In Theorems 3 and 4 analogous results are obtained for the remainder term in the asymptotic expansion of the function \(F_n(x)\).

The methods of the present paper are those of papers \((^1,^2)\).

1. We formulate the main results. Put

\[ \psi_{n,2}= \frac{1}{\sigma^2}\int_{|x|>\sigma\sqrt n}x^2\,dV(x) +\frac{1}{\sigma^3\sqrt n}\left|\int_{|x|\leq \sigma\sqrt n}x^3\,dV(x)\right| +\frac{1}{\sigma^4 n}\int_{|x|\leq \sigma\sqrt n}x^4\,dV(x). \]

Theorem 1. If \(\mathbf{E}|X_n|^3=\infty\) and \(\limsup_{|t|\to\infty}|v(t)|<1\), then

\[ \sup_x |F_n(x)-\Phi(x)|\asymp \psi_{n,2}. \tag{2} \]

* The relation \(a_n\asymp b_n\) means that the sequences \(a_n\) and \(b_n\) satisfy the relation
\(0<\liminf_{n\to\infty} a_n/b_n\leq \limsup_{n\to\infty} a_n/b_n<\infty\).

Theorem 2. If \(X_n\) has a lattice distribution, then

\[ \sup_x \left|F_n(x)-\Phi(x)\right| \preccurlyeq \left(\psi_{n,2}+1/\sqrt{n}\right). \tag{3} \]

Let us note that the assertion of Theorem 2 in the case when \(\mathbf E|X_n|^3<\infty\) follows from the stronger relation (1).

Suppose now that \(\mathbf E|X_n|^3<\infty\) for some integer \(k\geq 3\). Introduce the notation:

\[ \Lambda_{n,\nu} = \frac{1}{\sigma^\nu n^{(\nu-2)/2}} \int_{|x|>\sigma\sqrt n} x^\nu\,dV(x) \quad (\nu=1,\ldots,k), \]

\[ L_{n,\nu} = \frac{1}{\sigma^\nu n^{(\nu-2)/2}} \int_{|x|<\sigma\sqrt n} x^\nu\,dV(x) \quad (\nu=1,2,\ldots), \]

\[ \psi_{n,k}= \begin{cases} \Lambda_{n,k}+|L_{n,k+1}|+L_{n,k+2}, & \text{if } k \text{ is even},\\ \Lambda_{n,k-1}+|\Lambda_{n,k}|+L_{n,k+1}, & \text{if } k \text{ is odd}. \end{cases} \]

Theorem 3. Let \(\limsup_{|t|\to\infty}|v(t)|<1\), and let there exist an integer \(k\geq 3\) such that \(\mathbf E|X_n|^k<\infty\), \(\mathbf E|X_n|^{k+1}=\infty\). Then, for odd \(k\),

\[ \sup_x \left| F_n(x)-\Phi(x)- \sum_{\nu=1}^{k-2} \frac{P_\nu(-\Phi)}{n^{\nu/2}} \right| \preccurlyeq \psi_{n,k}, \]

and, for even \(k\),

\[ \sup_x \left| F_n(x)-\Phi(x)- \sum_{\nu=1}^{k-2} \frac{P_\nu(-\Phi)}{n^{\nu/2}} - \frac{P'_{k-1}(-\Phi)}{n^{(k-1)/2}} \right| \preccurlyeq \psi_{n,k}. \]

Here \(P_\nu(-\Phi)\) are the functions known in the theory of probabilistic asymptotic expansions (see \(({}^3)\)). Namely,

\[ P_\nu(-\Phi) = \sum \prod_{m=1}^{\nu} \frac{1}{r_m!} \left( \frac{\gamma_{m+2}}{\sigma^{m+2}(m+2)!} \right)^{r_m} W_{3r+\cdots+(\nu+2)r_\nu}(x), \tag{4} \]

where the summation is over all nonnegative integer solutions of the equation
\(r_1+2r_2+\cdots+\nu r_\nu=\nu\), \(\gamma_\nu\) is the cumulant of order \(\nu\) of the random variable \(X_n\),

\[ W_s(x)= -\frac{s!}{\sqrt{2\pi}}e^{-x^2/2} \sum_{m=0}^{[s/2]} \frac{(-1)^m x^{s-2m}}{m!(s-2m)!2^m}. \]

The cumulants of the random variable \(X_n\) are expressed in the following way through its moments \(\alpha_m=\mathbf E X_n^m\):

\[ \gamma_\nu = \nu! \sum (-1)^{r_1+\cdots+r_\nu-1} (r_1+\cdots+r_\nu-1)! \prod_{m=1}^{\nu} \frac{1}{r_m!} \left(\frac{\alpha_m}{m!}\right)^{r_m}, \tag{5} \]

where the summation is over all nonnegative integers \(r_1,r_2,\ldots,r_\nu\) satisfying the equation
\(r_1+2r_2+\cdots+\nu r_\nu=\nu\). Under the conditions of Theorem 3, \(\gamma_\nu\) are defined only for \(\nu\leq k\), and the functions \(P_\nu(-\Phi)\) only for \(\nu\leq k-2\). For large values of \(\nu\) we shall regard (4) and (5) as formal equalities. We set the function \(P'_{k-1}(-\Phi)\) equal to the sum (4) for \(\nu=k-1\), in which, from \(\gamma_{k+1}\), we omit the term containing \(\alpha_{k+1}\).

Let us formulate an analogous result for lattice distributions. Introduce the functions \(S_\nu(x)\), setting

\[ S_1(x)=\sum_{m=1}^{\infty}\frac{\sin 2\pi mx}{\pi m},\qquad S_2(x)=\sum_{m=1}^{\infty}\frac{\cos 2\pi mx}{2(\pi m)^2},\ldots \]

\[ \ldots,\quad S_{2l}(x)=\sum_{m=1}^{\infty}\frac{\cos 2\pi mx}{2^{2l-1}(\pi m)^{2l}},\qquad S_{2l+1}(x)\sum_{m=1}^{\infty}\frac{\sin 2\pi mx}{2^{2l}(\pi m)^{2l+1}},\ldots \]

Next, let

\[ \Pi_{n,2}(x)=\Phi(x),\qquad \Pi_{n,l}(x)=\Phi(x)+\sum_{\nu=1}^{l-2}\frac{P_\nu(-\Phi)}{n^{\nu/2}} \qquad (l=3,\ldots,k), \]

\[ \delta_\nu= \begin{cases} 1, & \text{if } \nu=4m+1,\ 4m+2,\\ -1, & \text{if } \nu=4m+3,\ 4m. \end{cases} \]

Theorem 4. Suppose \(X_n\) assumes, with positive probabilities, only values of the form \(a+sh\) \((s=0,\pm1,\ldots)\), where \(a\) is some real number and \(h\) is the maximal span of the distribution. If \(\mathbf E|X_n|^k<\infty\) for some integer \(k\geq 3\), then

\[ \sup_x\left|F_n(x)-\Pi_{n,k}(x)- \right. \]

\[ \left. -\sum_{\nu=1}^{k-2}\delta_\nu \left(\frac{h}{\sigma\sqrt n}\right)^\nu S_\nu\left(\frac{x\sigma\sqrt n}{h}-\frac{na}{h}+\left[\frac{na}{h}\right]\right) \frac{d^\nu}{dx^\nu}\Pi_{n,k-\nu}(x) \right| \succ \left(\psi_{n,k}+n^{-(k-1)/2}\right). \]

In the case where \(\mathbf E|X_n|^{k+1}<\infty\), Theorem 4 follows from the known estimate of the remainder term in the asymptotic expansion of \(F_n(x)\) \((^1)\).

3. We outline the proof of Theorems 1 and 2. By \(C_1,C_2,\ldots\) we shall denote certain positive constants not depending on \(n\). Put

\[ \Delta_n=\sup_x|F_n(x)-\Phi(x)|. \]

The proof of the upper estimate for \(\Delta_n\) in Theorem 1 is based on applying the theorem of C. G. Esseen \((^1,\text{ p. }32)\) with \(F(x)=F_n(x)\), \(G(x)=\Phi(x)\), \(T=n\), and on the following expansion of the logarithm of the function \(f_n(t)=v^n(t/\sigma\sqrt n)\):

\[ \ln f_n(t)=-t^2/2+(t^2+t^4)O(\psi_{n,2}+1/n) \qquad (n\to\infty) \]

uniformly in \(t\) in the interval \(|t|<\sqrt n\). The proof of the upper estimate for \(\Delta_n\) in Theorem 2 is carried out analogously; in this case we put \(T=C_1\sqrt n\).

In proving the lower estimates for \(\Delta_n\), the methods of the work of I. A. Ibragimov \((^2)\) are used. Consider bounded functions \(A(x)\) and \(B(t)\) such that

\[ \int |A(x)|\,dx<\infty,\qquad B(t)=\int e^{itx}A(x)\,dx. \]

It is not difficult to see that the functions \(F_n(x)-\Phi(x)\) and \((f_n(t)-e^{-t^2/2})/-it\) belong to \(L_2(-\infty,\infty)\) and constitute a Fourier-transform pair. According to Parseval’s identity, we have

\[ -2\pi\int (F_n(x)-\Phi(x))\overline{A(x)}\,dx = \int (f_n(t)-e^{-t^2/2})\frac{\overline{B(t)}}{it}\,dt. \]

Consequently,

\[ \Delta_n \succ C_2\left|\int (f_n(t)-e^{-t^2/2})\frac{\overline{B(t)}}{it}\,dt\right|. \tag{6} \]

Further, uniformly with respect to \(t\) in the interval \(|t|<T_n=\min(\psi_{n,2}^{-1/4};n^{1/4})\),

\[ f_n(t)-e^{-t^2/2} = e^{-t^2/2} \left[ n\left( v\left(\frac{t}{\sigma\sqrt n}\right)-1+\frac{t^2}{2n} \right) + (t^4+t^8)O\left(\psi_{n,2}^2+\frac1n\right) \right]. \]

Let \(|B(t)|<C_3e^{-t^2/4}\). Then the integral on the right-hand side of (6) is equal to

\[ n\int \frac{\overline{B(t)}}{it}e^{-t^2/2} \left( v\left(\frac{t}{\sigma\sqrt n}\right)-1+\frac{t^2}{2n} \right)\,dt + O\left(\psi_{n,2}^2+\frac1n\right) = \]

\[ = n\iint \frac{\overline{B(t)}}{it}e^{-t^2/2} \left( e^{itx/\sigma\sqrt n}-1+\frac{t^2x^2}{2\sigma^2n} \right)\,dt\,dV(x) + O\left(\psi_{n,2}^2+\frac1n\right) = \]

\[ = I+O\left(\psi_{n,2}^2+\frac1n\right). \]

Putting here \(B(t)=-ite^{-t^2/2}\), we find

\[ I=n\sqrt\pi\int\left(e^{-x^2/4\sigma^2n}-1+\frac{x^2}{4\sigma^2n}\right)\,dV(x) > C_4(\Lambda_{n,2}+L_{n,4}); \]

for \(B(t)=t^2e^{-t^2/2}\) we obtain

\[ |I| = n\sqrt\pi \left| \int \frac{x}{2\sigma\sqrt n}e^{-x^2/4\sigma^2n}\,dV(x) \right| > \sqrt\pi\left(\frac18|L_{n,3}|-\Lambda_{n,2}-L_{n,4}\right). \]

Hence it follows that

\[ \Delta_n>C_5\psi_{n,2}+O(1/n). \tag{7} \]

Under the conditions of Theorem 1, \(n\psi_{n,2}\to\infty\) as \(n\to\infty\), and, consequently, we have \(\Delta_n>C_6\psi_{n,2}\). From this (2) follows.

Suppose now that \(X_n\) has a lattice distribution with maximal span \(h\). Putting in (6)
\(B(t)=\exp[-\tfrac12(t+\frac{2\pi}{h}\sigma\sqrt n)^2]\), we easily find that

\[ \Delta_n\ge C_2 \left| \int_{|t|<T_n} \frac{f_n(t)e^{-t^2/2}}{t-\frac{2\pi}{h}\sigma\sqrt n} \,dt \right| + O\left(\psi_{n,2}+\frac1{\sqrt n}\right) > \frac{C_7}{\sqrt n} + o\left(\frac1{\sqrt n}\right). \]

The last inequality and inequality (7) complete the proof of relation (3).

I express my sincere gratitude to my adviser V. V. Petrov for his constant attention to this work.

Leningrad State University
named after A. A. Zhdanov

Received
6 IV 1967

REFERENCES

¹ C. G. Essen, Acta Math., 77, 1 (1945).
² I. A. Ibragimov, Theory of Probability and Its Applications, 11, 632 (1966).
³ V. V. Petrov, Vestnik Leningrad. Univ., No. 19, 150 (1962).