UDC 519.21

MATHEMATICS

A. BIKYALIS

ON REMAINDER TERMS

IN MULTIDIMENSIONAL LIMIT THEOREMS

(Presented by Academician Yu. V. Linnik, 30 IX 1965)

At the present time several estimates are known for the rate of convergence of the probability function \(P_n(A)\) of the normalized sum of independent identically distributed \(k\)-dimensional random vectors to the probability function \(\Phi(A)\) of the \(k\)-dimensional normal law. When the set \(A\) is a \(k\)-dimensional interval, the best estimate is due to G. Bergström \((^1)\): if the summands have finite third moments, then

\[ P_n(A)-\Phi(A)=O\left(\frac{1}{\sqrt n}\right). \]

When \(A\) belongs to the class of convex or Borel sets, it is known only that the remainder term has the form \(O(n^{-1/2}\ln^\alpha n)\), \(\alpha>0\) (see \((^2,^3)\)). The natural question arises: under the same conditions, is it possible to get rid of the factor \(\ln^\alpha n\)? The first two theorems of the present paper are devoted to this question.

Consider a sequence of independent identically distributed \(k\)-dimensional random vectors \(\vec{\xi}^{(1)}, \vec{\xi}^{(2)}, \ldots, \vec{\xi}^{(n)}\) of Euclidean space \(R_k\). Suppose that \(\vec{\xi}^{(1)}\) has finite moments of order \(s\) \((s>0)\). Moreover, let \(M\vec{\xi}^{(1)}=0\).

We shall denote by \((\mathbf{x},\mathbf{y})\) the scalar product of vectors \(\mathbf{x},\mathbf{y}\in R_k\); by \(\mathfrak A\) the class of all Borel sets in \(R_k\); \(P_n(\mathbf{m})=P\{\vec{\xi}^{(1)}+\cdots+\vec{\xi}^{(n)}=\mathbf{m}\}\); \(P_n(A)\) the probability function of the sum

\[ S^{(n)}=\frac{1}{\sqrt n}\sum_{j=1}^n \vec{\xi}^{(j)}; \]

\(V\) the covariance matrix of the vector \(\vec{\xi}^{(1)}\); \(\lambda_1^{\nu_1}\lambda_2^{\nu_2}\ldots\lambda_k^{\nu_k}\) the semi-invariants of order \((\nu_1,\nu_2,\ldots,\nu_k)\) of the random vector \(\vec{\xi}^{(1)}\). Following R. Rao \((^2)\), we define \(\widetilde P_j(\vec{\omega})\) by means of the formal identity

\[ \exp\left\{\sum_{j=3}^{\infty}\frac{(\vec{\lambda},\vec{\omega})^j}{j!} \left(\frac{1}{\sqrt n}\right)^{j-2}\right\} = \sum_{j=0}^{\infty}\left(\frac{1}{\sqrt n}\right)^j \widetilde P_j(\vec{\omega}), \]

and \(P_j(-\varphi)(\mathbf{x})\) and \(P_j(-\Phi)(\mathbf{x})\) by the equalities

\[ P_j(-\varphi)(\mathbf{x}) = \frac{1}{(2\pi)^k} \int_{\hat R_k} \widetilde P_j(i\mathbf{t})e^{-i(\mathbf{t},\mathbf{x})-\frac12 \mathbf{t}V\mathbf{t}'}\,d\mathbf{t}, \]

\[ P_j(-\Phi)(\mathbf{x}) = \int_{\substack{y_1\le x_1\\ \cdots\\ y_k\le x_k}} P_j(-\varphi)(\mathbf{y})\,d\mathbf{y}. \]

Theorem 1. If the distribution function of the vector $\vec{\xi}^{(1)}$ has an absolutely continuous component, then

\[ \sup_{A\in \mathfrak A}\left|P_n(A)-\sum_{j=0}^{s-2} n^{-j/2}\int_A dP_j(-\Phi)\right|=o\left(n^{-(s-2)/2}\right). \]

Theorem 2. If $\vec{\xi}^{(1)}$ has the identity covariance matrix and takes values in the lattice $L=\{\mathbf m\}$, $\mathbf m=(m_1,m_2,\ldots,m_k)$; $m_1,m_2,\ldots,m_k=0,\pm1,\ldots$, with maximal span of the distribution equal to 1, then for all $k$, uniformly in $A\in\mathfrak A$,

\[ P_n(A)= \sum_{\mathbf m/\sqrt n\in A}\sum_{j=0}^{s-2} n^{-(j+k)/2}P_j(-\varphi)\left(\frac{\mathbf m}{\sqrt n}\right) +o\left(n^{-(s-2)/2}\right). \tag{1} \]

In the case where $\vec{\xi}^{(1)}$ has finite third moments, from (1), by means of the multidimensional Euler–Maclaurin summation formula, we obtain that for all $k$, uniformly in $A\in\mathfrak A$,

\[ P_n(A)=\int_A dQ(\mathbf x)+\frac{1}{\sqrt n}\int_A dP_1(-\Phi) +o\left(\frac{1}{\sqrt n}\right). \]

In particular, uniformly in $\mathbf x\in R_k$,

\[ F_n(\mathbf x)=\Phi(\mathbf x)+\frac{1}{\sqrt n}P_1(-\Phi)(\mathbf x) -\sum_{j=1}^k S(x_j\sqrt n)\frac{\partial\Phi(\mathbf x)}{\partial x_j} +o\left(\frac{1}{\sqrt n}\right). \]

Here $F_n(\mathbf x)$ is the distribution function of the sum $S^{(n)}$; $\Phi(\mathbf x)$ is the distribution function of the $k$-dimensional normal law with zero vector of mathematical expectations and identity covariance matrix,

\[ Q(\mathbf x)=\prod_{j=1}^k \left[1-\frac{1}{\sqrt n}S(x_j\sqrt n)\frac{\partial}{\partial x_j}\right]\Phi(\mathbf x) \]

and $S(u)=[u]-u+\tfrac12$, where $[u]$ is the integer part of $u$.

Theorems 1 and 2 for all $k<2s$ are proved by the method of characteristic functions, and after this, with the help of the inequalities obtained, the proofs of the theorems for arbitrary $k$ are completed by the method of truncation of random vectors.

Theorem 3. Under the conditions of Theorem 2, for all $k<2s$,

\[ \sum_{\mathbf m\in L}\left|P_n(\mathbf m)- \sum_{j=0}^{s-2} n^{-(j+k)/2}P_j(-\varphi) \left(\frac{\mathbf m}{\sqrt n}\right)\right| =o\left(n^{-(s-2)/2}\right). \tag{2} \]

Let us note that relation (2) with a remainder term $O\left(n^{-(s-2)/2}\ln^{k/2}n\right)$ was obtained in (²).

Theorem 4. Under the conditions of Theorem 2, for all $k$,

\[ \sum_{\mathbf m\in L}\left|\frac{\mathbf m}{\sqrt n}\right|^{2\nu} \left|P_n(\mathbf m)- \sum_{j=0}^{s-2} n^{-(j+k)/2}P_j(-\varphi) \left(\frac{\mathbf m}{\sqrt n}\right)\right|^2 =o\left(n^{-s-k/2+2}\right) \]

for $\nu=0,1,\ldots,s$.

Theorems 3 and 4 are proved by the method of characteristic functions.

Vilnius State University
named after V. Kapsukas

Received
27 IX 1965

REFERENCES

¹ H. Bergstrom, Skand. Aktuar., 33, 37 (1949).
² R. R. Rao, Bull. Am. Math. Soc., 67, 359 (1961).
³ M. K. Khalikov, Izv. AN UzSSR, ser. phys.-math., No. 2, 3 (1959).