UDC 519.214

MATHEMATICS

S. M. SADIKOVA

A DISTANCE BETWEEN DISTRIBUTIONS CONNECTED WITH THEIR VALUES ON CONVEX SETS

(Presented by Academician A. N. Kolmogorov, December 17, 1966)

In the paper \((^1)\) the author obtained an inequality (a two-dimensional analogue of Esseen’s inequality \((^2)\), § 39) that makes it possible to estimate, in terms of the difference of characteristic functions, the difference of the values \(P(A)-Q(A)\) of the corresponding distributions for all rectangles \(A\) with sides parallel to the coordinate axes. The purpose of the present note is to derive inequality (4), which gives an upper estimate for the distance
\[ \rho(P,Q)=\sup_A |P(A)-Q(A)| \]
(where the supremum is taken over all convex sets of the space \(R^s\)) in terms of characteristic functions. Let us note that a method for estimating the difference \(P(A)-Q(A)\) for balls was indicated in \((^3)\) (see also \((^4)\)).

The starting point in this and similar cases may be the following relation, easily derived from the inversion formula for Fourier integrals. Let \(\xi,\eta\), and \(\zeta\) be \(s\)-dimensional random vectors, \(\zeta\) independent of \(\xi\) and \(\eta\) and having an absolutely integrable characteristic function \(h(t)\). Then
\[ \Delta_A=Pr\{\xi+\zeta\in A\}-Pr\{\eta+\zeta\in A\} =\frac{1}{(2\pi)^s}\int_{R^s}\overline{\tau_A(t)}(f(t)-g(t))h(t)\,dt . \]

Here \(A\) is an arbitrary bounded Borel set;
\[ \tau_A(t)=\int_A e^{i(t,x)}\,dx; \]
\(f\) and \(g\) are the characteristic functions of the vectors \(\xi\) and \(\eta\), respectively.

Representing the quantity \((2\pi)^s\Delta_A\) as the sum \(I_1+I_2+I_3\), where
\[ I_1=\int_{|t|\le 1},\qquad I_2=\int_{1<|t|<T},\qquad I_3=\int_{|t|\ge T}, \]
and applying the Cauchy—Bunyakovsky inequality to each integral, we obtain
\[ \begin{aligned} (2\pi)^s|\Delta_A| &\le \left(\int_{|t|\le 1}|t|^2|\tau_A|^2\right)^{1/2}J_1 +\left(\int_{1<|t|<T}|\tau_A|^2\right)J_2 \\ &\quad +2\left(\int_{|t|\ge T}|\tau_A|^2\right)^{1/2}J_3, \end{aligned} \tag{1} \]
where
\[ J_1=\left(\int_{|t|\le 1}\frac{|f-g|^2}{|t|^2}\right)^{1/2},\qquad J_2=\left(\int_{1<|t|<T}|f-g|^2\right)^{1/2},\qquad J_3=\left(\int_{|t|\ge T}|h|^2\right)^{1/2}. \]

Let \(O_r\) denote the ball of radius \(r\) centered at the origin, and let
\[ \Delta_r=\sup_{A\subset O_r}|\Delta_A|, \]
where the supremum is taken over all convex \(A\subset O_r\). As is easy to see,
\[ \gamma_1=\sup_A|\Delta_A|\le 2\Delta_r+2Pr\{\eta+\zeta\in \overline{O}_r\}. \]

The quantity \(\Lambda_r\) can be estimated with the aid of inequality (1). Indeed, if \(A\) is a convex set lying in the ball \(O_r\), and \(S(A)\) is the magnitude of its surface, then

\[ S(A)\leq S(O_r)\leq 2\pi^{s/2}\Gamma^{-1}(s/2)r^{s-1}. \]

Therefore, taking into account the inequalities for the integrals of \(\tau_A(t)\) (formulas of the paper \((^1)\), p. 379), we obtain, for \(A\subset O_r\),

\[ |\Delta_A|\leq \frac{\sqrt{\lambda_s S(O_r)}}{(2\pi)^s} \left[ J_1+2\sqrt{2}J_2+\frac{4\sqrt{2}}{\sqrt{T}}J_3 \right], \]

where

\[ \lambda_s=\frac{(2\pi)^{s+1}}{4} \left(\int_0^\pi \sin^s\alpha\,d\alpha\right)^{-1}. \tag{2} \]

Denote the right-hand side of (2) by \(\Lambda_r^*\). Then

\[ \gamma_1\leq 2\Lambda_r^*+2Pr\{\eta+\zeta\in \overline{O}_r\}. \]

Here the constants \(r\) and \(T\) and the random variable \(\zeta\) are at our disposal. The lemma stated below serves for passing from \(\gamma_1\) to \(\gamma=\rho(P,Q)\). Before formulating it, introduce the following notation: \(A\) is a convex set; \(\dot A\) is its boundary; \((\dot A)^\delta\) is the \(\delta\)-neighborhood of \(A\); \(\omega_\eta(\delta)=\sup_A Pr\{\eta\in(\dot A)^\delta\}\). Let us note that, for any \(A\), the inequalities

\[ Pr\{\xi\in A\}\leq Pr\{\xi+\zeta\in A\cup(\dot A)^\delta\}+Pr\{|\zeta|\geq\delta\}, \tag{3} \]

\[ Pr\{\xi\in A\}\geq Pr\{\xi+\zeta\in A\setminus(\dot A)^\delta\}-Pr\{|\zeta|\geq\delta\}. \]

It is also easy to prove that \(\omega_{\eta+\zeta}(\delta)\leq \omega_\eta(\delta)\).

Lemma. The inequalities hold

\[ \gamma_1\leq \gamma,\qquad \gamma\leq \gamma_1+\omega_\eta(\delta)+2Pr\{|\zeta|\geq\delta\}. \]

We omit the proof of the lemma.

Let us now note that

\[ Pr\{\eta+\zeta\in\overline{O}_r\} \leq Pr\{(\eta+\zeta\in\overline{O}_r)\cap(|\zeta|<\delta)\}+Pr\{|\zeta|\geq\delta\}\leq \]

\[ \leq Pr\{|\eta|\geq r\}+\omega_\eta(\delta)+Pr\{|\zeta|\geq\delta\}. \]

Therefore

\[ \gamma=\rho(P,Q)\leq 2\Lambda_r^*+2Pr\{|\eta|\geq r\}+3\omega_\eta(\delta)+4Pr\{|\zeta|\geq\delta\}, \]

\[ \rho(P,Q)\leq C_s r^{(s-1)/2} \left[ J_1+2\sqrt{2}J_2+\frac{4\sqrt{2}}{\sqrt{T}}J_3 \right]+ \]

\[ +3\omega_\eta(\delta)+2Pr\{|\eta|\geq r\}+4Pr\{|\zeta|\geq\delta\}, \tag{4} \]

where \(C_s=\sqrt{\lambda_s S(O_1)}/(2\pi)^s\).

From Stirling’s formula it is not difficult to derive that

\[ C_s=\sqrt[4]{2}\,[s(s+1)]^{1/4} \left(\frac{e}{2\pi s}\right)^{s/4} e^{\theta/s+\theta'/12s},\qquad |\theta|,\ |\theta'|\leq 1. \]

The function \(h(t)\) and the constants \(r,\delta\), and \(T\) entering inequality (4) may be chosen depending on our aims.

We shall now show how the function \(\omega_\eta(\delta)\) can be estimated in the example of the normal distribution in \(R^s\) with density

\[ p_\eta(x)=\frac{1}{(2\pi)^{s/2}}e^{-|x|^2/2}. \]

Let \(A\) be an arbitrary convex set in \(R^s\). Without loss of generality, one may assume that \(A\) does not lie in any subspace of \(R^s\) of dimension \(s' < s\). Consider the system of concentric spheres \(O_n\), \(n = 0, 1, 2, \ldots\), with centers at the origin and such that the radius of \(O_n\) is equal to \(n\). The part of the surface \(\dot A\) lying between the \(n\)-th and \((n+1)\)-st spheres does not exceed its part lying inside the \((n+1)\)-st sphere. Suppose that \(\delta < 1\). Then the measure of the part of the \(\delta\)-neighborhood of \(\dot A\) lying between the \(n\)-th and \((n+1)\)-st spheres does not exceed the measure of the \(\delta\)-neighborhood of the \((n+1)\)-st sphere, i.e., in any case, does not exceed \(2\delta S(O_{n+2})\). The probability of hitting the mentioned part of the \(\delta\)-neighborhood of \(\dot A\) does not exceed

\[ \frac{1}{(2\pi)^{s/2}} e^{-(n-1)^2/2} 2\delta \frac{2\pi^{s/2}}{\Gamma(s/2)} (n+2)^{s-1} \]

(for \(n \geqslant 1\); for \(n=0\) the factor \(e^{-(n-1)^2/2}\) must be replaced by \(1\)). Therefore

\[ Pr\{\eta \in (\dot A)^\delta\} \leqslant \frac{4\delta}{2^{s/2}\Gamma(s/2)} \left[ 2^{s-1} + \sum_{n=1}^{\infty} e^{-(n-1)^2/2}(n+2)^{s-1} \right]. \]

From this it is not difficult to derive that for \(\delta < 1\)

\[ \omega_\eta(\delta) \leqslant \delta \frac{4}{2^{s/2}\Gamma(s/2)} \left[ 2^{s-1} + 3^{s-1} + 2^{3s/2-3}\Gamma(s/2) + 2^{3s-4}\frac{\sqrt{2\pi}}{2} \right]. \]

From the last inequality and Stirling’s formula it follows that

\[ \omega_\eta(\delta) \leqslant \delta C_{\mathrm{abs}} \cdot 2^s, \]

where \(C_{\mathrm{abs}}\) is an absolute constant.

Moscow Engineering Physics
Institute

Received
13 XII 1966

REFERENCES

1. S. M. Sadikova, Theory of Probability and Its Applications, 11, 3 (1966).
2. B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Moscow, 1949.
3. K. G. Esseen, Acta Math., 77, 1 (1944).
4. G. de Barra, Proc. Cambridge Phil. Soc., 59, 2 (1963).