UDC 517.512+519.21

MATHEMATICS

V. V. PETROV

ESTIMATING THE CLOSENESS OF FUNCTIONS OF BOUNDED VARIATION FROM THE CLOSENESS OF THEIR FOURIER–STIELTJES TRANSFORMS

(Presented by Academician Yu. V. Linnik on 1 XII 1969)

An important role in probability theory is played by Esseen’s inequalities ((^{1,2})), which make it possible to estimate the closeness of a nondecreasing bounded function (F(x)) and a function of bounded variation (G(x)) from the closeness of the corresponding Fourier–Stieltjes transforms on some finite interval. It is assumed here that either (G(x)) has a derivative uniformly bounded in (x), or (F(x)) is a purely discontinuous function such that the functions (F(x)) and (G(x)) may have discontinuities only at points (x=x_\nu) ((x_\nu0), with (|G'(x)|\le A) everywhere, except at the points (x=x_\nu). These results of Esseen can be encompassed by a single formulation. The following theorem is a generalization of Esseen’s theorems.

Theorem 1. Let (F(x)) be a nondecreasing function, and (G(x)) a function of bounded variation on the real line, (F(-\infty)=G(-\infty)), (F(+\infty)=G(+\infty)). Let

[
f(t)=\int_{-\infty}^{\infty} e^{itx}\,dF(x), \qquad
g(t)=\int_{-\infty}^{\infty} e^{itx}\,dG(x),
]

and let (T) be an arbitrary positive number. Then, for any number (b>1/2\pi), the inequality

[
\sup_x |F(x)-G(x)|
\le
b\int_{-T}^{T}\left|\frac{f(t)-g(t)}{t}\right|\,dt
+
bT\sup_x \int_{|y|\le c(b)/T}|G(x+y)-G(x)|\,dy,
\tag{1}
]

holds, where (c(b)) is a positive constant depending only on (b).

In inequality (1) one may take (c(b)) equal to the root of the equation

[
\int_0^{\,1/4\,c(b)} \frac{\sin^2 u}{u^2}\,du
=
\frac{\pi}{4}+\frac{1}{8b}.
]

In the special case when (F(x)) and (G(x)) are distribution functions, a similar result was obtained by A. C. Feinleib ((^3)). A uniform estimate of the difference between a distribution function (F(x)) and a certain function of bounded variation (G(x)) that is not a distribution function is of considerable interest for applications (for example, in the study of asymptotic expansions in limit theorems for sums of independent random variables).

If the conditions of Theorem 1 are satisfied and the function (G(x)) satisfies the following Lipschitz condition:

[
|G(x)-G(y)|\le K|x-y|^\alpha
]

for all (x) and (y) and for some positive constants (K) and (\alpha), then the second term on the right-hand side of inequality (1) may be replaced by (2bK(c(b))^{1+\alpha}(1+\alpha)^{-1}T^{-\alpha}).

We give one more immediate consequence of Theorem 1, which is also a generalization of Esseen’s theorem.

Theorem 2. Let (F(x)) be a nondecreasing function, (G(x)) a function of bounded variation, and let (f(t)) and (g(t)) be the corresponding Fourier–Stieltjes transforms; let (T) be an arbitrary positive number, and
[
F(-\infty)=G(-\infty), \qquad F(+\infty)=G(+\infty).
]
Suppose (|G'(x)| \le A) everywhere, except at points of discontinuity of the function (G(x)).

Then, for any number (b>1/2\pi), the inequality
[
\sup_x |F(x)-G(x)| \le
b \int_{-T}^{T} \left|\frac{f(t)-g(t)}{t}\right|\,dt
+ r(b)\frac{A}{T},
\tag{2}
]
holds, where (r(b)) is a positive constant depending only on (b).

In inequality (2) one may take (r(b)=bc^2(b)), where (c(b)) is the constant from Theorem 1.

Leningrad State University
named after A. A. Zhdanov

Received
1 XII 1969

REFERENCES

({}^{1}) C. G. Esseen, Acta Math., 77, 1 (1945).
({}^{2}) B. V. Gnedenko, A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Moscow–Leningrad, 1949.
({}^{3}) A. C. Feinleib, Izv. Acad. Sci. USSR, Ser. Math., 32, No. 4, 859 (1968).