On Improvement of the Estimate for the Distance Between Sequential Sums of Independent Random Variables
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ON IMPROVEMENT OF THE ESTIMATE FOR THE DISTANCE BETWEEN SEQUENTIAL SUMS OF INDEPENDENT RANDOM VARIABLES Ia. S. Golikova∗
UDC 519.2
The aim of the paper is to improve previously obtained estimate of the constant in the inequality for the uniform distance between n- and (n + 1)-fold convolutions of one-dimensional probability distributions in the case where 0 is the q-quantile of a distribution F . Bibliography: 6 titles.
In the present paper, we give an estimate from above for the absolute constant in a special case of the Kolmogorov–Rogozin inequality and improve an estimate for the constant in the inequality for the uniform distance between n- and (n+1)-fold convolutions of one-dimensional probability distributions in the case where 0 is the q–quantile of a distribution F . Let ξ1 , ξ2 , . . . , ξn , . . . be independent random variables, Fk the distribution function of ξk , and Fk the distribution function of symmetrized random variable ξk . Estimates from above n Fk , τ with of the concentration function for the convolution of the distributions Q k=1
Q(F, τ ) = sup F {[x, x + τ ]} were obtained by Nagaev and Khodzhabagyan [1]. x∈R
Let lk and mk be positive numbers (k = 1, . . . , n). Set n n 2 2 σk (lk ) = x2 dFk (x), Bn = k=1
γn =
n
γk (lk ) =
k=1
k=1|x|≤l k n
|x|3 dFk (x).
(1)
(2)
k=1|x|≤l k
Theorem 1. For any nonnegative τ ,
√ n −1/2
τ γn τ 2π −2 > mk ) Fk , τ ≤ min c1 + c2 3 , c3 1+ P(|ξ| , (3) Q Bn Bn mk k=1 k=1 √ n n −1/2 τ γk (lk ) −2 τ 2π −2 > lk ) + c2 3 Fk , τ ≤ c1 + c3 1 + P(|ξ| , (4) Q σk (lk ) σn (lk ) lk n
k=1
k=1
where c4 ≤ 2.128815, c2 ≤ 6.6383125, and c3 ≤ e−3/8 ≈ 0.687289. Theorem 2. For any τ ≥ 0, Q
n k=1
Fk , τ ≤
n
−2
(c4 τ + c5 lk )
k=1
σk2 (lk )
√ −1/2 τ 2π −2 > lk ) + c3 1 + P(|ξ| , lk
(5)
where c4 ≤ 2.34584 and c5 ≤ 4.86119. In the context of the present paper, it is of fundamental importance that all the constants ci in Theorems 1, 2 are better than the constants obtained by Siegel in [2]. ∗
St.Petersburg State University, St.Petersburg, Russia; Baltic State Technical University “VOENMEH”, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 118–123. Original article submitted November 28, 2018. 74 1072-3374/20/2511-0074 ©2020 Springer Science+Business Media, LLC
For i.i.d. random variables ξk with distribution F , a special case of Kolmogorov–Rogozin inequality with improved constant can be obtained from Theorem 2 by simple transformations of inequality (5). Corollary 1. Let ξ1 , ξ2 , . . . , ξn , . . . be i.i.d. random variables with distribution F . Then c0 , (6) Q(F n , τ ) ≤ n (1 − Q(F, τ )) where c0 =
√ 1+2 2π 3/8 e
≈ 4.132847.
To prove this corollary, it suffices to apply Theorem 2 for independent identically distributed random variables and take l = τ2 . In this case, Bn and γn from (1), (2) can be represented in the following form: Bn2 =n σk2 (l) = n x2 dF(x), |x|≤l
γn =n γ(l) = n
|x|3 dF
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