Estimates for Closeness of Convolutions of Probability Distributions on Convex Polyhedra
- PDF / 177,658 Bytes
- 7 Pages / 594 x 792 pts Page_size
- 23 Downloads / 193 Views
ESTIMATES FOR CLOSENESS OF CONVOLUTIONS OF PROBABILITY DISTRIBUTIONS ON CONVEX POLYHEDRA F. G¨ otze∗ and A. Yu. Zaitsev†
UDC 519.2
The aim of the present work is to show that previously obtained results on approximation of the distributions of sums of independent summands by the accompanying compound Poisson laws and the estimates of closeness of the sequential convolutions of multidimensional distributions are transferred to the estimates for closeness of the convolutions of probability distributions on convex polyhedra. Bibliography: 14 titles.
Let us first introduce some notation. Let Fd denote the set of probability distributions defined on the Borel σ-algebra of subsets of the Euclidean space Rd , and let L(ξ) ∈ Fd be the distribution of a d-dimensional random vector ξ. Let Fsd ⊂ Fd be the set of symmetric distributions. For F ∈ Fd , we denote the corresponding characteristic and distribution functions by F (t), t ∈ Rd , and F (x) = F {(−∞, x1 ] × · · · × (−∞, xd ]}, x = (x1 , . . . , xd ) ∈ Rd , respectively. The uniform Kolmogorov distance is defined by ρ(G, H) = sup G(x) − H(x), G, H ∈ Fd . x∈Rd
We denote by c and c( · ) not necessarily equal positive absolute constants and quantities depending only on the arguments in parentheses. For 0 ≤ α ≤ 2, we set (α) (1) Fd = F ∈ Fsd : F(t) ≥ −1 + α, for all t ∈ Rd , F+ d = Fd . The products and powers of measures are understood in the convolution sense: GH = G ∗ H, H m = H m∗ , and H 0 = E = E0 , where Ex is the distribution concentrated at x ∈ Rd . A n Fi is the accompanying compound natural approximating infinitely divisible distribution for Poisson distribution
n i=1
i=1
e(Fi ), where −1
e(H) = e
∞ Hk k=0
and, more generally, e(αH) = e−α
k!
,
∞ αk H k k=0
k!
H ∈ Fd ,
,
α > 0.
(1)
It is well known that the distribution e(αH) is infinitely divisible. Arak [1] showed that if F is a symmetric one-dimensional distribution with nonnegative characteristic function for all t ∈ R, then ρ(F n , e(nF )) ≤ c n−1 ,
(2)
He introduced and developed the so-called method of triangular functions (see [2, Chap. 3, Secs. 2–4]). ∗
Universit¨ at Bielefeld, Germany, e-mail: [email protected].
†
St.Petersburg Department of Steklov Mathematical Institute, and St.Petersburg State University, St.Petersburg, Russia, e-mail: [email protected].
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 474, 2018, pp. 108–117. Original article submitted November 19, 2018. 1072-3374/20/2511-0067 ©2020 Springer Science+Business Media, LLC 67
Zaitsev [8] applied the Arak methods while proving inequality (2) (see [2, Chap. 5, Secs. 2, 5–7]). Later, he managed to modify these methods by adapting them to the multidimensional case (see [9–13]). In particular, a multidimensional analog of inequality (2) was obtained in [12]. The method of triangular functions and its generalizations allowed to find several bounds of the type ρ(G, H) ≤ c(d) ε, (3) where 0 < ε < 1 is small, G, H ∈ Fd , and the inequalities − H(t) ≤ cε sup G(t)
(4)
t∈Rd
are
Data Loading...
