On the rate of convergence in the global central limit theorem for random sums of uniformly strong mixing random variabl
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Lithuanian Mathematical Journal
On the rate of convergence in the global central limit theorem for random sums of uniformly strong mixing random variables Jonas Kazys Sunklodas Faculty of Mathematics and Informatics, Vilnius University, Akademijos 4, LT-08663 Vilnius, Lithuania (e-mail: [email protected]) Received June 20, 2019; revised January 12, 2020
∞ Abstract. We present upper bounds of the integral −∞ |x|l |P{Z N < x}−Φ(x)| dx for 0 l 1+δ, where 0 < δ 1, 2 is the normalized random sum with ES 2 > 0 Φ(x) is a standard normal distribution function, and ZN = SN / ESN N (SN = X1 + · · · + XN ) of centered random variables X1 , X2 , . . . satisfying the uniformly strong mixing condition. The number of summands N is a nonnegative integer-valued random variable independent of X1 , X2 , . . . . MSC: 60F05 Keywords: global central limit theorem, random sum, normal approximation, uniformly strong mixing random variables, τ -shifted distributions
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Introduction and main results
Let X1 , X2 , . . . be a sequence of real centered random variables (r.v.s). For a b, we denote by Fab the σ -algebra of events generated by r.v.s Xa , Xa+1 , . . . , Xb . As usual, R is the real line, N = {1, 2, . . . }, N0 = {0, 1, 2, . . . }, and 1A is the indicator of an event A. We consider the weak dependence condition defined between the “past” and “future” in terms of the uniformly strong mixing coefficient ϕ(τ ) introduced by Ibragimov (1959): We say that a sequence of r.v.s X1 , X2 , . . . satisfies the uniformly strong mixing (u.s.m.) condition (or the ϕ-mixing condition) with the u.s.m. coefficient ϕ(τ ) if ϕ(τ ) = sup
sup
t∈N A∈F , B∈F P(A)>0 t 1
∞ t+τ
|P(AB) − P(A)P(B)| → 0 τ →∞ P(A)
(1.1)
(see [4] or [5]). In what follows, Φ(x) is the standard normal distribution function. By C(·) with an index or without it we denote a positive finite factor depending only on the quantities indicated in the parentheses (not necessarily the same at different occurrences). c 2020 Springer Science+Business Media, LLC 0363-1672/20/6002-0001
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J.K. Sunklodas
Recall the following result for sums with a fixed number n of random summands satisfying the u.s.m. condition (1.1), which will be used to prove the corresponding results for random sums. Theorem A. (See [15, Cor. 3].) Let a sequence of r.v.s X1 , X2 , . . . with EXi = 0 and E|Xi |2+δ < ∞, where 0 < δ 1, i = 1, . . . , n, satisfy the u.s.m. condition (1.1) with coefficient ϕ(τ ) Ke−μτ , where 0 < K < ∞ and μ > 0 are constants. Denote Sn Zn = , ESn2
Sn =
n
∞ Xi ,
Il,n =
i=1
|x|l P{Zn < x} − Φ(x)dx,
−∞
λl,n = E|Zn |l − E|Y |l ,
Lr,n =
n 1 E|Xi |r , (ESn2 )r/2 i=1
where ESn2 > 0, and Y is a standard normal r.v. Then Il,n C0 L2+δ,n ln1+δ (1 + n)
(1.2)
if (i) 0 l 1 or (ii) 1 < l 1 + δ and L2,n C∗ ; and λl,n C0 L2+δ,n ln1+δ (1 + n)
if (i) 1 l 2 or (ii) 2 < l 2 + δ and L2,n C∗ . Here C0 = C(K, μ, l) in cases (i), and C0 = C(K, μ, l, C∗ ) in cases (ii). Recall that to prove Theorem A, we used the powerful and gen
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