On One Inequality for Characteristic Functions
This paper deals with an inequality for characteristic functions. This inequality (see (3) below) founds connection between “measure of almost normality” and characteristic functions. Also an analysis of accuracy in the local limit theorem and connection
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Abstract This paper deals with an inequality for characteristic functions. This inequality (see (3) below) founds connection between “measure of almost normality” and characteristic functions. Also an analysis of accuracy in the local limit theorem and connection between the central limit and local limit theorem are given. Keywords Characteristic functions • Limit theorems • Central limit theorem • Local limit theorem
Mathematics Subject Classification (2010): 60E10, 60E15
A good deal of probability theory consists of the study of limit theorems, because “in reality the epistemological value of the theory of probability is revealed only by limit theorems” (see [4]). Important part of this area consists of the upper estimation of the rate of convergence in the central limit theorem. It is quite reasonable turn to the construction of the lower estimates. Unfortunately, many mathematicians working in the field of theory of limit theorems pay less attention to such a kind of problems. This paper focuses around one inequality for characteristic function. It should be noted that this one don’t demands from limit theorems. Taking account of this notion at first we introduce following
N. Gamkrelidze () Gubkin Russian State University of Oil and Gas, Leninsky Prospekt, 65, 119991, Moscow, Russia e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 15, © Springer-Verlag Berlin Heidelberg 2013
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N. Gamkrelidze
Definition. An integer valued random variable is said .A; B; / normal if there are some constants A; B 1; ; .0 < < B/ and integer k.1 < k < 1/ such that ˇ n .k A/2 oˇ ˇ ˇ (1) supˇP . D k/ .2/1=2 B 1 exp ˇ : 2 2B B k We will show that may be estimate from below by 1 4
Z jf .t; k /j2 dt
ı > 0:
where
ıjt j 0 we have p 1 X 2 X m2 t 1 2 p exp .s C 2j/ D e 2 cos.ms/ 2t t j mD1 (5) 1 X 2 m2 t D1C2 e cos.ms/: mD1
Put s D 2A; t D 4 B and write the left-hand side (5) as 2
2
1 X 1 S1 D p exp 2 .j A/2 : 2B 2B j On the right-hand side (5) we get 1C2
1 X
e 2m
22B2
cos.m 2A/
(6)
mD1
Since B 1 we can write 2e 2
1 P
e 2.m 1/ mD1 2 2 2e 2 B 1 C 2B2
2
2e 2 2 e4 : 2 2
2B2
2B2
1 P 2 2 1C .e 2 B /m mD2
1e
The evaluation c WD e 4 =1 e 2 gives c < 1017 . Consequently 2
2
S1 D 1 C 1 2; 01e 2
2B2
;
j1 j < 1:
In the same way we estimate S2 . At first consider j -th term of the sum S2 i 1 1 h 2 2 : exp 2 .k C j A/ C .j A/ 2B 2 2B Note that 2 k 2 .k C j A/2 C .j A/2 D 2 j .A k=2/ C : 2
(7)
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N. Gamkrelidze
So 2 2 X k2 1 1 1 1 k2 4B exp 2 j .a k=2/ D p S2 D p e e 4B 2 S10 ; 2B B 2 B 2 B j (8) p k 0 where S1 is the same as S1 but with changing B to B= 2 and A to A 2 . According (8) we have S10 D 1 C 2 2; 01e
2B 2
j2 j 1:
;
(9)
Taking in account (1) and (6)–(9) we receive the statement (2). Remark 1. Applying Eule
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