An Extension of the Concept of Slowly Varying Function with Applications to Large Deviation Limit Theorems
Karamata’s integral representation for slowly varying functions is extended to a broader class of the so-called ψ-locally constant functions, i.e. functions f(x) > 0 having the property that, for a given non-decreasing function ψ(x) and any fixed v, f(
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Abstract Karamata’s integral representation for slowly varying functions is extended to a broader class of the so-called -locally constant functions, i.e. functions f .x/ > 0 having the property that, for a given non-decreasing function .x/ and any fixed v, f .x C v .x//=f .x/ ! 1 as x ! 1. We consider applications of such functions to extending known theorems on large deviations of sums of random variables with regularly varying distribution tails. Keywords Slowly varying function • Locally constant function • Large deviation probabilities • Random walk
Mathematics Subject Classification (2010): Primary 60F10; Secondary 26A12
1 Introduction Let L.x/ be a slowly varying function (s.v.f.), i.e. a positive measurable function such that, for any fixed v 2 .0; 1/ holds L.vx/ L.x/ as x ! 1: lim
x!1
L.vx/ D 1: L.x/
(1)
A.A. Borovkov () Sobolev Institute of Mathematics, Russian Federation and Novosibirsk State University, Ac. Koptyug, pr. 4, 630090 Novosibirsk, Russia e-mail: [email protected] K.A. Borovkov Department of Mathematics and Statistics, The University of Melbourne, Parkville 3010, Melbourne, Australia 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 7, © Springer-Verlag Berlin Heidelberg 2013
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A.A. Borovkov and K.A. Borovkov
Among the most important and often used results on s.v.f.’s are the Uniform Convergence Theorem (see property (U) below) and the Integral Representation Theorem (property (I)), the latter result essentially relying on the former. These theorems, together with their proofs, can be found e.g. in monographs [1] (Theorems 1.2.1 and 1.3.1) and [2] (see 1.1). (U) For any fixed 0 < v1 < v2 < 1, convergence (1) is uniform in v 2 Œv1 ; v2 . (I) A function L.x/ is an s.v.f. iff the following representation holds true: Z
x
L.x/ D c.x/ exp 1
".t/ dt ; t
x 1;
(2)
where c.t/ > 0 and ".t/ are measurable functions, c.t/ ! c 2 .0; 1/ and ".t/ ! 0 as t ! 1. The concept of a s.v.f. is closely related to that of a regularly varying function (r.v.f.) R.x/, which is specified by the relation R.x/ D x ˛ L.x/;
˛ 2 R;
where L is an s.v.f. and ˛ is called the index of the r.v.f.‘R. The class of all r.v.f.’s we will denote by R. R.v.f.’s are characterised by the relation lim
x!1
R.vx/ D v˛ ; R.x/
v 2 .0; 1/:
(3)
For them, convergence (3) is also uniform in v on compact intervals, while representation (2) holds for r.v.f.’s with ".t/ ! ˛ as t ! 1. In Probability Theory there exists a large class of limit theorems on large deviations of sums of random variable whose distributions F have the property that their right tails FC .x/ WD F Œx; 1/ are r.v.f.’s. The following assertion (see e.g. Theorem 4.4.1 in [2]) is a typical representative of such results. Let ; 1 ; 2 ; : : : be independent identically distributed random variables, E D 0, E 2 < 1, Pn Sn WD kD1 k and S n WD maxkn Sk . Theorem A. If FC .t/ D P. t/ is an r.v.f.
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