| Title |
On the asymptotic of the maximal weighted increment of a random walk with regularly varying jumps: the boundary case / |
| Authors |
Račkauskas, Alfredas ; Suquet, Charles |
| DOI |
10.1214/21-EJP691 |
| Full Text |
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| Is Part of |
Electronic journal of probability.. Cleveland, OH : Institute of Mathematical Statistics. 2021, vol. 26, art. no. 122, p. [1-31].. ISSN 1083-6489 |
| Keywords [eng] |
random walk ; maximal increment ; regularly varying random variables. |
| Abstract [eng] |
Let (Xi)i1 be i.i.d. random variables with EX1 = 0, regularly varying with exponent a > 2 and taP(jX1j > t) L(t) slowly varying as t ! 1. We give the limit distribution of Tn( )=max0j<kn jXj+1 + + Xkj(kj) in the threshold case a :=1=21=a which separates the Brownian phase corresponding to 0 < a where the limit of Tn( ) is T( ), with 2 = EX2 1 , T( ) is the -Hölder norm of a standard Brownian motion and the Fréchet phase corresponding to a < < 1 where the limit of Tn( ) is Ya with Fréchet distribution P(Ya x) = exp(xa), x > 0. We prove that c1 n (Tn( a) n), converges in distribution to some random variable Z if and only if L has a limit a 2 [0;1] at infinity. In such case, there are A > 0, B 2 R such that Z = AVa;; + B in distribution, where for 0 < < 1, Va;; := max(T( a); Ya) with T( a) and Ya independent and Va;;0 := T( a), Va;;1 := Ya. When < 1, a possible choice for the normalization is cn = n1=a and n = 0, with Z = Va;; . We also build an example where L has no limit at infinity and (Tn( ))n1 has for each 2 [0;1] a subsequence converging after normalization to Va;; . |
| Published |
Cleveland, OH : Institute of Mathematical Statistics |
| Type |
Journal article |
| Language |
English |
| Publication date |
2021 |
| CC license |
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