Title :

On the asymptotic of the maximal weighted increment of a random walk with regularly varying jumps: the boundary case

Author(s) :

Račkauskas, Alfredas [Auteur]
Suquet, Charles [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

Electronic Journal of Probability

Pages :

122

Publisher :

Institute of Mathematical Statistics (IMS)

Publication date :

2021

ISSN :

1083-6489

English keyword(s) :

random walk
maximal increment
regularly varying random variables

HAL domain(s) :

Mathématiques [math]/Probabilités [math.PR]

English abstract : [en]

Let (Xi)i≥1 be i.i.d. random variables with E X1 = 0, regularly varying with exponent a > 2 and taP (|X1| > t) ∼ L(t) slowly varying as t → ∞. We give the limit distribution of Tn(γ) = max0≤j<k≤n |Xj+1 + · · · + Xk |(k ...
Show more >Let (Xi)i≥1 be i.i.d. random variables with E X1 = 0, regularly varying with exponent a > 2 and taP (|X1| > t) ∼ L(t) slowly varying as t → ∞. We give the limit distribution of Tn(γ) = max0≤j<k≤n |Xj+1 + · · · + Xk |(k −j)−γ in the threshold case γa := 1/2−1/a which separates the Brownian phase corresponding to 0 ≤ γ < γa where the limit ofTn(γ) is σT (γ), with σ2 = E X2 1 , T (γ) is the γ-Hölder norm of a standard Brownian motion and the Fréchet phase corresponding to γa < γ < 1 where the limit of Tn(γ) is Ya with Fréchet distribution P (Ya ≤ x) = exp(−x−a), x > 0. We prove that c−1 n (Tn(γa) − μn), converges in distribution to some random variable Z if and only if L has a limit τ a ∈ [0, ∞] at infinity. In such case, there are A > 0, B ∈ R such that Z = AVa,σ,τ + B in distribution, where for 0 < τ < ∞, Va,σ,τ := max(σT (γa), τ Ya) with T (γa) and Ya independent and Va,σ,0 := σT (γa), Va,σ,∞ := Ya. When τ < ∞, apossible choice for the normalization is cn = n−1/a and μn = 0, with Z = Va,σ,τ . We also build an example where L has no limit at infinity and (Tn(γ))n≥1 has for each τ ∈ [0, ∞] a subsequence converging after normalization to Va,σ,τ .Show less >

Language :

Anglais

Popular science :

Non

Collections :

• Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL