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

Abstract

Let (Xi) i≥1 be i.i.d. random variables with E X1 = 0, regularly varying with exponent a > 2 and t a P (|X1| > t) ∼ L(t) slowly varying as t → ∞. We give the limit distribution of Tn(γ) = max 0≤j<k≤n |Xj+1 + • • • + X k |(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 of Tn(γ) is σT (γ), with σ 2 = E X 2 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 < ∞, a possible 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 .