TY - JOUR

T1 - Edgeworth expansions for profiles of lattice branching random walks

AU - Grübel, Rudolf

AU - Kabluchko, Zakhar

N1 - Publisher Copyright: © Association des Publications de l'Institut Henri Poincaré, 2017.

PY - 2017/11

Y1 - 2017/11

N2 - Consider a branching random walκ on Z in discrete time. Denote by Ln(κ) the number of particles at site κ ∈ Z at time n ∈ N0. By the profile of the branching random walκ (at time n) we mean the function κ → Ln(κ). We establish the following asymptotic expansion of Ln(κ), as n→∞: (Equation presented) The expansion is valid uniformly in κ ∈ Z with probability 1 and the Fj 's are polynomials whose random coefficients can be expressed through the derivatives of Φ and the derivatives of the limit of the Biggins martingale at 0. Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walκ except its extreme values. As an application of this expansion for r = 0, 1, 2 we recover in a unified way a number of κnown results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers Ln(κn), where κn ∈ Z depends on n in some regular way. We also prove a.s. limit theorems for the mode argmaxκ∈Z Ln(κ) and the height maxκ∈Z Ln(κ) of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter Φ (0) is integer, non-integer rational, or irrational. Applications of our results to profiles of random trees including binary search trees and random recursive trees will be given in a separate paper.

AB - Consider a branching random walκ on Z in discrete time. Denote by Ln(κ) the number of particles at site κ ∈ Z at time n ∈ N0. By the profile of the branching random walκ (at time n) we mean the function κ → Ln(κ). We establish the following asymptotic expansion of Ln(κ), as n→∞: (Equation presented) The expansion is valid uniformly in κ ∈ Z with probability 1 and the Fj 's are polynomials whose random coefficients can be expressed through the derivatives of Φ and the derivatives of the limit of the Biggins martingale at 0. Using exponential tilting, we also establish more general expansions covering the whole range of the branching random walκ except its extreme values. As an application of this expansion for r = 0, 1, 2 we recover in a unified way a number of κnown results and establish several new limit theorems. In particular, we study the a.s. behavior of the individual occupation numbers Ln(κn), where κn ∈ Z depends on n in some regular way. We also prove a.s. limit theorems for the mode argmaxκ∈Z Ln(κ) and the height maxκ∈Z Ln(κ) of the profile. The asymptotic behavior of these quantities depends on whether the drift parameter Φ (0) is integer, non-integer rational, or irrational. Applications of our results to profiles of random trees including binary search trees and random recursive trees will be given in a separate paper.

KW - Biggins martingale

KW - Branching random walk

KW - Central limit theorem

KW - Edgeworth expansion

KW - Height

KW - Mod-Φ-convergence

KW - Mode

KW - Profile

KW - Random analytic function

UR - http://www.scopus.com/inward/record.url?scp=85035346082&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1503.04616

DO - 10.48550/arXiv.1503.04616

M3 - Article

AN - SCOPUS:85035346082

VL - 53

SP - 2103

EP - 2134

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 4

ER -