The height and range of watermelons without wall

TL;DR

This paper investigates the asymptotic behavior of the height and range of p-watermelons without wall restrictions, providing their limiting distributions and moment asymptotics as the number of steps grows large.

Contribution

It establishes the weak limit distributions for height and range of p-watermelons without wall and derives asymptotic formulas for their moments.

Findings

Weak limit distribution of height and range identified.

Asymptotic formulas for moments of height derived.

Results applicable to large-step asymptotics of p-watermelons.

Abstract

We determine the weak limit of the distribution of the random variables "height" and "range" on the set of p-watermelons without wall restriction as the number of steps tends to infinity. Additionally, we provide asymptotics for the moments of the random variable "height".