Non standard functional limit laws for the increments of the compound empirical distribution function

TL;DR

This paper establishes a nonstandard functional limit law for the increments of the compound empirical process, revealing almost sure convergence to a set related to large deviations of compound Poisson processes.

Contribution

It introduces a novel nonstandard limit law for the compound empirical process increments under logarithmic scaling, extending the understanding of their asymptotic behavior.

Findings

Almost sure convergence to a compact set of functions

Limit law related to large deviations of compound Poisson processes

Applicable under conditions on exponential moments of the data

Abstract

Let $(Y_i,Z_i)_{i\geq 1}$ be a sequence of independent, identically distributed (i.i.d.) random vectors taking values in $\RRR^k\times\RRR^d$, for some integers $k$ and $d$. Given $z\in \RRR^d$, we provide a nonstandard functional limit law for the sequence of functional increments of the compound empirical process, namely $$\mathbf{\Delta}_{n,\cc}(h_n,z,\cdot):= \frac{1}{nh_n}\sliin 1_{[0,\cdot)}\poo \frac{Z_i-z}{{h_n}^{1/d}}\pff Y_i.$$ Provided that $nh_n\sim c\log n $ as $\nif$, we obtain, under some natural conditions on the conditional exponential moments of $Y\mid Z=z$, that $$\mathbf{\Delta}_{n,\cc}(h_n,z,\cdot)\leadsto \Gam\text{almost surely},$$ where $\leadsto$ denotes the clustering process under the sup norm on $\Idd$. Here, $\Gam$ is a compact set that is related to the large deviations of certain compound Poisson processes.