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

Abstract

Let $(Y_i,Z_i)_{i≥1}$ be a sequence of independent, identically distributed (i.i.d.) random vectors taking values in $ℝ^k×ℝ^d$, for some integers $k$ and $d$. Given $z∈ℝ^d$, we provide a nonstandard functional limit law for the sequence of functional increments of the compound empirical process, namely $$\mathbf{\Delta}_{n,\mathfrak{c}}(h_{n},z,\cdot):=\frac{1}{nh_{n}}\sum_{i=1}^{n}1_{[0,\cdot)}\Big(\frac{Z_{i}-z}{{h_{n}}^{1/d}}\Big)Y_{i}.$$ Provided that $nh_n∼c\log n$ as $n→∞$, we obtain, under some natural conditions on the conditional exponential moments of $Y\mid Z=z$, that $$\mathbf{\Delta}_{n,\mathfrak{c}}(h_{n},z,\cdot)\leadsto \Gamma \text{ almost surely},$$ where $↝$ denotes the clustering process under the sup norm on $[0,1)^d$. Here, $Γ$ is a compact set that is related to the large deviations of certain compound Poisson processes.