Renewal theory for asymmetric $U$-statistics

Svante Janson

doi:10.1214/18-ejp252

Abstract

We extend a functional limit theorem for symmetric U-statistics [Miller and Sen, 1972] to asymmetric U-statistics, and use this to show some renewal theory results for asymmetric U-statistics. Some ...

Highlights

Let X, X1, X2, . . . , be an i.i.d. sequence of random variables taking values in an arbitrary measurable space S = (S, S). (In most cases, S = R or perhaps Rk, or a Borel subset of one of these, but we can just as well consider the general case.) let d 1 and let f : Sd → R be a given measurable function
I.e. the case d = 1 in our setting, this was studied in [14]; we extend the main result there to U -statistics
For moment convergence in the renewal theory theorems, we assume for simplicity that f and fhave finite moments of all orders; see Remark 6.1. (For the case d = d = 1, see e.g. [19], [14], and [12, Section 3.8 and Theorem 4.2.3].)

Summary

Introduction

We find it more convenient for our purposes to use the unnormalized version above It is common, following Hoeffding [16], to assume that f is a symmetric function of its d variables. One of the purposes of this paper is to generalize a result by [26] on functional convergence from the symmetric case to the general, asymmetric case We use this result to derive some renewal theory results for the sequence Un. We use this result to derive some renewal theory results for the sequence Un One motivation for this is some applications to random restricted permutations, see Section 5. The proofs are given in Section 4; they use standard methods, in particular the decomposition and projection method of Hoeffding [16], but some complications arise in the asymmetric case; this includes applications to random restricted permutations that gave the initial motivation to write the present paper. See Remark 6.3 for further comments on the degenerate case

Some notation

Limit theorems

Moment convergence

Renewal theory

Examples and applications

Further comments and open problems