Weak convergence of nonparametric estimators of the multidimensional and multidimensional-multivariate renewal functions on Skorohod topology spaces

• Published in: Statistical inference for stochastic processes : an international journal devoted to time series analysis and the statistics of continuous time processes and dynamic systems Vol. 25; no. 3; pp. 485 - 504
• Main Authors: Harel, Michel, Ngatchou-Wandji, Joseph, Andriamampionona, Livasoa, Harison, Victor
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-10-2022; Springer Nature B.V
• Subjects: Convergence; Distribution functions; Estimators; Mathematics; Mathematics and Statistics; Multivariate analysis; Nonparametric statistics; Probability Theory and Stochastic Processes; Statistical Theory and Methods; Topology; Weak convergence; Primary 60F05; Renewal process; 62G20; Skorohod topology; Renewal function; 60F17; Secondary 62M86
• ISSN: 1387-0874; 1572-9311
• DOI: 10.1007/s11203-021-09263-3

This paper deals with the weak convergence of nonparametric estimators of the multidimensional and multidimensional-multivariate renewal functions on Skorohod topology spaces. It is an extension of Harel et al. (J Math Anal Appl 189:240–255, 1995) from the one-dimensional case to the multivariate and multidimensional case. The estimators are based on a sequence of non-negative independent and identically distributed (iid) random vectors. They are expressed as infinite sums of
k
-
folds
convolutions of the empirical distribution function. Their weak convergence study heavily rests on that of the empirical distribution function.