Limit theorems for multidimensional renewal sets

DOI10.1007/S10474-018-0806-YzbMATH Open1438.60038arXiv1708.03779OpenAlexW2964320198MaRDI QIDQ1714993

Erdal Karapınar

Publication date: 1 February 2019

Published in: Acta Mathematica Hungarica (Search for Journal in Brave)

Abstract: Consider multiple sums $S_n$ on the $d$-dimensional integer grid,which are generated by i.i.d. random variables with a positive expectation. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional limit theorem for random sets ${mathcalM}_t$ that appear as inversion of the multiple sums, that is, as the set of all arguments $xin{mathbbR}_{+}^d$ such that the interpolated multiple sum $S_x$ exceeds $t$. The moment conditions are identical to those imposed in the almost sure limit theorems for multiple sums. The results are expressed in terms of set inclusions and using distances between sets.

Full work available at URL: https://arxiv.org/abs/1708.03779

zbMATH Keywords

strong law of large numbersrandom fieldlaw of the iterated logarithmrandom setexcursion setmultiple sumrenewal set

Mathematics Subject Classification ID

Random fields (60G60) Geometric probability and stochastic geometry (60D05) Sums of independent random variables; random walks (60G50) Strong limit theorems (60F15) Functional limit theorems; invariance principles (60F17) Renewal theory (60K05)

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