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Limit theorems for multidimensional renewal sets

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Abstract

Consider the multiple sums Sn of i.i.d. random variables with a positive expectation on the d-dimensional integer grid. We prove the strong law of large numbers, the law of the iterated logarithm and the distributional limit theorem for random sets which appear as upper excursion sets of the interpolated multiple sums, that is, as the set of all arguments \({x\in\mathbb{R}_+^d}\) such that the interpolated multiple sums Sx exceed t. The results are expressed in terms of set inclusions and using distances between sets.

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Authors and Affiliations

  1. Department of Mathematical Analysis and Probability Theory, Igor Sikorsky Kyiv Polytechnic Institute, Peremogy avenue 37, 03056, Kyiv, Ukraine

    A. Ilienko

  2. Institute of Mathematical Statistics and Actuarial Science, University of Bern, Alpeneggstrasse 22, CH-3012, Bern, Switzerland

    I. Molchanov

Authors
  1. A. Ilienko
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  2. I. Molchanov
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Correspondence to I. Molchanov.

Additional information

Supported by the Swiss National Science Foundation in the framework of the SCOPES programme, Grant No. IZ73Z0_152292.

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Ilienko, A., Molchanov, I. Limit theorems for multidimensional renewal sets. Acta Math. Hungar. 156, 56–81 (2018). https://doi.org/10.1007/s10474-018-0806-y

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  • DOI: https://doi.org/10.1007/s10474-018-0806-y

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