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On Exact Laws of Large Numbers for Oppenheim Expansions with Infinite Mean

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Abstract

In this work, we investigate the asymptotic behaviour of weighted partial sums of a particular class of random variables related to Oppenheim series expansions. More precisely, we verify convergence in probability as well as almost sure convergence to a strictly positive and finite constant without assuming any dependence structure or the existence of means. Results of this kind are known as exact weak and exact strong laws.

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Acknowledgements

R. Giuliano wishes to thank Prof. Tasos Christofides for his kind invitation at the University of Cyprus; the present paper was started during the permanence there. The authors would like to thank the anonymous referee and the associate editor, for their careful reading and insightful comments that led to an improved version of the paper.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italy

    Rita Giuliano

  2. School of Sciences, University of Central Lancashire, Cyprus Campus 12-14 University Avenue, Pyla, 7080, Larnaka, Cyprus

    Milto Hadjikyriakou

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  1. Rita Giuliano
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  2. Milto Hadjikyriakou
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Correspondence to Milto Hadjikyriakou.

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Giuliano, R., Hadjikyriakou, M. On Exact Laws of Large Numbers for Oppenheim Expansions with Infinite Mean. J Theor Probab 34, 1579–1606 (2021). https://doi.org/10.1007/s10959-020-01010-3

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  • DOI: https://doi.org/10.1007/s10959-020-01010-3

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