2017/03/15 by Oleg Klesov, Ilya Molchanov, Klesov, Oleg +1
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60G55 60F15 60D05 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.1703.04994
openalex publication_date 2017/03/15 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
The validity of the strong law of large numbers for multiple sums Sn of\nindependent identically distributed random variables Zk, k\≤ n, with\nr-dimensional indices is equivalent to the integrability of\n|Z|(\log+|Z|)r-1, where Z is the typical summand. We consider the\nstrong law of large numbers for more general normalisations, without assuming\nthat the summands Zk are identically distributed, and prove a multiple sum\ngeneralisation of the Brunk--Prohorov strong law of large numbers. In the case\nof identical finite moments of irder 2q with integer q\≥1, we show that\nthe strong law of large numbers holds with the normalisation \‖n1\⋯\nnr\‖1/2(\log n1\⋯\log nr)1/(2q)+\ε for any\n\ε>0. The obtained results are also formulated in the setting of\nergodic theorems for random measures, in particular those generated by marked\npoint processes.\n