Moment conditions in strong laws of large numbers for multiple sums and random measures

Klesov, Oleg; Molchanov, Ilya (2017). Moment conditions in strong laws of large numbers for multiple sums and random measures. Statistics and Probability Letters, 131, pp. 56-63. Elsevier 10.1016/j.spl.2017.08.007

Text 1-s2.0-S0167715217302675-main.pdf - Published Version Restricted to registered users only Available under License Publisher holds Copyright. Download (527kB)

The validity of the strong law of large numbers for multiple sums Sn of independent identically
distributed random variables Zk , k ≤ n, with r-dimensional indices is equivalent to the integrability of |Z|(log+|Z|)r⁻¹, where Z is the generic summand. We consider the strong law of large numbers for more general normalizations, without assuming that the summands Zk are identically distributed, and prove a multiple sum generalization of the Brunk–Prohorov strong law of large numbers. In the case of identical finite moments of order 2q with integer q ≥ 1, we show that the strong law of large numbers holds with the normalization (n₁ · · · nr)1/2(log n₁ · · · log nr)1/(2q)+ε for any ε > 0. The obtained results are also formulated in the setting of ergodic theorems for random measures, in particular those generated by marked point processes.

Interest & Impact

Downloads

2 since deposited on 20 Mar 2018
0 in the past 12 months

Citations

5 Citations in Web of Science ®
6 Citations in Scopus

Search

in Google Scholar™

Services

Actions (login required)

Edit item

Actions (login required)

Edit item