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

[img] Text
1-s2.0-S0167715217302675-main.pdf - Published Version
Restricted to registered users only
Available under License Publisher holds Copyright.

Download (527kB) | Request a copy

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.

Item Type:

Journal Article (Original Article)

Division/Institute:

08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematical Statistics and Actuarial Science

UniBE Contributor:

Molchanov, Ilya

Subjects:

500 Science > 510 Mathematics

ISSN:

1669-1676

Publisher:

Elsevier

Funders:

[4] Swiss National Science Foundation

Language:

English

Submitter:

Ilya Molchanov

Date Deposited:

20 Mar 2018 10:46

Last Modified:

29 Oct 2019 18:05

Publisher DOI:

10.1016/j.spl.2017.08.007

BORIS DOI:

10.7892/boris.109665

URI:

https://boris.unibe.ch/id/eprint/109665

Actions (login required)

Edit item Edit item
Provide Feedback