English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The validity of the strong law of large numbers for multiple sums $S_n$ ofindependent identically distributed random variables $Z_k$, $k\leq n$, with$r$-dimensional indices is equivalent to the integrability of$|Z|(\log^+|Z|)^{r-1}$, where $Z$ is the typical summand. We consider thestrong law of large numbers for more general normalisations, without assumingthat the summands $Z_k$ are identically distributed, and prove a multiple sumgeneralisation of the Brunk--Prohorov strong law of large numbers. In the caseof identical finite moments of irder $2q$ with integer $q\geq1$, we show thatthe strong law of large numbers holds with the normalisation $\|n_1\cdotsn_r\|^{1/2}(\log n_1\cdots\log n_r)^{1/(2q)+\varepsilon}$ for any$\varepsilon>0$. The obtained results are also formulated in the setting ofergodic theorems for random measures, in particular those generated by markedpoint processes.

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