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 Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

In Nevzorov's $F^\alpha$-scheme, one deals with a sequence of independent random variables whose distribution functions are all powers of a common continuous distribution function. A key property of the $F^\alpha$-scheme is that the record indicators for such a sequence are independent. This allows one to obtain several important limit theorems for the total number of records in the sequence up to time $n\to\infty$. We extend these theorems to a much more general class of sequences of random variables obeying a "threshold $F^\alpha$-scheme" in which the distribution functions of the variables are close to the powers of a common $F$ only in their right tails, above certain non-random non-decreasing threshold levels. Of independent interest is the characterization of the growth rate for extremal processes that we derived in order to be able to verify the conditions of our main theorem. We also establish the asymptotic pair-wise independence of record indicators in a special case of threshold $F^\alpha$-schemes.

Limit Theorems for Record Indicators in Threshold $F^\alpha$-Schemes