Recent results of author for sums and maxima of non-stationary random length sequences of regularly varying
distributed random variables (r.v.s) are presented. A doubly-indexed array of regularly varying r.v.s in which the "row index" corresponds to time, and the "column index" corresponds to the level is considered. Each "column" series is assumed to be stationary distributed with some tail and extremal indices. We focus on sums and maxima of weighted row sequences of random length.
In Markovich and Rodionov (2020) it is assumed that there is a unique series with minimum tail index. The
novelty is that "column" series are arbitrary mutually dependent. Conditions when tail and extremal indices of mentioned sums and maxima
are the same are obtained.
The random length is assumed to be regularly varying with a lighter tail than r.v.s in the series.
In Markovich (2021a) d "column" series with minimum tail index are assumed.
All elements in pairs of the most heavy-tailed "column" sequences have to show the same mutual dependence. Then the sums and maxima have the same minimum tail index.
If d>1 is fixed and d "column" series are mutually independent, and independent of the rest of "column" series, then
the sums and maxima have the same extremal index. If the independence does not hold, then their extremal indices may not exist. If d is a bounded discrete r.v., then the extremal indices of the sums and maxima sequences do not exist.
Theorems in Markovich and Rodionov (2020), Markovich (2021a) are valid if there are non-zero
elements in each "row" sequence corresponding to "column" sequences with minimum tail index.
Particularly, if the most heavy-tailed column is unique and at least one of its elements
is equal to zero, then the sums and maxima are non-stationary. This property plays a role for graphs.
The sums and maxima may be associated with graphical model that allows us to obtain tail and extremal indices of PageRank (PR) and the Max-Linear Model (MLM), Markovich (2021b). The latter are used as node influence measures in complex networks.
The "row" sequences serve as coordinates of points in R^d. Then the PR of a newly attached node (a webpage)
is determined as sums of PRs of node parents taken as the row elements. The MLM of the node
is determined by
maximum of the row elements. It is shown that the tail and extremal indices of the PR and MLM of attached nodes are determined by the graph community with minimum tail index.
The PR of a webpage is often considered as the solution to the fixed-point problem (see, Jelenkovic and Olvera-Cravioto 2010, Volkovich and Litvak 2010 among others), where PRs of in-coming nodes are assumed to be iid. We allow these PRs to be dependent and non-stationary which is more plausible for real networks.