Asymptotics of running maxima for \(\varphi \)-subgaussian random double arrays. (English) Zbl 1498.60112
Let \(\{X_{k,n}, k \geq 1, n \geq 1\}\) be a double array of \(\phi\)-subgaussian random variables that are not necessarily i.i.d., defined on the same probability space. The authors study sufficient conditions on the tail distributions of \(X_{k,n}\) that guarantee the existence of a sequence \(\{a_{m,j}, m \geq 1, j \geq 1\}\) of real numbers such that \(Y_{m,j} = \max_{1\leq k \leq m, 1 \leq n \leq j} X_{k,n} - a_{m,j}\) converge to 0 almost surely as the number of random variables \(X_{k,n}\) tends to infinity. The array \(\{Y_{m,j}\}\) is split into positive part and negative part and the results are proved separately for the two. Almost sure and lim (max) convergence of random functionals of the double arrays are investigated. After discussing some examples, some numerical examples are provided that confirm the obtained theoretical results.
Reviewer: Ravi Sreenivasan (Mysore)
MSC:
| 60F15 | Strong limit theorems |
| 60G60 | Random fields |
| 60G70 | Extreme value theory; extremal stochastic processes |
Keywords:
\(\phi\)-subgaussian random variables; almost sure convergence; random double array; rate of convergence; running maxima; strong theoremsReferences:
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