On some applications of the Archimedean copulas in the proofs of the almost sure central limit theorems for certain order statistics
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Our goal is to establish and prove the almost sure central limit theorems for some order statistics $\left\{ M_{n}^{\left( k\right) }\right\} $, $k=1,2,\ldots$, formed by stochastic processes $\left( X_{1},X_{2},\ldots,X_{n}\right) $, $n\in N$, the distributions of which are defined by certain Archimedean copulas. Some properties of generators of such the copulas are intensively used in our proofs. The first class of theorems stated and proved in the paper concerns sequences of ordinary maxima $\left\{ M_{n}\right\} $, the second class of the presented results and proofs applies for sequences of the second largest maxima $\left\{ M_{n}^{\left( 2\right) }\right\} $ and the third (and the last) part of our investigations is devoted to the proofs of the almost sure central limit theorems for the $k$-th largest maxima $\left\{ M_{n}^{\left( k\right) }\right\} $ in general. The assumptions imposed in the first two of the mentioned groups of claims significantly differ from the conditions used in the last - the most general - case.
Abstract
Our goal is to establish and prove the almost sure central limit theorems for some order statistics $\left\{ M_{n}^{\left( k\right) }\right\} $, $k=1,2,\ldots$, formed by stochastic processes $\left( X_{1},X_{2},\ldots,X_{n}\right) $, $n\in N$, the distributions of which are defined by certain Archimedean copulas. Some properties of generators of such the copulas are intensively used in our proofs. The first class of theorems stated and proved in the paper concerns sequences of ordinary maxima $\left\{ M_{n}\right\} $, the second class of the presented results and proofs applies for sequences of the second largest maxima $\left\{ M_{n}^{\left( 2\right) }\right\} $ and the third (and the last) part of our investigations is devoted to the proofs of the almost sure central limit theorems for the $k$-th largest maxima $\left\{ M_{n}^{\left( k\right) }\right\} $ in general. The assumptions imposed in the first two of the mentioned groups of claims significantly differ from the conditions used in the last - the most general - case.
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