engleski

Weak convergence of partial maxima processes

For a sequence of independent and identically distributed random variables (X_n) the regular variation condition is equivalent to weak convergence of partial maxima M_n = max{; ; ; X_1, ..., X_n}; ; ; , appropriately scaled. A functional version of this is known to be true as well, the limit process being an extremal process, and the convergence takes place in the space D[0, 1] of csdlag functions endowed with the Skorohod J_1 topology. Recently, in [3] under the properties of weak dependence and joint regular variation the asymptotic distributional behavior of extremes M_n and processes M_n(t) = max_{; ; ; i=1, ...⌊nt⌋}; ; ; X_i, t \in [0, 1], was investigated. We present the results of [3]. Material and methods: The obtained results rest on point processes convergence results obtained in [2] and [1], and the used methods are partly based on the work of [1] for partial sums. In the functional case the continuous mapping theorem and the standard "finite dimensional convergence plus tightness" procedure were used. Results and conclusion(s): First it is shown that weak convergence of partial maxima M_n holds for a class of weakly dependent sequences under the joint regular variation condition. Then using this result a corresponding functional version for the processes of partial maxima M_n(.
) was obtained, with the convergence taking place with respect to the Skorohod M_1 topology, which is weaker than the more usual J_1 topology. The limiting process is an extremal process given in terms of regular variation and extremal indices of the sequence (X_n). It is also shown that the M_1 convergence generally can not be replaced by the J_1 convergence. References: [1] B. Basrak, D. Krizmanic and J. Segers, A functional limit theorem for partial sums of dependent random variables with innite variance, Ann. Probab. 40 (2012), 2008-2033. [2] R. A. Davis and T. Hsing, Point process and partial sum convergence for weakly dependent random variables with innite variance, Ann. Probab. 23 (1995), 879-917. [3] D. Krizmanic, Weak convergence of partial maxima processes in the M1 topology, Extremes 17 (2014), 447-465.

partial maxima process ; Skorohod M1 topology ; regular variation ; functional limit theorem

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