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Extremal processes with one jump. (English) Zbl 0976.60046

A stochastic process \(Y\) with right-continuous increasing sample paths is called an extremal process if it has “independent max-increments”, i.e. for any \(0=t_0<t_1<\dots<t_m\) there exist independent r.v.’s \(U_0\), …, \(U_m\) such that the vector \((Y(t_0),\dots,Y(t_m))\) has the same distribution as \((U_0,\max(U_0,U_1),\dots,\max(U_0,\dots,U_m))\). The paper is devoted to the description of such processes and their convergence in the Skorokhod topology \(D([0,\infty))\). E.g., it is shown that any univariate extremal process is a limit in \(D([0,\infty))\) of stochastically continuous extremal processes. One of the presented results is the following (the function \(g(t)=E e^{-Y(t)}\) is called the weight of the extremal process \(Y\)): Let \(Y_n\) be extremal processes for \(n>0\) and let \(Y_n\) converge weakly to \(Y_0\) in \(D([0,\infty))\). Then the fixed jumps of \(Y_n\) converge to \(Y_0\) iff the weights of \(Y_n\) converge in \(D([0,\infty))\). (It is said that the fixed jumps of \(Y_n\) converge to \(Y_0\) iff for any fixed discontinuity \(t\) of \(Y_0\) there exists \(t_n\to t\) such that \((Y_n(t_n-0),Y_n(t_n))\) converges weakly to \((Y(t-0),Y(t))\) as \(n\to\infty\)).

MSC:

60G51 Processes with independent increments; Lévy processes
60G70 Extreme value theory; extremal stochastic processes
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