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1976•The Annals of ProbabilityOpen access

On Weak Convergence of Extremal Processes

Ishay Weissman

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Abstract

Lamperti in 1964 showed that the convergence of the marginals of an extremal process generated by independent and identically distributed random variables implies the full weak convergence in the Skorohod $J_1$-topology. This result is generalized to the $k$th extremal process and to random variables which need not be identically distributed. The proof here is based on the weak convergence of a certain point-process (which counts the number of up-crossings of the variables) to a two-dimensional nonhomogeneous Poisson process.

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What this paper is about

Lamperti in 1964 showed that the convergence of the marginals of an extremal process generated by independent and identically distributed random variables implies the full weak convergence in the Skorohod $J_1$-topology. This result is generalized to the $k$th extremal process and to random variables which need not be identically distributed. The proof here is based on the weak convergence of a certain point-process (which counts the number of up-crossings of the variables) to a two-dimensional nonhomogeneous Poisson process.

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Available abstract

Lamperti in 1964 showed that the convergence of the marginals of an extremal process generated by independent and identically distributed random variables implies the full weak convergence in the Skorohod $J_1$-topology. This result is generalized to the $k$th extremal process and to random variables which need not be identically distributed. The proof here is based on the weak convergence of a certain point-process (which counts the number of up-crossings of the variables) to a two-dimensional nonhomogeneous Poisson process.

Key concepts: Independent and identically distributed random variables, Mathematics, Weak convergence, Random variable, Convergence (economics), Point process, Convergence of random variables, Poisson point process

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