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The Generalized Trotter Operator and Weak Convergence Of Dependent Random Variables in Different Probability Metrics

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Abstract

The Trotter operator-theoretic method for establishing weak convergence of sums of real-valued random variables X i i∈N on a probability space (Ω,A,P) is extended to the situation that the Xi are dependent. For this goal, the generalized Trotter-operator is defined for functions f∈Cb and a sub- σ algebra G ⊂ A by VX¦G f (y) ≔ ∫Rf(x+y) dPX¦G(x,w). General results concerning the weak convergence of dependent random variables (r.vs.) with rates are presented. Further, the distance V between the distributions P,Q of two random variables X,Y with respect to a function f and a sub-σ algebra G, namely the Trotter-distance V(P,Q;f), defined via the generalized Trotter-operator by V(P,Q;f) = supy¦∫Rf(x+y) d(P−q)(x)¦, is compared with other well-known probability-metrics, which metrize weak convergence, such as the Zolotarev-metric ξ, the Gudynas-metric η, the Levy-metric L or the Prohorov-metric ϱ. Lastly, the Trotter-distance with rates is shown to be equivalent to weak convergence with rates.

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  1. Lehrstuhl A für Mathematik, Aachen University of Technology, 5100, Aachen, W. Germany

    Heribert Kirschfink

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Kirschfink, H. The Generalized Trotter Operator and Weak Convergence Of Dependent Random Variables in Different Probability Metrics. Results. Math. 15, 294–323 (1989). https://doi.org/10.1007/BF03322619

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