On Some Properties of Randomly Indexed Sequences of Random Elements
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On Some Properties of Randomly Indexed Sequences of Random Elements A. Krajka
Published online: 23 March 2007 © Springer Science + Business Media B.V. 2007
Abstract Let (, A, P ) be a probability space with a nonatomic measure P and let (S, ρ) be a separable complete metric space. Let {Nn , n ≥ 1} be an arbitrary sequence of positiveinteger valued random variables. Let {Fk , k ≥ 1} be a family of probability laws and let X be some random element defined on (, A, P ) and taking values in (S, ρ). In this paper we present necessary and sufficient conditions under which one can construct an array of random elements {Xn,k , n, k ≥ 1} defined on the same probability space D
P
and taking values in (S, ρ), and such that Xn,k ∼ Fk (), n, k ≥ 1, and moreover Xn,Nn −→ a.s.
Lr
(−→, −→)X, as n → ∞. Furthermore, we consider the speed of convergence Xn,Nn to X as n → ∞. Keywords Randomly indexed sequence of random elements · Strong limit law · Convergence in Lr space Mathematics Subject Classification (2000) 60F25 · 60F15
1 Introduction Let {Nn , n ≥ 1} be an arbitrary sequence of positive integer-valued random variables defined on a probability space (, A, P ) with a nonatomic measure P . Let (S, ρ) be a separable metric space and let {Fk , k ≥ 1} be a family of probability laws defined on (S, ρ). Let X be an arbitrary random element in (S, ρ). Thus, we can ask under what conditions concerning {Nn , n ≥ 1}, {Fk (), k ≥ 1}, and X, there exists an array {Xn,k , n, k ≥ 1} of S-valued random elements, defined on the same probability space (, A, P ), such that P a.s. Lr Xn,Nn −→ −→, −→ X,
as n → ∞,
A. Krajka () Maria Curie-Skłodowska University, Pl. Marii Curie-Skłodowskiej 1, 20-031 Lublin, Poland e-mail: [email protected]
(1.1)
328
A. Krajka
and D
Xn,k ∼ Fk ().
(1.2)
D
Here, and in what follows, X ∼ F () denotes that the law of X is F (). In this paper we present some sufficient and necessary conditions under which for a given sequence {Nn , n ≥ 1}, S-valued random element X and a family of probability laws {Fk (), k ≥ 1}, there exists an array {Xn,k , n, k ≥ 1} of S-valued random elements, defined on the same probability space, and such that (1.1) and (1.2) hold. We shall also present the rate of convergence in (1.1). We have to consider an array {Xn,k , k, n ≥ 1}, instead of a sequence, because the construction will run for every random variable Nn , n ≥ 1, so that the random variables defined for N2 may be inappropriate for N1 . But if one wants to have a sequence, the following additional assumption on Nn , n ≥ 1 is needed: supω∈ Nn−1 (ω) < infω∈ Nn (ω). Randomly indexed sequences are applicable in a variety of situations, for example, in chromatographic methods, cryptomachines, age replacement policies, insurance risk theory and counter models. Applications of sequences of random variables with random indices may be found, for example, in Gnedenko [4] or Gut [6]. Following the work of Richter [8], many authors have investigated the strong asymptotic behaviour of Xn,Nn as n → ∞. Strong conve
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