Simple approximation techniques are developed exploiting relationships between generalized convex orders and appropriate probability metrics. In particular, the distance between s-convex ordered random variables is investigated. Results connecting positive or negative dependence concepts and convex ordering are also presented. These results lead to approximations and bounds for the distributions of sums of positively or negatively dependent random variables. Applications and extensions of the main results pertaining to compound Poisson, normal and exponential approximation are provided as well.
Let X 1 , . . . , X n be a sequence of r.v.s produced by a stationary Markov chain with state space an alphabet Ω = {ω 1 , . . . , ω q }, q > 2. We consider a set of words {A 1 , . . . , A r } , r > 2, with letters from the alphabet Ω. We allow the words to have self-overlaps as well as overlaps between them. Let E denote the event of the appearance of a word from the set {A 1 , . . . , A r } at a given position. Moreover, define by N the number of non-overlapping (competing renewal) appearances of E in the sequence X 1 , . . . , X n . We derive a bound on the total variation distance between the distribution of N and a Poisson distribution with parameter EN. The Stein-Chen method and combinatorial arguments concerning the structure of words are employed. As a corollary, we obtain an analogous result for the i.i.d. case. Furthermore, we prove that, under quite general conditions, the r.v. N converges in distribution to a Poisson r.v. A numerical example is presented to illustrate the performance of the bound in the Markov case.
Simple approximation techniques are developed exploiting relationships between generalized convex orders and appropriate probability metrics. In particular, the distance between s-convex ordered random variables is investigated. Results connecting positive/negative dependence concepts and convex ordering are also presented. These results lead to approximations and bounds for the distributions of sums of positive/negative dependent random variables. Applications and extensions of the main results pertaining to compound Poisson, normal and exponential approximation are provided as well.
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