On the length of the longest head run

We evaluate the accuracy of approximation to the distribution of the length of the longest head run in a Markov chain with a discrete state space. An estimate of the accuracy of approximation in terms of the total variation distance is established for the first time

Compound Poisson approximation: a user's guide

Compound Poisson approximation is a useful tool in a variety of applications, including insurance mathematics, reliability theory, and molecular sequence analysis. In this paper, we review the ways in which Stein’s method can currently be used to derive bounds on the error in such approximations. The theoretical basis for the construction of error bounds is systematically discussed, and a number of specific examples are used for illustration. We give no numerical comparisons in this paper, contenting ourselves with references to the literature, where compound Poisson approximations derived using Stein’s method are shown frequently to improve upon bounds obtained from problem specific, ad hoc methods

On a conjecture by Eriksson concerning overlap in strings

Consider a finite alphabet Ω and strings consisting of elements from Ω. For a given string w, let cor(w) denote the autocorrelation, which can be seen as a measure of the amount of overlap in w. Furthermore, let aw(n) be the number of strings of length n that do not contain w as a substring. Eriksson [4] stated the following conjecture: if cor(w)>cor(w′), then aw(n)>aw′(n) from the first n where equality no longer holds. We prove that this is true if [mid R:]Ω[mid R:][gt-or-equal, slanted]3, by giving a lower bound for aw(n)−aw′(n)

Compound Poisson approximation in systems reliability

The compound Poisson "local" formulation of the Stein-Chen method is applied to problems in reliability theory. Bounds for the accuracy of the approximation of the reliability by an appropriate compound Poisson distribution are derived under fairly general conditions, and are applied to consecutive-2 and connected-s systems, and the 2-dimensional consecutive-k-out-ofn system, together with a pipeline model. The approximations are usually better than the Poisson "local" approach would give

Compound Poisson approximation for long increasing sequences

Consider a sequence X 1, . . ., X n of independent random variables with the same continuous distribution and the event {X i-r+1 < . . . < X i} of the appearance of an increasing sequence with length r, for i = r, . . . , n. Denote by W the number of overlapping appearances of the above event in the sequence of n trials. In this work, we derive bounds for the total variation and Kolmogorov distances between the distribution of W and a suitable compound Poisson distribution. Via these bounds, an associated theorem concerning the limit distribution of W is obtained. Moreover, using the previous results we study the asymptotic behaviour of the length of the longest increasing sequence. Finally, we suggest a non-parametric test based on W for checking randomness against local increasing trend

Compound poisson approximation: A user's guide

Compound Poisson approximation is a useful tool in a variety of applications, including insurance mathematics, reliability theory, and molecular sequence analysis. In this paper, we review the ways in which Stein's method can currently be used to derive bounds on the error in such approximations. The theoretical basis for the construction of error bounds is systematically discussed, and a number of specific examples are used for illustration. We give no numerical comparisons in this paper, contenting ourselves with references to the literature, where compound Poisson approximations derived using Stein's method are shown frequently to improve upon bounds obtained from problem specific, ad hoc methods