A connection between supermodular ordering and positive/negative association

In this paper, we show that a vector of positively/negatively associated random variables is larger/smaller than the vector of their independent duplicates with respect to the supermodular order. In that way, we solve an open problem posed by Hu (Chinese J. Appl. Probab. Statist. 16 (2000) 133) refering to whether negative association implies negative superadditive dependence, and at the same time to an open problem stated in Müller and Stoyan (Comparison Methods for Stochastic Modes and Risks, Wiley, Chichester, 2002) whether association implies positive supermodular dependence. Therefore, some well-known results concerning sums and maximum partial sums of positively/negatively associated random variables are obtained as an immediate consequence. The aforementioned result can be exploited to give useful probability inequalities. Consequently, as an application we provide an improvement of the Kolmogorov-type inequality of Matula (Statist. Probab. Lett. 15 (1992) 209) for negatively associated random variables. Moreover, a Rosenthal-type inequality for associated random variables is presented.     

A New Method for Bounding the Distance Between Sums of Independent Integer-Valued Random Variables

Let X ₁, X ₂,..., X _n and Y ₁, Y ₂,..., Y _n be two sequences of independent random variables which take values in ? and have finite second moments. Using a new probabilistic method, upper bounds for the Kolmogorov and total variation distances between the distributions of the sums \(\sum_{i=1}^{n}X_{i}\) and \(\sum_{i=1}^{n}Y_{i}\) are proposed. These bounds adopt a simple closed form when the distributions of the coordinates are compared with respect to the convex order. Moreover, they include a factor which depends on the smoothness of the distribution of the sum of the X _i ’s or Y _i ’s, in that way leading to sharp approximation error estimates, under appropriate conditions for the distribution parameters. Finally, specific examples, concerning approximation bounds for various discrete distributions, are presented for illustration.     

On the Number of Appearances of a Word in a Sequence of I.I.D. Trials

Let X ₁,...,X _n be a sequence of i.i.d. random variables taking values in an alphabet =₁,...,_q,q 2, with probabilities P(X _a=_i)=p _i,a=1,...,n,i=1,...,q. We consider a fixed h-letter word W=w₁...w_h which is produced under the above scheme. We define by R(W) the number of appearances of W as Renewal (which is equal with the maximum number of non-overlapping appearances) and by N(W) the number of total appearances of W (overlapping ones) in the sequence X _{a 1} a1n under the i.i.d. hypothesis. We derive a bound on the total variation distance between the distribution (R(W)) of the r.v. R(W) and that of a Poisson with parameter E(R(W)). We use the Stein-Chen method and related results from Barbour et al. (1992), as well as, combinatorial results from Schbath (1995b) concerning the periodic structure of the word W. Analogous results are obtained for the total variation distance between the distribution of the r.v. N(W) and that of an appropriate Compound Poisson r.v. Related limit theorems are obtained and via numerical computations our bounds are presented in tables.     

Compound Poisson Approximation for Multiple Runs in a Markov Chain

We consider a sequence X ₁, ..., X _n of r.v.'s generated by a stationary Markov chain with state space A = {0, 1, ..., r}, r 1. We study the overlapping appearances of runs of k _i consecutive i's, for all i = 1, ..., r, in the sequence X ₁,..., X _n. We prove that the number of overlapping appearances of the above multiple runs can be approximated by a Compound Poisson r.v. with compounding distribution a mixture of geometric distributions. As an application of the previous result, we introduce a specific Multiple-failure mode reliability system with Markov dependent components, and provide lower and upper bounds for the reliability of the system.