Annals of the Institute of Statistical Mathematics

Many large engineering systems can be viewed (or imbedded) as a series system in time. In this paper, we introduce the structure of a repairable system and the reliabilities of these large systems are studied systematically by studying the ergodicities of certain non-homogeneous Markov chains. It shows that if the failure probabilities of components satisfy certain conditions, then the reliability of the large system is approximately exp (-β) for some β>0. In particular, we demonstrate how the repairable system can be used for studying the reliability of a large linearly connected system. Several practical examples of large consecutive-k-out-of-n:F systems are given to illustrate our results. The Weibull distribution is derived under our natural set-up.

The total number of successes in success runs of length greater than or equal to k in a sequence of n two-state trials is a statistic that has been broadly used in statistics and probability. For Bernoulli trials with k equal to one, this statistic has been shown to have binomial and normal distributions as exact and limiting distributions, respectively. For the case of Markov-dependent two-state trials with k greater than one, its exact and limiting distributions have never been considered in the literature. In this article, the finite Markov chain imbedding technique and the invariance principle are used to obtain, in general, the exact and limiting distributions of this statistic under Markov dependence, respectively. Numerical examples are given to illustrate the theoretical results.

When using an auxiliary Markov chain (AMC) to compute sampling distributions, the computational complexity is directly related to the number of Markov chain states. For certain complex pattern statistics, minimal deterministic finite automata (DFA) have been used to facilitate efficient computation by reducing the number of AMC states. For example, when statistics of overlapping pattern occurrences are counted differently than non-overlapping occurrences, a DFA consisting of prefixes of patterns extended to overlapping occurrences has been generated and then minimized to form an AMC. However, there are situations where forming such a DFA is computationally expensive, e.g., with computing the distribution of spaced seed coverage. In dealing with this problem, we develop a method to obtain a small set of states during the state generation process without forming a DFA, and show that a huge reduction in the size of the AMC can be attained.

In this paper, we establish the limit of empirical spectral distributions of quaternion sample covariance matrices. Motivated by Bai and Silverstein (Spectral analysis of large dimensional random matrices, Springer, New York, 2010) and Marčenko and Pastur (Matematicheskii Sb, 114:507–536, 1967), we can extend the results of the real or complex sample covariance matrix to the quaternion case. Suppose $$\mathbf X_n = ({x_{jk}^{(n)}})_{p\times n}$$ is a quaternion random matrix. For each $$n$$ , the entries $$\{x_{ij}^{(n)}\}$$ are independent random quaternion variables with a common mean $$\mu $$ and variance $$\sigma ^2>0$$ . It is shown that the empirical spectral distribution of the quaternion sample covariance matrix $$\mathbf S_n=n^{-1}\mathbf X_n\mathbf X_n^*$$ converges to the Marčenko–Pastur law as $$p\rightarrow \infty $$ , $$n\rightarrow \infty $$ and $$p/n\rightarrow y\in (0,+\infty )$$ .

To estimate the finite population mean, $$\bar Y$$ , a two-phase sample may be selected. A simple random sample of sizen′ is chosen, and a concomitant variable,X, is measured for all units. Then, a simple random subsample of sizen (0