The existence and properties of a general ARMA (p, q) process, whose autocovariances, up to lag p, and impulse coefficients, up to lag q, coincide with some given values, are shown. A closed-form solution is obtained. Based on this, the maximal entropy property for ARMA process and the relation and difference between ARMA maximal entropy approximation and extended Padé approximation are discussed. This method may be used for the design of digital filters and for ARMA spectral estimation.

Books being placed on a shelf, a book bi is selected with probability pi, it is removed and replaced at the left-hand end of the shelf prior to next removal. The different arrangements of books on the shelf are considered as states in a Markov chain, known as a Tsetlin library. In this paper, we establish the necessary and sufficient condition of transience for some extensions of this chain.

We generalize a two-type mutation process in which particles reproduce by binary fission, inheriting the parental type, but which can mutate with small probability during their lifetimes to the opposite type. The generalization allows an arbitrary offspring distribution. The branching process structure of this scheme is exploited to obtain a variety of limit theorems, some of which extend known results for the binary case. In particular, practically usable asymptotic normality results are obtained when the initial population size is large.

The asymptotic moments of the concentration of a pollutant subject to rainout and dry removal are given. Both the lengths of the wet and dry periods and the wet and dry removal rates are allowed to be random, although the removal rate is assumed to be fixed within any one wet or dry period. The results hold in a more general setting than removal processes, so its is hoped that they will be applicable to other problems.

In this paper we prove that the limiting distribution of the general bulk queue exists and is independent of the initial conditions if and only if the traffic intensity is less than one. We further generalize the following heavy traffic results of the GI/G/1 model to the general bulk queue model. When ρ > 1 or ρ = 1 the waiting time is distributed approximately as a Gaussian variable and the absolute value of a Gaussian variable, respectively. The exponential approximation is derived from the Wiener–Hopf matrix equations established in a previous paper while the unstable case ρ ≧ 1 is treated by means of functional central limit theorems for mixing processes.

The horizon ξ T(x) of a random field ζ (x, y) of right circular cones on a plane is investigated. It is supposed that bases of cones are centered at points sn = (xn, yn), n = 1, 2, ···, on the (X, Y)-plane, constituting a Poisson point process S with intensity λ0 > 0 in a strip ΠT = {(x, y): – ∞< x <∞, 0 ≦ y ≦ T}, while altitudes of the cones h1, h2, · ·· are of the form hn = hn* + f(yn), n = 1, 2, ···, where f(y) is an increasing continuous function on [0,∞), f(0) = 0, and h1*, h2*, · ·· is a sequence of i.i.d. positive random variables which are independent of the Poisson process S and have a distribution function F(h) with density p(h).

Let denote the expected mean number of local maxima of the process ξ T(x) per unit length of the X-axis. We obtain an exact formula for under an arbitrary trend function f(y). Conditions sufficient for the limit being infinite are obtained in two cases: (a) h1* has the uniform distribution in [0, H], f(y) = kyγ; (b) h1* has the Rayleigh distribution, f(y) = k[log(y + 1)]γ. (In both cases γ 0 and 0 < k∞.) The corresponding sufficient conditions are: 0 < γ< 1 in case (a), 0 < γ< 1/2 in case (b).

In this paper, we present the distribution of the coalescence time of two DNA sequences (or genes) subject to symmetric migration between two islands, and conditional on the observed number of segregating sites in the sequences. The distribution for the segregating-site pattern is also obtained. Some surprising results emerge when both genes are initially on the same island. First, the post-data mean coalescence time is shown to be dependent on the migration parameter, as opposed to the pre-data mean. Second, both the post-data density and expectation for the coalescence time are shown to converge, in the weak-migration limit, to the corresponding panmictic results, as opposed to the pre-data situation where there is convergence in the density but not in the expectation. Finally, it is shown that there is convergence in the weak-migration limit in the distribution of the number of segregating sites but not in the expectation and variance. Numerical and graphical results for samples of size greater than two are also presented.

We consider the repairman problem which corresponds to the finite population M/M/1 queue. Asymptotic approximations for the transient probability distribution of the number of broken machines constructed when the number M of machines is large and the service (repair) rate is also large, specifically, O(M). The approximations are constructed by using singular perturbation techniques such as the ray method, boundary layer theory, and the method of matched asymptotic expansions. Extensive numerical comparisons show the quality of our approximations.

Existence and finiteness of the sample-mean limit of sojourn times of jobs in a queueing system are investigated. The queueing system operates under rather general multiprocessor disciplines allowing job classes and priorities. The input stream of jobs consisting of job classes and interarrival and processing times is stationary and ergodic and may contain batch arrivals. Existence of the sample-mean limit is proved by means of the superadditive ergodic theorem, and its finiteness is controlled by uniform mixing of the input stream.

In this paper we develop maximum likelihood procedures for parameter estimation and hypothesis testing for three classes of point processes that have been used to model rainfall occurrences; renewal processes, Neyman-Scott processes, and RCM processes (which are members of the family of Cox processes). The statistical inference procedures developed in this paper are based on the intensity process