Sharma and Autar [6] have formed a functional equation in two variables containing two parameters which, as a particular case, includes Daróczy's [1] generalized functional equation used in information theory. The solutions of our functional equation, under suitable boundary conditions, lead to an inaccuracy function of a new type. A characterization of this inaccuracy function and the convexity of Daróczy's entropy of type-ß have been given in this paper.

Level crossings in a stationary dam process with additive input and arbitrary release are considered and an explicit expression for the expected number of downcrossings (and also overcrossings) of a fixed level, per time unit, is obtained. This leads to a short derivation of a basic relation which the stationary distribution of a general dam must satisfy.

We choose a uniform random point in a given convex body K in n-dimensional Euclidean space and through that point the secant of K with random direction chosen independently and isotropically. Given the volume of K, the expectation of the length of the resulting random secant of K was conjectured by Enns and Ehlers [5] to be maximal if K is a ball. We prove this, and we also treat higher-dimensional sections defined in an analogous way. Next, we consider a finite number of independent isotropic uniform random flats meeting K, and we prove that certain geometric probabilities connected with these again become maximal when K is a ball.

A central limit theorem is proved for the sum of random variables Xr all having the same form of distribution and each of which depends on a parameter which is the number occurring in the rth cell of a multinomial distribution with equal probabilities in N cells and a total n, where nN–1 tends to a non-zero constant. This result is used to prove the asymptotic normality of the distribution of the fractional volume of a large cube which is not covered by N interpenetrating spheres whose centres are at random, and for which NV–1 tends to a non-zero constant. The same theorem can be used to prove asymptotic normality for a large number of occupancy problems.

We consider weak lumpability of finite homogeneous Markov chains, which is when a lumped Markov chain with respect to a partition of the initial state space is also a homogeneous Markov chain. We show that weak lumpability is equivalent to the existence of a direct sum of polyhedral cones that is positively invariant by the transition probability matrix of the original chain. It allows us, in a unified way, to derive new results on lumpability of reducible Markov chains and to obtain spectral properties associated with lumpability.

Many random variables arising in problems of geometric probability have intractable densities, and it is very difficult to find probabilities or percentage points based on these densities. A simple approximation, a generalization of the chi-square distribution, is suggested, to approximate such densities; the approximation uses the first three moments. These may be theoretically derived, or may be obtained from Monte Carlo sampling.

The approximation is illustrated on random variables (the area, the perimeter, and the number of sides) associated with random polygons arising from two processes in the plane. Where it can be checked theoretically, the approximation gives good results. It is compared also with Pearson curve fits to the densities.

Suppose that n persons each know a different piece of information, and that whenever a pair of persons talk on the telephone each tells the other all the information that he knows at the time. If calls are made at random, we show that the expected number of calls necessary until everyone knows all n pieces of information is asymptotically 1.5 n log n. This sharpens an earlier result of J. W. Moon.

In this paper we study the bias and the overtaking optimality criteria for continuous-time jump Markov decision processes in general state and action spaces. The corresponding transition rates are allowed to be unbounded, and the reward rates may have neither upper nor lower bounds. Under appropriate hypotheses, we prove the existence of solutions to the bias optimality equations, the existence of bias optimal policies, and an equivalence relation between bias and overtaking optimality.

The concept of strong ergodicity for discrete-time homogeneous Markov chains has been characterized in several ways (Dobrushin (1956), Lin (1975), Isaacson and Tweedie (1978)). In this paper the characterization using mean visit times (Huang and Isaacson (1977)) is extended to continuous-time Markov chains. From this it follows that for a certain subclass of continuous-time Markov chains, X(t), is strongly ergodic if and only if the associated embedded chain is Cesaro strongly ergodic.

We study the limiting behaviour of a manpower system where the non-homogeneous Markov chain model proposed by Young and Vassiliou (1974) is applicable. This is done in the cases where the input is a time-homogeneous and time-inhomogeneous Poisson random variable. It is also found that the number in the various grades are asymptotically mutually independent Poisson variates.

We show that there exists a pair of Poisson processes which are not stochastically independent, and whose sum is still a Poisson process.

Let {Xi}i=1n be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable Sn,m to be the maximum number of 1s within any m consecutive trials in {Xi}i=1n. The random variable Sn,m is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {Xi}i=1n consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of Sn,m for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.

Consider the random graph model of Barabási and Albert, where we add a new vertex in every step and connect it to some old vertices with probabilities proportional to their degrees. If we connect it to only one of the old vertices then this will be a tree. These graphs have been shown to have a power-law degree distribution, the same as that observed in some large real-world networks. We are interested in the width of the tree and we show that it is at the nth step; this also holds for a slight generalization of the model with another constant. We then see how this theoretical result can be applied to directory trees.

We propose a new approach to utilities in (state) complete markets that is consistent with state-dependent utilities. Full solutions of the optimal consumption and portfolio problem are obtained in a very general setting which includes several functional forms for utilities used in the current literature, and consider general restrictions on allowable wealths. As a secondary result, we obtain a suitable representation for straightforward numerical computations of the optimal consumption and investment strategies. In our model, utilities reflect the level of consumption satisfaction of flows of cash in future times as they are (uniquely) valued by the market when the economic agents are making their consumption and investment decisions. The theoretical framework used for the model is the one proposed in Londoño (2008). We develop the martingale methodology for the solution of the problem of optimal consumption and investment in this setting.

When the number of candidates is unknown, the problem of selecting the best during a sequence of interviews has many reasonable solutions. A simple condition for admissibility is established and it is shown that the class of Bayes solutions obtained by treating the number of candidates as a random variable is by no means complete. On the other hand, there is a single improper prior distribution for which the extended Bayes solutions constitute the whole family of admissible procedures.

Prediction of future values of a shot noise process observed on a discrete lattice of points is considered. The shot magnitudes are assumed to be independent and identically distributed and to arrive via a Poisson process; the effect of each shot dissipates and/or accumulates according to a known shot function. Conditional mean and linear point predictors of future process values are developed. Distributional prediction, obtained through saddlepoint approximation of the conditional distributions of the process, is also explored.

We propose two distance-based topological indices (level index and Gini index) as measures of disparity within a single tree and within tree classes. The level index and the Gini index of a single tree are measures of balance within the tree. On the other hand, the Gini index for a class of random trees can be used as a comparative measure of balance between tree classes. We establish a general expression for the level index of a tree. We compute the Gini index for two random classes of caterpillar trees and see that a random multinomial model of trees with finite height has a countable number of limits in [0, ⅓], whereas a model with independent level numbers fills the spectrum (0, ⅓].

The two-sided nonlinear boundary crossing probabilities for one-dimensional Brownian motion and related processes have been studied in Fu and Wu (2010) based on the finite Markov chain imbedding technique. It provides an efficient numerical method to computing the boundary crossing probabilities. In this paper we extend the above results for high-dimensional Brownian motion. In particular, we obtain the rate of convergence for high-dimensional boundary crossing probabilities. Numerical results are also provided to illustrate our results.

The use of Erlangian distributions has been proposed for the approximation of more general types of distributions of interarrival and service times in single-server queueing systems. Any Erlangian approximation should have the same mean and variance as the distribution it approximates, but it is not obvious what effect the various possible approximants have on the behaviour of the system. A major difference between approximants is their degree of skewness and accordingly, numerical results for various approximants are obtained for (a) the mean time spent by a customer in a simple single-server system, and (b) the mean queue length in a system with bulk service. Skewness is shown to have little effect on these quantities.