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A class of games for finding a leader among a group of candidates is studied in detail. This class covers games based on coin tossing and rock-paper-scissors as special cases and its complexity exhibits similar stochastic behaviors: either of logarithmic mean and bounded variance or of exponential mean and exponential variance. Many applications are also discussed.

In this note we re-examine the analysis of the paper "On the martingale property of stochastic exponentials" by B. Wong and C.C. Heyde, Journal of Applied Probability, 41(3):654-664, 2004. Some counterexamples are presented and alternative formulations are discussed., Comment: To appear in Journal of Applied Probability, 11 pages

Both paper and author (D. R. McNeil, (1968)) will be referred to below as DRM. The said paper deals with the following situation: an intersection is controlled by a traffic light with a fixed cycle time, T; the possibility of other delays, e.g., due to turning vehicles, is ignored; arrivals at the light form a compound Poisson process; if vehicles arrive to find the light green and the queue empty they are not delayed, while in the contrary case they depart when they reach the head of the queue, providing the light is green, each vehicle taking a constant time to move off. The length of the effective red period is R. For further details and discussion, DRM may be consulted.

In a recent paper Weiss (1965) has suggested a simple model for a carrier-borne epidemic such as typhoid. He considers a population (of size m) of susceptibles into which a number (k) of carriers is introduced. These carriers exhibit no overt symptoms and are only detectable by the discovery of infected persons. He supposed that after the initial introduction of the carriers, the population remains entirely closed and no new carriers arise. The epidemic then progresses until either all the carriers have been traced and isolated or until the entire population has succumbed to the disease.

The utilisation factor for a discrete dam, defined as the stationary probability of non-emptiness of the dam just before release, is obtained for a class of models that includes the Odoom–Lloyd model, the Anis–Lloyd model, the model of Herbert, etc. The method used points to a different approach to measuring dam utilisation via the actual utilisation, which appears to be more fruitful, and this is discussed in the last section.

The purpose of this note is to point out some difficulties which arise in connection with the stochastic calculus used in the previous paper and to suggest some ways in which these difficulties may be overcome. It turns out that if one wishes to restrict the application of the stochastic calculus to stochastic processes whose sample functions are step-functions with probability one, then difficulties of the following sort arise. Suppose X(ω, t) is a stochastic process whose sample functions are, with probability one, right-continuous step-functions in t on the interval [a, b). Then for every ω outside a null set N and every t ɛ [a, b), there is a δ(ω, t) such that < 5(w, t) implies X(ω, t + h) = X(ω, t) which in turn implies (X(ω, t + h) — X(ω, t))/h → 0 with probability one as Since convergence with probability one implies convergence in probability, it follows that the right-hand derivative of X(ω, t) in probability is zero at every t ɛ [a, b). Consequently, if the left-hand t ɛ [a, b), then it is also zero, but in general left-hand derivatives in probability on [a, b) may not even exist as finite-valued random functions. It is also worthwhile to observe that if a process is not differentiable in probability, then it is not differentiable in quadratic mean. From these observations it follows that the claims in Section 6 of the previous paper regarding the existence of non-zero derivatives in quadratic mean are in error, for if a derivative in quadratic mean exists, then it is a derivative in probability and is therefore zero. Incidentally, it turns out that the Poisson process has a zero derivative in probability but is not differentiable in quadratic mean. The fact that the Poisson process has a derivative in probability seems to depend rather crucially on the fact that it is also a process with independent increments.

In this paper, to model cascading failures, a new stochastic failure model is proposed. In a system subject to cascading failures, after each failure of the component, the remaining component suffers from increased load or stress. This results in shortened residual lifetimes of the remaining components. In this paper, to model this effect, the concept of the usual stochastic order is employed along with the accelerated life test model, and a new general class of stochastic failure models is generated.

In this paper, we first use the distribution of the number of records to demonstrate that the right tail probabilities of counts of rare events are generally better approximated by the right tail probabilities of Poisson distribution than {those} of normal distribution. We then show the moderate deviations in Poisson approximation generally require an adjustment and, with suitable adjustment, we establish better error estimates of the moderate deviations in Poisson approximation than those in \cite{CFS}. Our estimates contain no unspecified constants and are easy to apply. We illustrate the use of the theorems in six applications: Poisson-binomial distribution, matching problem, occupancy problem, birthday problem, random graphs and 2-runs. The paper complements the works of \cite{CC92,BCC95,CFS}., 29 pages and 5 figures

This paper generalizes the Kunita–Watanabe decomposition of an $L^2$ space. The generalization comes from using nonlinear stochastic integrals where the integrator is a family of continuous martingales bounded in $L^2$ . This result is also the solution of an optimization problem in $L^2$ . First, martingales are assumed to be stochastic integrals. Then, to get the general result, it is shown that the regularity of the family of martingales with respect to its spatial parameter is inherited by the integrands in the integral representation of the martingales. Finally, an example showing how the results of this paper, with the Clark–Ocone formula, can be applied to polynomial functions of Brownian integrals.

We study a Markovian agent-based model (MABM) in this paper. Each agent is endowed with a local state that changes over time as the agent interacts with its neighbours. The neighbourhood structure is given by a graph. In a recent paper [Simon et al. 2011], the authors used the automorphisms of the underlying graph to generate a lumpable partition of the joint state space ensuring Markovianness of the lumped process for binary dynamics. However, many large random graphs tend to become asymmetric rendering the automorphism-based lumping approach ineffective as a tool of model reduction. In order to mitigate this problem, we propose a lumping method based on a notion of local symmetry, which compares only local neighbourhoods of vertices. Since local symmetry only ensures approximate lumpability, we quantify the approximation error by means of Kullback-Leibler divergence rate between the original Markov chain and a lifted Markov chain. We prove the approximation error decreases monotonically. The connections to fibrations of graphs are also discussed., Comment: 28 pages, 4 figures

This paper considers two variants of M/G/1 queues with impatient customers, which are denoted by M/G/1+Gw and M/G/1+Gs. In the M/G/1+Gw queue customers have deadlines for their waiting times, and they leave the system immediately if their services do not start before the expiration of their deadlines. On the other hand, in the M/G/1+Gs queue customers have deadlines for their sojourn times, where customers in service also immediately leave the system when their deadlines expire. In this paper we derive comparison results for performance measures of these models. In particular, we show that if the service time distribution is new better than used in expectation, then the loss probability in the M/G/1+Gs queue is greater than that in the M/G/1+Gw queue.

The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.

In a defaultable market, an investor trades having only partial information about the behavior of the market. Taking into account the intraday stock movements, the risky asset prices are modelled by marked point processes. Their dynamics depend on an unobservable process, representing the amount of news reaching the market. This is a marked point process, which may have common jump times with the risky asset price processes. The problem of hedging a defaultable claim is studied. In order to discuss all these topics, in this paper we examine stochastic control problems using backward stochastic differential equations (BSDEs) and filtering techniques. The goal of this paper is to construct a sequence of functions converging to the value function, each of these is the unique solution of a suitable BSDE.

In this paper we investigate the limiting behavior of the failure rate for the convolution of two or more life distributions. In a previous paper on mixtures, Block, Mi and Savits (1993) showed that the failure rate behaves like the limiting behavior of the strongest component. We show a similar result here for convolutions. We also show by example that unlike a mixture population, the ultimate direction of monotonicity does not necessarily follow that of the strongest component.

Let 𝓈 be a finite or countable set. Given a matrix F = (F ij ) i,j∈𝓈 of distribution functions on R and a quasistochastic matrix Q = (q ij ) i,j∈𝓈 , i.e. an irreducible nonnegative matrix with maximal eigenvalue 1 and associated unique (modulo scaling) positive left and right eigenvectors u and v, the matrix renewal measure ∑ n≥0 Q n ⊗ F *n associated with Q ⊗ F := (q ij F ij ) i,j∈𝓈 (see below for precise definitions) and a related Markov renewal equation are studied. This was done earlier by de Saporta (2003) and Sgibnev (2006, 2010) by drawing on potential theory, matrix-analytic methods, and Wiener-Hopf techniques. In this paper we describe a probabilistic approach which is quite different and starts from the observation that Q ⊗ F becomes an ordinary semi-Markov matrix after a harmonic transform. This allows us to relate Q ⊗ F to a Markov random walk {(M n , S n )} n≥0 with discrete recurrent driving chain {M n } n≥0. It is then shown that renewal theorems including a Choquet-Deny-type lemma may be easily established by resorting to standard renewal theory for ordinary random walks. The paper concludes with two typical examples.

Suppose that X 1, …, X n are random variables with the same known marginal distribution F but unknown dependence structure. In this paper we study the smallest possible value of P(X 1 + · · · + X n < s) over all possible dependence structures, denoted by m n,F (s). We show that m n,F (ns) → 0 for s no more than the mean of F under weak assumptions. We also derive a limit of m n,F (ns) for any s ∈ R with an error of at most n -1/6 for general continuous distributions. An application of our result to risk management confirms that the worst-case value at risk is asymptotically equivalent to the worst-case expected shortfall for risk aggregation with dependence uncertainty. In the last part of this paper we present a dual presentation of the theory of complete mixability and give dual proofs of theorems in the literature on this concept.

We consider possibly nonlinear distributional fixed-point equations on weighted branching trees, which include the well-known linear branching recursion. In Jelenković and Olvera-Cravioto (2012), an implicit renewal theorem was developed that enables the characterization of the power-tail asymptotics of the solutions to many equations that fall into this category. In this paper we complement the analysis in our 2012 paper to provide the corresponding rate of convergence.

This paper provides a simple proof for the fact that the hitting time to an infrequently visited subset for a one-dependent regenerative process converges weakly to an exponential distribution. Special cases are positive recurrent Harris chains and Harris processes. The paper further extends this class of limit theorems to ‘rewards’ that are cumulated to the hitting time of such a rare set.

This paper is concerned with statistical inference for both continuous and discrete phase-type distributions. We consider maximum likelihood estimation, where traditionally the expectation-maximization (EM) algorithm has been employed. Certain numerical aspects of this method are revised and we provide an alternative method for dealing with the E-step. We also compare the EM algorithm to a direct Newton–Raphson optimization of the likelihood function. As one of the main contributions of the paper, we provide formulae for calculating the Fisher information matrix both for the EM algorithm and Newton–Raphson approach. The inverse of the Fisher information matrix provides the variances and covariances of the estimated parameters.

In this paper, the componentwise increasing convex order, the upper orthant order, the upper orthant convex order, and the increasing directionally convex order for random vectors are generalized to hierarchical classes of integral stochastic order relations. The elements of the generating classes of functions possess nonnegative partial derivatives up to some given degrees. Some properties of these new stochastic order relations are studied. Particular attention is paid to the comparison of weighted sums of the respective components of ordered random vectors. By providing a unified derivation of standard multivariate stochastic orderings, the present paper shows how some well-known results derive from a common principle.

This thesis is concerned with the study of stochastic epidemic models for infectious diseases in heterogeneous populations. All diseases treated are of SIR type, i.e. individuals are either Susceptible, Infectious or Recovered (and immune). The transitions between these states are according to S to I to R. The thesis consists of five papers. Papers I and II treat approximations for the distribution of the time to extinction. In Paper I, a sub-community version of the SIR model with demography is considered. The interest is in how the distribution of the time to extinction is affected by varying the degree of interaction between the sub-communities. Paper II is concerned with a two-type version of Bartlett's model. The distribution of the time to extinction is studied when the difference in susceptibility/infectivity between the types of individuals is varied. Papers III and IV treat random intersection graphs with tunable clustering. In Paper III a Reed-Frost epidemic is run on such a random intersection graph. The critical parameter R_0 and the probability of a large outbreak are derived and it is investigated how these quantities are affected by the clustering in the graph. In Paper IV the interest is in the component structure of such a graph, i.e. the size and the emergence of a giant component is studied. The last paper, Paper V, treats the situation when a simple epidemic is running in a varying environment. A varying environment is in this context any external factor that affects the contact rate in the population, but is itself unaffected by the population. The model treated is a term-time forced version of the stochastic general epidemic where the contact rate is modelled by an alternating renewal process. A threshold parameter R_* and the probability of a large outbreak are derived and studied.

This paper is concerned with the analysis of Markov decision processes in which a natural form of termination ensures that the expected future costs are bounded, at least under some policies. Whereas most previous analyses have restricted attention to the case where the set of states is finite, this paper analyses the case where the set of states is not necessarily finite or even countable. It is shown that all the existence, uniqueness, and convergence results of the finite-state case hold when the set of states is a general Borel space, provided we make the additional assumption that the optimal value function is bounded below. We give a sufficient condition for the optimal value function to be bounded below which holds, in particular, if the set of states is countable.

Let X = (X1, X2, …, Xn) be an exchangeable random vector, and write X(1:i) = min{X1, X2, …, Xi}, 1 ≤ i ≤ n. In this paper we obtain conditions under which X(1:i) decreases in i in the hazard rate order. A result involving more general (that is, not necessarily exchangeable) random vectors is also derived. These results are applied to obtain the limiting behaviour of the hazard rate function of the lifetimes of various coherent systems in reliability theory. The notions of the Samaniego signatures and the minimal signatures of such systems are extensively used in the paper. An interesting relationship between these two signatures is obtained. The results are illustrated in a series of examples.

We study the uncertain dichotomous choice model. In this model, a group of expert decision makers is required to select one of two alternatives. The applications of this model are relevant to a wide variety of areas. A decision rule translates the individual opinions of the members into a group decision, and is optimal if it maximizes the probability of the group making a correct choice. In this paper, we assume the correctness probabilities of the experts to be independent random variables selected from some given distribution. Moreover, the ranking of the members in the group is (at least partly) known. Thus, one can follow rules based on this ranking. The extremes are the expert rule and the majority rule. The probabilities of the two extreme rules being optimal were compared in a series of early papers, for a variety of distributions. In most cases, the asymptotic behaviours of the probabilities of the two extreme rules followed the same patterns. Do these patterns hold in general? If not, what are the ranges of possible asymptotic behaviours of the probabilities of the two extreme rules being optimal? In this paper, we provide satisfactory answers to these questions.

This thesis consists of four papers.In paper 1, we prove central limit theorems for Markov chains under (local) contraction conditions. As a corollary we obtain a central limit theorem for Markov chains associated with iterated function systems with contractive maps and place-dependent Dini-continuous probabilities.In paper 2, properties of inverse subordinators are investigated, in particular similarities with renewal processes. The main tool is a theorem on processes that are both renewal and Cox processes.In paper 3, distributional properties of supercritical and especially immortal branching processes are derived. The marginal distributions of immortal branching processes are found to be compound geometric.In paper 4, a description of a dynamic population model is presented, such that samples from the population have genealogies as given by a Lambda-coalescent with mutations. Depending on whether the sample is grouped according to litters or families, the sampling distribution is either regenerative or non-regenerative.

In this paper, we describe a class of Wiener functionals that are ‘indeterminate by their moments’, that is, whose distributions are not uniquely determined by their moments. In particular, it is proved that the integral of a geometric Brownian motion is indeterminate by its moments and, moreover, shown that previous proofs of this result are incorrect. The main result of this paper is based on geometric inequalities in Gauss space and on a generalization of the Krein criterion due to H. L. Pedersen.

This paper studies a discrete-time single-server queue with two independent inputs and service interruptions. One of the inputs to the queue is an independent and identically distributed process. The other is a much more general process and it is not required to be Markov nor is it required to be stationary. The service interruption process is also general and it is not required to be Markov or to be stationary. This paper shows that a stochastic decomposition property for the virtual waiting-time process holds in the discrete-time single-server queue with service interruptions. To the best of the author's knowledge, no stochastic decomposition results for virtual waiting-time processes in non-work-conserving queues, such as queues with service interruptions, have been obtained before and only work-conserving queues have been studied in the literature.

In this paper, we study the on-line parameter estimation problem for a partially observable system subject to deterioration and random failure. The state of the system evolves according to a continuous-time homogeneous Markov process with a finite state space. The state of the system is hidden except for the failure state. When the system is operating, only the information obtained by condition monitoring, which is related to the working state of the system, is available. The condition monitoring observations are assumed to be in continuous range, so that no discretization is required. A recursive maximum likelihood (RML) algorithm is proposed for the on-line parameter estimation of the model. The new RML algorithm proposed in the paper is superior to other RML algorithms in the literature in that no projection is needed and no calculation of the gradient on the surface of the constraint manifolds is required. A numerical example is provided to illustrate the algorithm.

In this paper we review some recent work on limit results on realised power variation, that is, sums of powers of absolute increments of various semimartingales. A special case of this analysis is realised variance and its probability limit, quadratic variation. Such quantities often appear in financial econometrics in the analysis of volatility. The paper also provides some new results and discusses open issues.

This thesis, consisting of six papers and a summary, studies the area of continuous time financial mathematics. A unifying theme for many of the problems studied is the implications of possible mis-specifications of models. Intimately connected with this question is, perhaps surprisingly, convexity properties of option prices. We also study qualitative behavior of different optimal stopping boundaries appearing in option pricing.In Paper I a new condition on the contract function of an American option is provided under which the option price increases monotonically in the volatility. It is also shown that American option prices are continuous in the volatility.In Paper II an explicit pricing formula for the perpetual American put option in the Constant Elasticity of Variance model is derived. Moreover, different properties of this price are studied.Paper III deals with the Russian option with a finite time horizon. It is shown that the value of the Russian option solves a certain free boundary problem. This information is used to analyze the optimal stopping boundary.A study of perpetual game options is performed in Paper IV. One of the main results provides a condition under which the value of the option is increasing in the volatility.In Paper V options written on several underlying assets are considered. It is shown that, within a large class of models, the only model for the stock prices that assigns convex option prices to all convex contract functions is geometric Brownian motion.Finally, in Paper VI it is shown that the optimal stopping boundary for the American put option is convex in the standard Black-Scholes model.

Suppose that {X s , 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝ m → [0,∞) is a (weighted) l q (ℝ m )-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the L p (μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(X s ), 0 ≤ s ≤ T. By using various geometric inequalities in Wiener space, this paper gives upper and lower bounds for the distribution function of Y and proves that the distribution function is log-concave and absolutely continuous on every open subset of the distribution's support. Moreover, the paper derives tail probabilities, presents sharp moment inequalities, and shows that Y is indetermined by its moments. The paper will also discuss the so-called moment-matching method for the pricing of Asian-styled basket options.

In this paper, we study the classification of matrix GI/M/1-type Markov chains with a tree structure. We show that the Perron–Frobenius eigenvalue of a Jacobian matrix provides information for classifying these Markov chains. A fixed-point approach is utilized. A queueing application is presented to show the usefulness of the classification method developed in this paper.

In this paper, we consider denumerable-state continuous-time Markov decision processes with (possibly unbounded) transition and reward rates and general action space under the discounted criterion. We provide a set of conditions weaker than those previously known and then prove the existence of optimal stationary policies within the class of all possibly randomized Markov policies. Moreover, the results in this paper are illustrated by considering the birth-and-death processes with controlled immigration in which the conditions in this paper are satisfied, whereas the earlier conditions fail to hold.

This paper presents an algorithmic procedure to calculate the delay distribution of a type k customer in a first-come-first-served (FCFS) discrete-time queueing system with multiple types of customers, where each type has different service requirements (the MMAP[K]/PH[K]/1 queue). First, we develop a procedure, using matrix analytical methods, to handle arrival processes that do not allow batch arrivals to occur. Next, we show that this technique can be generalized to arrival processes that do allow batch arrivals to occur. We end the paper by presenting some numerical examples.

In this paper we study the allocation problem of relevations in coherent systems. The optimal allocation strategies are obtained by implementing stochastic comparisons of different policies according to the usual stochastic order and the hazard rate order. As special cases of relevations, the load-sharing and minimal repair policies are further investigated. Sufficient (and necessary) conditions are established for various stochastic orderings. Numerical examples are also presented as illustrations.

Drawdown/regret times feature prominently in optimal stopping problems, in statistics (CUSUM procedure), and in mathematical finance (Russian options). Recently it was discovered that a first passage theory with more general drawdown times, which generalize classic ruin times, may be explicitly developed for spectrally negative Lévy processes [9, 20]. In this paper we further examine the general drawdown-related quantities in the (upward skip-free) time-homogeneous Markov process, and then in its (general) tax process by noticing the pathwise connection between general drawdown and the tax process.

This paper investigates the ordering properties of largest claim amounts in heterogeneous insurance portfolios in the sense of some transform orders, including the convex transform order and the star order. It is shown that the largest claim amount from a set of independent and heterogeneous exponential claims is more skewed than that from a set of independent and homogeneous exponential claims in the sense of the convex transform order. As a result, a lower bound for the coefficient of variation of the largest claim amount is established without any restrictions on the parameters of the distributions of claim severities. Furthermore, sufficient conditions are presented to compare the skewness of the largest claim amounts from two sets of independent multiple-outlier scaled claims according to the star order. Some comparison results are also developed for the multiple-outlier proportional hazard rates claims. Numerical examples are presented to illustrate these theoretical results.

Consider a renewal process. The renewal events partition the process into i.i.d. renewal cycles. Assume that on each cycle, a rare event called 'success’ can occur. Such successes lend themselves naturally to approximation by Poisson point processes. If each success occurs after a random delay, however, Poisson convergence may be relatively slow, because each success corresponds to a time interval, not a point. In 1996, Altschul and Gish proposed a finite-size correction to a particular approximation by a Poisson point process. Their correction is now used routinely (about once a second) when computers compare biological sequences, although it lacks a mathematical foundation. This paper generalizes their correction. For a single renewal process or several renewal processes operating in parallel, this paper gives an asymptotic expansion that contains in successive terms a Poisson point approximation, a generalization of the Altschul-Gish correction, and a correction term beyond that.

The paper considers one of the standard processes for modeling returns in finance, the stochastic volatility process with regularly varying innovations. The aim of the paper is to show how point process techniques can be used to derive the asymptotic behavior of the sample autocorrelation function of this process with heavy-tailed marginal distributions. Unlike other non-linear models used in finance, such as GARCH and bilinear models, sample autocorrelations of a stochastic volatility process have attractive asymptotic properties. Specifically, in the infinite variance case, the sample autocorrelation function converges to zero in probability at a rate that is faster the heavier the tails of the marginal distribution. This behavior is analogous to the asymptotic behavior of the sample autocorrelations of independent identically distributed random variables.

This paper studies the dependence structure and bounds of several basic prototypical parallel queueing systems with correlated arrival processes to different queues. The marked feature of our systems is that each queue viewed alone is a standard single-server queuing system extensively studied in the literature, but those queues are statistically dependent due to correlated arrival streams. The major difficulty in analysing those systems is that the presence of correlation makes the explicit computation of a joint performance measure either intractable or computationally intensive. In addition, it is not well understood how and in what sense arrival correlation will improve or deteriorate a system performance measure. The objective of this paper is to provide a better understanding of the dependence structure of correlated queueing systems and to derive computable bounds for the statistics of a joint performance measure. In this paper, we obtain conditions on arrival processes under which a performance measure in two systems can be compared, in the sense of orthant and supermodular orders, among different queues and over different arrival times. Such strong comparison results enable us to study both spatial dependence (dependence among different queues) and temporal dependence (dependence over different time instances) for a joint performance measure. Further, we derive a variety of upper and lower bounds for the statistics of a stationary joint performance measure. Finally, we apply our results to synchronized queueing systems, using the ideas combined from the theory of orthant and supermodular dependence orders and majorization with respect to weighted trees (Xu and Li (2000)). Our results reveal how a performance measure can be affected, favourably or adversely, by different types of dependencies.

Consider a janken game (scissors-paper-rock game) started by n players such that (1) the first round is played by n players, (2) the losers of each round (if any) retire from the rest of the game, and (3) the game ends when only one player (winner) is left. Let W n be the number of rounds played through the game. Among other things, it is proved that (2/3) n W n is asymptotically (as n → ∞) distributed according to the exponential distribution with mean ⅓, provided that each player chooses one of the three strategies (scissors, paper, rock) with equal probability and independently from other players in any round.

This paper arose from interest in assessing the quality of random number generators. The problem of testing randomness of a string of binary bits produced by such a generator gained importance with the wide use of public key cryptography and the need for secure encryption algorithms. All such algorithms are based on a generator of (pseudo) random numbers; the testing of such generators for randomness became crucial for the communications industry where digital signatures and key management are vital for information processing. The concept of approximate entropy has been introduced in a series of papers by S. Pincus and co-authors. The corresponding statistic is designed to measure the degree of randomness of observed sequences. It is based on incremental contrasts of empirical entropies based on the frequencies of different patterns in the sequence. Sequences with large approximate entropy must have substantial fluctuation or irregularity. Alternatively, small values of this characteristic imply strong regularity, or lack of randomness, in a sequence. Pincus and Kalman (1997) evaluated approximate entropies for binary and decimal expansions of e, π, √2 and √3 with the surprising conclusion that the expansion of √3 demonstrated much less irregularity than that of π. Tractable small sample distributions are hardly available, and testing randomness is based, as a rule, on fairly long strings. Therefore, to have rigorous statistical tests of randomness based on this approximate entropy statistic, one needs the limiting distribution of this characteristic under the randomness assumption. Until now this distribution remained unknown and was thought to be difficult to obtain. To derive the limiting distribution of approximate entropy we modify its definition. It is shown that the approximate entropy as well as its modified version converges in distribution to a χ2-random variable. The P-values of approximate entropy test statistics for binary expansions of e, π and √3 are plotted. Although some of these values for √3 digits are small, they do not provide enough statistical significance against the randomness hypothesis.

In this paper, we give necessary and sufficient conditions to ensure the validity of confidence intervals, based on the central limit theorem, in simulations of highly reliable Markovian systems. We resort to simulations because of the frequently huge state space in practical systems. So far the literature has focused on the property of bounded relative error. In this paper we focus on ‘bounded normal approximation’ which asserts that the approximation of the normal law, suggested by the central limit theorem, does not deteriorate as the reliability of the system increases. Here we see that the set of systems with bounded normal approximation is (strictly) included in the set of systems with bounded relative error.

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.