Showing papers in "Journal of Applied Probability in 1988"
TL;DR: In this article, the Lagrange multiplier associated with this constraint defines an index which reduces to the Gittins index when projects not being operated are static, and arguments are advanced to support the conjecture that, for m and n large in constant ratio, the policy of operating the m projects of largest current index is nearly optimal.
Abstract: We consider a population of n projects which in general continue to evolve whether in operation or not (although by different rules). It is desired to choose the projects in operation at each instant of time so as to maximise the expected rate of reward, under a constraint upon the expected number of projects in operation. The Lagrange multiplier associated with this constraint defines an index which reduces to the Gittins index when projects not being operated are static. If one is constrained to operate m projects exactly then arguments are advanced to support the conjecture that, for m and n large in constant ratio, the policy of operating the m projects of largest current index is nearly optimal. The index is evaluated for some particular projects.
1,092 citations
TL;DR: In this article, the hazard rate of a distribution function on [0, oc] with finite expectation is analyzed in terms of subexponentiality of its integrated tail distribution.
Abstract: Let F be a distribution function on [0, oc) with finite expectation. In terms of the hazard rate of F several conditions are given which simultaneously imply subexponentiality of F and of its integrated tail distribution Fl. These conditions apply to a wide class of longtailed distributions, and they can also be used in connection with certain random walks which occur in risk theory and queueing theory. INTEGRATED TAIL DISTRIBUTION; SUBEXPONENTIALITY; DOMINATED VARIATION; RANDOM WALK THEORY; RISK THEORY; QUEUEING THEORY
319 citations
TL;DR: In this article, Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions.
Abstract: Stein's method of obtaining rates of convergence, well known in normal and Poisson approximation, is considered here in the context of approximation by Poisson point processes, rather than their one-dimensional distributions. A general technique is sketched, whereby the basic ingredients necessary for the application of Stein's method may be derived, and this is applied to a simple problem in Poisson point process approximation.
210 citations
TL;DR: In this article, a detailed study of the asymptotic behavior of the first-passage-time p.d. and its moments is carried out for an unrestricted conditional Ornstein-Uhlenbeck process and for a constant boundary.
Abstract: A detailed study of the asymptotic behavior of the first-passage-time p.d.f. and its moments is carried out for an unrestricted conditional Ornstein-Uhlenbeck process and for a constant boundary. Explicit expressions are determined which include those earlier discussed by Sato [ 15] and by Nobile et al. [9]. In particular, it is shown that the first-passage-time p.d.f. can be expressed as the sum of exponential functions with negative exponents and that the latter reduces to a single exponential density as time increases, irrespective of the chosen boundary. The explicit expressions obtained for the first-passage-time moments of any order appear to be particularly suitable for computation purposes.
200 citations
100 citations
TL;DR: In this article, it has been shown that the same conditions imply the same conclusions without any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically.
Abstract: Foster's criterion for positive recurrence of irreducible countable space Markov chains is one of the oldest tools in applied probability theory. In various papers in JAP and AAP it has been shown that, under extensions of irreducibility such as ¢-irreducibility, analogues of and generalizations of Foster's criterion give conditions for the existence of an invariant measure Jr for general space chains, and for t to have a finite f-moment Jf (dy)f(y), where f is a general function. In the case f = 1 these cover the question of finiteness of n itself. In this paper we show that the same conditions imply the same conclusions without any irreducibility assumptions; Foster's criterion forces sufficient and appropriate regularity on the space automatically. The proofs involve detailed consideration of the structure of the minimal subinvariant measures of the chain. The results are applied to random coefficient autoregressive processes in order to illustrate the need to remove irreducibility conditions if possible. FOSTER'S CRITERIA; RECURRENCE; ERGODICITY; TIME SERIES
87 citations
TL;DR: In this article, it was shown that (T 1, · ··, T n ) belongs to various multivariate nonparametric life classes depending on the life class of the component.
Abstract: A non-homogeneous Poisson shock model has a continuous mean function Λ( t ). The k th shock S k causes simultaneous failure of the components j ∊ J ∊ {1, ···, n } with probability p J ( S k ). If T j is the lifetime of component j , it is shown that ( T 1 , · ··, T n ) belongs to various multivariate non-parametric life classes depending on the life class of .
82 citations
TL;DR: In this article, the authors consider the tail behavior of the usual almost-sure limit random variable in a supercritical simple branching process and distinguish two cases, the Schr6der case and the B6ttcher case.
Abstract: Let W be the usual almost-sure limit random variable in a supercritical simple branching process; we study its tail behaviour. For the left tail, we distinguish two cases, the 'Schroder' and 'B6ttcher' cases; both appear in work of Harris and Dubuc. The Schr6der case is related to work of Karlin and McGregor on embeddability in continuous-time (Markov) branching processes. New results are obtained for the B6ttcher case; there are links with recent work of Barlow and Perkins on Brownian motion on a fractal. The right tail is also considered. Use is made of recent progress in Tauberian theory.
74 citations
TL;DR: In this article, a sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations and the condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they considered.
Abstract: A sufficient condition is derived for the existence of a strictly stationary solution of the general bilinear time series equations. The condition is shown to reduce to the conditions of Pham and Tran (1981) and Bhaskara Rao et al. (1983) in the special cases which they consider. Under the condition specified, a solution is constructed which is shown to be causal, stationary and ergodic. It is moreover the unique causal solution and the unique stationary solution of the defining equations. In the special case when the defining equations contain no non-linear terms, our condition reduces to the well-known necessary and sufficient condition for existence of a causal stationary solution.
64 citations
TL;DR: The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated in this article, where it is shown that the expected number of vertices is O(logdn) for any polytopes, the expected vertex count is Q(logd)-n) for a simple polytopes, and the expected vertices count is O (logd' n) for an infinite polytope.
Abstract: The convex hull of n points drawn independently from a uniform distribution on the interior of a d-dimensional polytope is investigated. It is shown that the expected number of vertices is O(logdn) for any polytope, the expected number of vertices is Q(logd-' n) for any simple polytope, and the expected number of facets is O(logd' n) for any simple polytope. An algorithm is presented for constructing the convex hull of such sets of points in linear average time. EXTREME POINTS; GEOMETRIC PROBABILITY; AVERAGE-CASE ANALYSIS OF ALGORITHMS
61 citations
TL;DR: In this article, the notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established, where the limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.
Abstract: The notion of a quantum random walk in discrete time is formulated and the passage to a continuous time diffusion limit is established. The limiting diffusion is described in terms of solutions of certain quantum stochastic differential equations.
TL;DR: In this article, a set of necessary and sufficient conditions for the so-called exponential formula for failure intensities in the counting process and martingale framework is given. But their relationship to conditional survival functions does not seem to be equally well understood.
Abstract: Failure intensities in which the evaluation of hazard is based on the observation of an auxiliary random process have become very popular in survival analysis. While their definition is well known, either as the derivative of a conditional failure probability or in the counting process and martingale framework, their relationship to conditional survival functions does not seem to be equally well understood. This paper gives a set of necessary and sufficient conditions for the so-called exponential formula in this context.
TL;DR: A simple proof of the multivariate random time change theorem of Meyer (1971) is given in this article, which includes Watanabe's (1964) characterization of the Poisson process; even in this special case the present proof is simpler than existing proofs.
Abstract: A simple proof of the multivariate random time change theorem of Meyer (1971) is given. This result includes Watanabe's (1964) characterization of the Poisson process; even in this special case the present proof is simpler than existing proofs.
TL;DR: In this paper, it was shown that VI/Klog(n) - 1 a.s.d. where K_K(q, v) is a large deviation result for Markov chains in discrete time and in continuous time.
Abstract: A derivation of a law of large numbers for the highest-scoring matching subsequence is given. Let Xk, Yk be i.i.d. q = (q(i))i,s letters from a finite alphabet S and v = (v(i))E,s be a sequence of non-negative real numbers assigned to the letters of S. Using a scoring system similar to that of the game Scrabble, the score of a word w = i, i'. im is defined to be V(w) = v(il) + ? . + v(im). Let Vn denote the value of the highest-scoring matching contiguous subsequence between X1X2.. Xn and Y,Y2". Y,. In this paper, we show that VI/Klog(n) - 1 a.s. where K_K(q, v). The method employed here involves 'stuttering' the letters to construct a Markov chain and applying previous results for the length of the longest matching subsequence. An explicit form for fE Pr(S), where p(i) denotes the proportion of letter i found in the highest-scoring word, is given. A similar treatment for Markov chains is also included. Implicit in these results is a large-deviation result for the additive functional, H - E<,r v(Xn), for a Markov chain stopped at the hitting time r of some state. We give this large deviation result explicitly, for Markov chains in discrete time and in continuous time.
TL;DR: In this article, sufficient conditions on the parameters of a pure jump process under which Tx has increasing failure rate average (IFRA), increasing failure ratio (IFR) or logconcave density (PF2) are identified.
Abstract: Let Tx be the time it takes for a pure jump process, which starts at 0, to cross a threshold x > 0. Sufficient conditions on the parameters of this process under which Tx has increasing failure rate average (IFRA), increasing failure rate (IFR) or logconcave density (PF2) are identified. The conditions for IFRA are weaker than those of Drosen (1986). Sufficient conditions on the parameter of a pure jump process for Tx to the IFR or PF2 are not available in the literature.
TL;DR: In this article, a fonction de densite de probabilite importante dans le processus de Dirichlet de Poisson de la genetique des populations is studied.
Abstract: On etudie une fonction de densite de probabilite importante dans le processus de Dirichlet de Poisson de la genetique des populations. On donne un algorithme de calcul
TL;DR: In this article, a tandem queue with a FIFO multiserver system at each stage and a renewal process of external arrivals is shown to be regenerative by modeling it as a Harris-ergodic Markov chain.
Abstract: A tandem queue with a FIFO multiserver system at each stage, i.i.d. service times and a renewal process of external arrivals is shown to be regenerative by modeling it as a Harris-ergodic Markov chain. In addition, some explicit regeneration points are found. This generalizes the results of Nummelin (1981) in which a single server system is at each stage and the result of Charlot et al. (1978) in which the FIFO GI/GI/c queue is modeled as a Harris chain. In preparing for our result, we study the random assignment queue and use it to give a new proof of Harris ergodicity of the FIFO queue.
TL;DR: In this article, the arc-sine laws form one of the cornerstones of classical one-dimensional fluctuation theory, and they are considered in higher dimensions, where knowledge of fluctuation theories remains a great deal less complete.
Abstract: The arc-sine laws form one of the cornerstones of classical one-dimensional fluctuation theory. In higher dimensions, knowledge of fluctuation theory remains a great deal less complete. Motivated by this, we consider higher-dimensional analogues of the classical arc-sine laws.
TL;DR: This article showed that the number of non-overlapping occurrences of long recurrent patterns has approximately a Poisson distribution under general conditions, using a sequence of independent experiments, each producing a letter from a given alphabet.
Abstract: A sequence of independent experiments is performed, each producing a letter from a given alphabet. Using a result by Barbour and Eagleson (1984) we prove that under general conditions the number of non-overlapping occurrences of long recurrent patterns has approximately a Poisson distribution.
TL;DR: In this paper, sufficient conditions for strong ergodicity of discrete-time non-homogeneous Markov chains have been given using the left eigenvectors ψ n of Pn.
Abstract: Sufficient conditions for strong ergodicity of discrete-time non-homogeneous Markov chains have been given in several papers. Conditions have been given using the left eigenvectors ψ n of Pn (ψ nPn = ψ n ) and also using the limiting behavior of Pn. In this paper we consider the analogous results in the case of continuous-time Markov chains where one uses the intensity matrices Q(t) instead of P(s, t). A bound on the rate of convergence of certain strongly ergodic chains is also given.
TL;DR: This article present an identite qui peut parfois exprimer la distribution conjointe de plusieurs combinaisons lineaires d'espacements uniformes comme une somme de distributions plus simples.
Abstract: On presente une identite qui peut parfois exprimer la distribution conjointe de plusieurs combinaisons lineaires d'espacements uniformes comme une somme de distributions plus simples
TL;DR: In this paper, the reliability function of a parallel redundant system whose components share a common unknown environment cannot be characterized by any of the well-known classes of distributions that have been proposed in the mathematical theory of reliability.
Abstract: A multivariate distribution for describing the life-lengths of the components of a system which operates in an environment that is different from the test bench environment has been proposed by Lindley and Singpurwalla (1986). In this paper, the properties of the reliability function of such a system are studied and comparisons made with the reliability function obtained under the assumption of independence. It is interesting to note that the reliability function of parallel redundant systems whose components share a common unknown environment cannot be characterized by any of the well-known classes of distributions that have been proposed in the mathematical theory of reliability. This observation suggests the need for defining a new class of failure distributions. A formula for making Bayesian inferences for the reliability function is also given. RELIABILITY OF DEPENDENT COMPONENT SYSTEMS; CROSSING PROPERTIES; GOLDEN RATIO; BAYESIAN INFERENCE IN RELIABILITY; CLASSES OF LIFE DISTRIBUTIONS; ROBUSTNESS OF THE INDEPENDENCE ASSUMPTION
TL;DR: For a strong Markov process on the line with continuous paths, the Karlin-McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations.
Abstract: For a strong Markov process on the line with continuous paths the Karlin-McGregor determinant formula of coincidence probabilities for multiple particle systems is extended to allow the individual component processes to start at variable times and run for variable durations. The extended formula is applied to a variety of combinatorial problems including counts of non-crossing paths in the plane with variable start and end points, dominance orderings, numbers of dominated majorization orderings, and time-inhomogeneous random walks.
TL;DR: In this paper, it was shown that the limit distribution of γ n (λ /n) is a Poisson distribution with expectation if n → ∞ and 0 < λ < 1.
Abstract: Let Γ n (p) denote a random graph with n vertices in which any two vertices, independently of the others, are connected by an edge with probability p where . Denote by γ n (p) the total number of cycles in the graph Γ n (p). The main object of this paper is to prove that the limit distribution of γ n (λ /n) is a Poisson distribution with expectation if n → ∞and 0 < λ< 1.
TL;DR: In this article, it was shown that the gamma distribution with shape parameter a can be obtained through a p-thinning for every 0 1, while for any ε > 0, the distribution cannot be obtained by thinning.
Abstract: It is shown that the gamma distribution with shape parameter a can be obtained through a p-thinning for every 0 1, the gamma distribution cannot be obtained through thinning. The class of renewal processes with gamma-distributed times between events is considered. It is shown that an ordinary gamma renewal process is a Cox process if and only if 0 < a < 1. Necessary and sufficient conditions for delayed gamma renewal processes to be Cox are also given. Finally, a short description of the gamma renewal process as a Cox process is given.
TL;DR: In this article, for m = 0, 1,· · ··, 14 direct computations give critical values which are lower bounds for the critical value of the original basic contact process.
Abstract: Instead of the basic contact process on with infection rate λ we consider for m ≧ 0 the Markov process starting with ξ 0(k) = 1 for k ≧ 0 and ξ 0(k)= 0 for k < 0 and with changing only those k which are at most m places to the right of the left-most infected cell. For m = 0, 1,· ··, 14 direct computations give critical values which are lower bounds for the critical value of the original basic contact process.
TL;DR: In this paper, a shock model in which the time intervals between shocks are in the domain of attraction of a stable law of order less than 1 or relatively stable is considered, and weak limit theorems are established for the cumulative magnitude of the shocks.
Abstract: A shock model in which the time intervals between shocks are in the domain of attraction of a stable law of order less than 1 or relatively stable is considered. Weak limit theorems are established for the cumulative magnitude of the shocks and the first time the cumulative magnitude exceeds z without any assumption on the
TL;DR: In this paper, the distribution of the pattern of evolutionarily stable strategies for A will depend, if n - 3, on this underlying distribution. But this is not the case for n = 3, and some results are obtained for n - 4.
Abstract: Suppose the n X n matrix A gives the payoffs for some evolutionary game, and its entries are the values of independent, identically distributed, continuous random variables. The distribution of the pattern of evolutionarily stable strategies for A will depend, if n - 3, on this underlying distribution. A fairly complete picture for n = 3 is found, and some results are obtained for n _ 4.
TL;DR: In this article, the long run expected cost per unit time of running the system is obtained as well as the variance of the cost which are used to get optimal times of replacement of the system.
Abstract: A system is subject to shocks; each shock at time t increases the cumulative damage λ (t) by a constant amount, while the system is subject to repair in between the shocks which brings down λ (t) at a constant rate. The shock arrival process is an inhomogeneous Poisson process with intensity function λ (t) and each shock weakens the system making it more expensive to run. The long-run expected cost per unit time of running the system is obtained as well as the variance of the cost which are used to get optimal times of replacement of the system.
TL;DR: In this article, sufficient conditions on λ are found which imply that T is either a Block-Savits MIFRA (multivariate increasing failure rate average) or a Savits-MIFR (mixture of increasing failure rates).
Abstract: If T = ( T 1 , · ··, T n ) is a vector of random lifetimes then its distribution can be determined by a set λof multivariate conditional hazard rates. In this paper, sufficient conditions on λare found which imply that T is Block–Savits MIFRA (multivariate increasing failure rate average) or Savits MIFR (multivariate increasing failure rate). Applications for a multivariate reliability model of Ross and for load-sharing models are given. The relationship between Shaked and Shanthikumar model of multivariate imperfect repair and the MIFRA property is also discussed.