Mostrando recursos 1 - 20 de 1.831

  1. Correction: Fractional Poisson process: long-range dependence and applications in ruin theory

    Biard, R.; Saussereau, B.

  2. Improved Chen‒Stein bounds on the probability of a union

    Ross, Sheldon M.
    We improve the Chen‒Stein bounds when applied to the probability of a union. When the probability is small, the improvement in the distance from the lower to the upper bound is roughly a factor of 2. Further improvements are determined when the events of the union are either negatively or positively dependent.

  3. An alternative axiomatic characterisation of pricing operators

    Kassberger, Stefan; Liebmann, Thomas
    In the spirit of the axiomatic approach by Rogers (1998) we show the equivalence between a set of assumptions on the behaviour of prices and the existence of a representation of these prices as conditional expectations. We rely on only weak assumptions and avoid any a priori modelling of negligible events or of any market filtration. Rather, both endogenously emerge along with the representation as conditional expectations.

  4. A generalization of the Mabinogion sheep problem of D. Williams

    Lin, Yi-Shen
    In his well-known textbook Probability with Martingales, David Williams (1991) introduces the Mabinogion sheep problem in which there is a magical flock of sheep, some black, some white. At each stage n=1,2,..., a sheep (chosen randomly from the entire flock, independently of previous events) bleats; if this bleating sheep is white, one black sheep (if any remain) instantly becomes white; if the bleating sheep is black, one white sheep (if any remain) instantly becomes black. No births or deaths occur. Suppose that one may remove any number of white sheep from the flock at (the end of) each stage n=0,1,.......

  5. Synchronization and fluctuation theorems for interacting Friedman urns

    Sahasrabudhe, Neeraja
    We consider a model of N interacting two-colour Friedman urns. The interaction model considered is such that the reinforcement of each urn depends on the fraction of balls of a particular colour in that urn as well as the overall fraction of balls of that colour in all the urns combined together. We show that the urns synchronize almost surely and that the fraction of balls of each colour converges to the deterministic limit of one-half, which matches with the limit known for a single Friedman urn. Furthermore, we use the notion of stable convergence to obtain limit theorems for...

  6. Asymptotics for randomly reinforced urns with random barriers

    Berti, Patrizia; Crimaldi, Irene; Pratelli, Luca; Rigo, Pietro
    An urn contains black and red balls. Let Zn be the proportion of black balls at time n and 0≤Ln is drawn. If bn is black and Zn-1n is replaced together with a random number Bn of black balls. If bn is red and Zn-1>L, then bn is replaced together with a random number Rn of red balls. Otherwise, no additional balls are added, and bn alone is replaced. In this paper we assume that Rn=Bn. Then, under mild conditions, it is shown that Zna.s.Z for some random variable...

  7. On the emergence of random initial conditions in fluid limits

    Barbour, A. D.; Chigansky, P.; Klebaner, F. C.
    In the paper we present a phenomenon occurring in population processes that start near 0 and have large carrying capacity. By the classical result of Kurtz (1970), such processes, normalized by the carrying capacity, converge on finite intervals to the solutions of ordinary differential equations, also known as the fluid limit. When the initial population is small relative to the carrying capacity, this limit is trivial. Here we show that, viewed at suitably chosen times increasing to ∞, the process converges to the fluid limit, governed by the same dynamics, but with a random initial condition. This random initial condition...

  8. A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk

    Iksanov, Alexander; Kabluchko, Zakhar
    Let (Wn(θ))n∈ℕ0 be the Biggins martingale associated with a supercritical branching random walk, and denote by W_(θ) its limit. Assuming essentially that the martingale (Wn(2θ))n∈ℕ0 is uniformly integrable and that var W1(θ) is finite, we prove a functional central limit theorem for the tail process (W(θ)-Wn+r(θ))r∈ℕ0 and a law of the iterated logarithm for W(θ)-Wn(θ) as n→∞.

  9. Distributions of jumps in a continuous-state branching process with immigration

    He, Xin; Li, Zenghu
    We study the distributional properties of jumps in a continuous-state branching process with immigration. In particular, a representation is given for the distribution of the first jump time of the process with jump size in a given Borel set. From this result we derive a characterization for the distribution of the local maximal jump of the process. The equivalence of this distribution and the total Lévy measure is then studied. For the continuous-state branching process without immigration, we also study similar problems for its global maximal jump.

  10. On a coalescence process and its branching genealogy

    Grosjean, Nicolas; Huillet, Thierry
    We define and analyze a coalescent process as a recursive box-filling process whose genealogy is given by an ancestral time-reversed, time-inhomogeneous Bienyamé‒Galton‒Watson process. Special interest is on the expected size of a typical box and its probability of being empty. Special cases leading to exact asymptotic computations are investigated when the coalescing mechanisms are either linear fractional or quadratic.

  11. Asymptotic frequency of shapes in supercritical branching trees

    Plazzotta, Giacomo; Colijn, Caroline
    The shapes of branching trees have been linked to disease transmission patterns. In this paper we use the general Crump‒Mode‒Jagers branching process to model an outbreak of an infectious disease under mild assumptions. Introducing a new class of characteristic functions, we are able to derive a formula for the limit of the frequency of the occurrences of a given shape in a general tree. The computational challenges concerning the evaluation of this formula are in part overcome using the jumping chronological contour process. We apply the formula to derive the limit of the frequency of cherries, pitchforks, and double cherries...

  12. Nonergodic Jackson networks with infinite supply–local stabilization and local equilibrium analysis

    Sommer, Jennifer; Daduna, Hans; Heidergott, Bernd
    Classical Jackson networks are a well-established tool for the analysis of complex systems. In this paper we analyze Jackson networks with the additional features that (i) nodes may have an infinite supply of low priority work and (ii) nodes may be unstable in the sense that the queue length at these nodes grows beyond any bound. We provide the limiting distribution of the queue length distribution at stable nodes, which turns out to be of product form. A key step in establishing this result is the development of a new algorithm based on adjusted traffic equations for detecting unstable nodes....

  13. Universality of load balancing schemes on the diffusion scale

    Mukherjee, Debankur; Borst, Sem C.; van Leeuwaarden, Johan S. H.; Whiting, Philip A.
    We consider a system of N parallel queues with identical exponential service rates and a single dispatcher where tasks arrive as a Poisson process. When a task arrives, the dispatcher always assigns it to an idle server, if there is any, and to a server with the shortest queue among d randomly selected servers otherwise (1≤d≤N). This load balancing scheme subsumes the so-called join-the-idle queue policy (d=1) and the celebrated join-the-shortest queue policy (d=N) as two crucial special cases. We develop a stochastic coupling construction to obtain the diffusion limit of the queue process in the Halfin‒Whitt heavy-traffic regime, and...

  14. Steady-state analysis of a multiclass MAP/PH/c queue with acyclic PH retrials

    Dayar, Tuǧrul; Can Orhan, M.
    A multiclass c-server retrial queueing system in which customers arrive according to a class-dependent Markovian arrival process (MAP) is considered. Service and retrial times follow class-dependent phase-type (PH) distributions with the further assumption that PH distributions of retrial times are acyclic. A necessary and sufficient condition for ergodicity is obtained from criteria based on drifts. The infinite state space of the model is truncated with an appropriately chosen Lyapunov function. The truncated model is described as a multidimensional Markov chain, and a Kronecker representation of its generator matrix is numerically analyzed.

  15. Detailed computational analysis of queueing-time distributions of the BMAP/G/1 queue using roots

    Singh, Gagandeep; Gupta, U. C.; Chaudhry, M. L.
    In this paper we present closed-form expressions for the distribution of the virtual (actual) queueing time for the BMAP/R/1 and BMAP/D/1 queues, where `R' represents a class of distributions having rational Laplace‒Stieltjes transforms. The closed-form analysis is based on the roots of the underlying characteristic equation. Numerical aspects have been tested for a variety of arrival and service-time distributions and results are matched with those obtained using the matrix-analytic method (MAM). Further, a comparative study of computation time of the proposed method with the MAM has been carried out. Finally, we also present closed-form expressions for the distribution of the...

  16. Stability of the stochastic matching model

    Mairesse, Jean; Moyal, Pascal
    We introduce and study a new model that we call the matching model. Items arrive one by one in a buffer and depart from it as soon as possible but by pairs. The items of a departing pair are said to be matched. There is a finite set of classes ?? for the items, and the allowed matchings depend on the classes, according to a matching graph on ??. Upon arrival, an item may find several possible matches in the buffer. This indeterminacy is resolved by a matching policy. When the sequence of classes of the arriving items is independent...

  17. Sequential stochastic assignment problem with time-dependent random success rates

    Baharian, Golshid; Khatibi, Arash; Jacobson, Sheldon H.
    The sequential stochastic assignment problem (SSAP) allocates distinct workers with deterministic values to sequentially arriving tasks with stochastic parameters to maximize the expected total reward. In this paper we study an extension of the SSAP, in which the worker values are considered to be random variables, taking on new values upon each task arrival. Several SSAP models with different assumptions on the distribution of the worker values and closed-form expressions for optimal assignment policies are presented.

  18. A sharp lower bound for choosing the maximum of an independent sequence

    Allaart, Pieter C.; Islas, José A.
    In this paper we consider a variation of the full-information secretary problem where the random variables to be observed are independent but not necessary identically distributed. The main result is a sharp lower bound for the optimal win probability. Precisely, if X1,...,Xn are independent random variables with known continuous distributions and Vn(X1,...,Xn):=supτℙ(Xτ=Mn), where Mn≔max{X1,...,Xn} and the supremum is over all stopping times adapted to X1,...,Xn then Vn(X1,...,Xn)≥(1-1/n)n-1, and this bound is attained. The method of proof consists in reducing the problem to that of a sequence of random variables taking at most two possible values, and then applying Bruss' sum-the-odds...

  19. The deterministic Kermack‒McKendrick model bounds the general stochastic epidemic

    Wilkinson, Robert R.; Ball, Frank G.; Sharkey, Kieran J.
    We prove that, for Poisson transmission and recovery processes, the classic susceptible→infected→recovered (SIR) epidemic model of Kermack and McKendrick provides, for any given time t>0, a strict lower bound on the expected number of susceptibles and a strict upper bound on the expected number of recoveries in the general stochastic SIR epidemic. The proof is based on the recent message passing representation of SIR epidemics applied to a complete graph.

  20. A stochastic two-stage innovation diffusion model on a lattice

    Coletti, Cristian F.; de Oliveira, Karina B. E.; Rodriguez, Pablo M.
    We propose a stochastic model describing a process of awareness, evaluation, and decision making by agents on the d-dimensional integer lattice. Each agent may be in any of the three states belonging to the set {0, 1, 2. In this model 0 stands for ignorants, 1 for aware, and 2 for adopters. Aware and adopters inform its nearest ignorant neighbors about a new product innovation at rate λ. At rate α an agent in aware state becomes an adopter due to the influence of adopters' neighbors. Finally, aware and adopters forget the information about the new product, thus becoming ignorant,...

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