Mostrando recursos 1 - 20 de 1.082

  1. The number of two consecutive successes in a Hoppe-Pólya urn

    Holst, Lars
    In a sequence of independent Bernoulli trials the probability of success in the kth trial is pk = a / (a + b + k - 1). An explicit formula for the binomial moments of the number of two consecutive successes in the first n trials is obtained and some consequences of it are derived.

  2. Average-case analysis of cousins in m-ary tries

    Mahmoud, Hosam M.; Ward, Mark Daniel
    We investigate the average similarity of random strings as captured by the average number of `cousins' in the underlying tree structures. Analytical techniques including poissonization and the Mellin transform are used for accurate calculation of the mean. The string alphabets we consider are m-ary, and the corresponding trees are m-ary trees. Certain analytic issues arise in the m-ary case that do not have an analog in the binary case.

  3. Assessing a linear nanosystem's limiting reliability from its components

    Ebrahimi, Nader
    Nanosystems are devices that are in the size range of a billionth of a meter (1 x 10-9) and therefore are built necessarily from individual atoms. The one-dimensional nanosystems or linear nanosystems cover all the nanosized systems which possess one dimension that exceeds the other two dimensions, i.e. extension over one dimension is predominant over the other two dimensions. Here only two of the dimensions have to be on the nanoscale (less than 100 nanometers). In this paper we consider the structural relationship between a linear nanosystem and its atoms acting as components of the nanosystem. Using such information, we then assess the nanosystem's limiting reliability which is, of course, probabilistic...

  4. Impact of routeing on correlation strength in stationary queueing network processes

    Daduna, Hans; Szekli, Ryszard
    For exponential open and closed queueing networks, we investigate the internal dependence structure, compare the internal dependence for different networks, and discuss the relation of correlation formulae to the existence of spectral gaps and comparison of asymptotic variances. A central prerequisite for the derived theorems is stochastic monotonicity of the networks. The dependence structure of network processes is described by concordance order with respect to various classes of functions. Different networks with the same first-order characteristics are compared with respect to their second-order properties. If a network is perturbed by changing the routeing in a way which holds the routeing equilibrium fixed, the resulting perturbations of the network processes are evaluated.

  5. Pricing of catastrophe insurance options under immediate loss reestimation

    Biagini, Francesca; Bregman, Yuliya; Meyer-Brandis, Thilo
    We specify a model for a catastrophe loss index, where the initial estimate of each catastrophe loss is reestimated immediately by a positive martingale starting from the random time of loss occurrence. We consider the pricing of catastrophe insurance options written on the loss index and obtain option pricing formulae by applying Fourier transform techniques. An important advantage is that our methodology works for loss distributions with heavy tails, which is the appropriate tail behavior for catastrophe modeling. We also discuss the case when the reestimation factors are given by positive affine martingales and provide a characterization of positive affine local martingales.

  6. Estimates for the absolute ruin probability in the compound Poisson risk model with credit and debit interest

    Zhu, Jinxia; Yang, Hailiang
    In this paper we consider a compound Poisson risk model where the insurer earns credit interest at a constant rate if the surplus is positive and pays out debit interest at another constant rate if the surplus is negative. Absolute ruin occurs at the moment when the surplus first drops below a critical value (a negative constant). We study the asymptotic properties of the absolute ruin probability of this model. First we investigate the asymptotic behavior of the absolute ruin probability when the claim size distribution is light tailed. Then we study the case where the common distribution of claim sizes are heavy tailed.

  7. Functional large deviations and moderate deviations for Markov-modulated risk models with reinsurance

    Gao, Fuqing; Yan, Jun
    We establish a functional large deviation principle and a functional moderate deviation principle for Markov-modulated risk models with reinsurance by constructing an exponential martingale approach. Lundberg's estimate of the ruin time is also presented.

  8. First passage times for Markov additive processes with positive jumps of phase type

    Breuer, Lothar
    The present paper generalises some results for spectrally negative Lévy processes to the setting of Markov additive processes (MAPs). A prominent role is assumed by the first passage times, which will be determined in terms of their Laplace transforms. These have the form of a phase-type distribution, with a rate matrix that can be regarded as an inverse function of the cumulant matrix. A numerically stable iteration to compute this matrix is given. The theory is first developed for MAPs without positive jumps and then extended to include positive jumps having phase-type distributions. Numerical and analytical examples show agreement with existing results in special cases.

  9. Epidemic size in the SIS model of endemic infections

    Kessler, David A.
    We study the susceptible-infected-susceptible model of the spread of an endemic infection. We calculate an exact expression for the mean number of transmissions for all values of the population size (N) and the infectivity. We derive the large-N asymptotic behavior for the infectivitiy below, above, and in the critical region. We obtain an analytical expression for the probability distribution of the number of transmissions, n, in the critical region. We show that this distribution has an n-3/2 singularity for small n and decays exponentially for large n. The exponent decreases with the distance from the threshold, diverging to ∞ far below and approaching 0 far above.

  10. Epidemics on random graphs with tunable clustering

    Britton, Tom; Deijfen, Maria; Lagerås, Andreas N.; Lindholm, Mathias
    In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.

  11. Asymptotics of posteriors for binary branching processes

    Piau, Didier
    We compute the posterior distributions of the initial population and parameter of binary branching processes in the limit of a large number of generations. We compare this Bayesian procedure with a more naïve one, based on hitting times of some random walks. In both cases, central limit theorems are available, with explicit variances.

  12. On prolific individuals in a supercritical continuous-state branching process

    Bertoin, Jean; Fontbona, Joaquin; Martínez, Servet
    We describe the genealogy of individuals with infinite descent in a supercritical continuous-state branching process.

  13. Optimal co-adapted coupling for the symmetric random walk on the hypercube

    Connor, Stephen; Jacka, Saul
    Let X and Y be two simple symmetric continuous-time random walks on the vertices of the n-dimensional hypercube, Z2n. We consider the class of co-adapted couplings of these processes, and describe an intuitive coupling which is shown to be the fastest in this class.

  14. Random walk delayed on percolation clusters

    Comets, Francis; Simenhaus, François
    We study a continuous-time random walk on the d-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we discuss the diffusive behavior in the case of zero drift and weak attraction.

  15. The noisy veto-voter model: a recursive distributional equation on [0, 1]

    Jacka, Saul; Sheehan, Marcus
    We study a particular example of a recursive distributional equation (RDE) on the unit interval. We identify all invariant distributions, the corresponding `basins of attraction', and address the issue of endogeny for the associated tree-indexed problem, making use of an extension of a recent result of Warren.

  16. On the relationships between lumpability and filtering of finite stochastic systems

    Ledoux, James; White, Langford B.; Brushe, Gary D.
    The aim of this paper is to provide the conditions necessary to reduce the complexity of state filtering for finite stochastic systems (FSSs). A concept of lumpability for FSSs is introduced. In this paper we assert that the unnormalised filter for a lumped FSS has linear dynamics. Two sufficient conditions for such a lumpability property to hold are discussed. We show that the first condition is also necessary for the lumped FSS to have linear dynamics. Next, we prove that the second condition allows the filter of the original FSS to be obtained directly from the filter for the lumped FSS. Finally, we generalise an earlier published result for the...

  17. Hitting time and inverse problems for Markov chains

    de la Peña, Victor; Gzyl, Henryk; McDonald, Patrick
    Let Wn be a simple Markov chain on the integers. Suppose that Xn is a simple Markov chain on the integers whose transition probabilities coincide with those of Wn off a finite set. We prove that there is an M > 0 such that the Markov chain Wn and the joint distributions of the first hitting time and first hitting place of Xn started at the origin for the sets {-M, M} and {-(M + 1), (M + 1)} algorithmically determine the transition probabilities of Xn.

  18. A compact framework for hidden Markov chains with applications to fractal geometry

    Ruiz, Víctor
    We introduce a class of stochastic processes in discrete time with finite state space by means of a simple matrix product. We show that this class coincides with that of the hidden Markov chains and provides a compact framework for it. We study a measure obtained by a projection on the real line of the uniform measure on the Sierpinski gasket, finding that the dimension of this measure fits with the Shannon entropy of an associated hidden Markov chain.

  19. The generalised coupon collector problem

    Neal, Peter
    Coupons are collected one at a time from a population containing n distinct types of coupon. The process is repeated until all n coupons have been collected and the total number of draws, Y, from the population is recorded. It is assumed that the draws from the population are independent and identically distributed (draws with replacement) according to a probability distribution X with the probability that a type-i coupon is drawn being P(X = i). The special case where each type of coupon is equally likely to be drawn from the population is the classic coupon collector problem. We consider the asymptotic distribution Y (appropriately normalized) as the number of coupons...

  20. A collector's problem with renewal arrival processes

    Kella, Offer; Stadje, Wolfgang
    We study a collector's problem with K renewal arrival processes for different type items, where the objective is to collect complete sets. In particular, we derive the asymptotic distribution of the sequence of interarrival times between set completions.

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