bluuebunny/crossref_metadata_2025 huggingface dataset:
Ergodicity and mixing properties of the northeast model
Authors: Kordzakhia George , Lalley Steven P.
Published: July 2016

The northeast model is a spin system on the two-dimensional integer lattice that evolves according to the following rule: whenever a site's southerly and westerly nearest neighbors have spin 1, it may reset its own spin by tossing a p-coin; at all other times, its spin remains frozen. It is proved that the northeast model has a phase transition at p
c = 1 - βc, where βc is the critical parameter for oriented percolation. For p < p
c, the trivial measure, δ0, that puts mass one on the configuration with all spins set at 0 is the unique ergodic, translation-invariant, stationary measure. For p ≥ p
c, the product Bernoulli-p measure on configuration space is the unique nontrivial, ergodic, translation-invariant, stationary measure for the system, and it is mixing. For p > ⅔, it is shown that there is exponential decay of correlations.
A discrete multivariate distribution resulting from the law of small numbers
Authors: Hoshino Nobuaki
Published: July 2016

In the present article we derive a new discrete multivariate distribution using a limiting argument that is essentially the same as the law of small numbers. The distribution derived belongs to an exponential family, and randomly partitions positive integers. The facts shown about the distribution are useful in many fields of application involved with count data. The derivation parallels that of the Ewens distribution from the gamma distribution, and the new distribution is produced from the inverse Gaussian distribution. The method employed is regarded as the discretization of an infinitely divisible distribution over nonnegative real numbers.
Bounds for perpetual American option prices in a jump diffusion model
Authors: Ekström Erik
Published: July 2016

We provide bounds for perpetual American option prices in a jump diffusion model in terms of American option prices in the standard Black–Scholes model. We also investigate the dependence of the bounds on different parameters of the model.
Central limit theorems for generalized Pólya urn models
Authors: Higueras I. , Moler J. , Plo F. , San Miguel M.
Published: July 2016

In this paper we obtain central limit theorems for generalized Pólya urn models with L ≥ 2 colors where one out of K different replacements (actions) is applied randomly at each step. Each possible action constitutes a row of the replacement matrix, which can be nonsquare and random. The actions are chosen following a probability distribution given by an arbitrary function of the proportions of the balls of different colors present in the urn. Moreover, under the same hypotheses it is proved that the covariance matrix of the asymptotic distribution is the solution of a Lyapunov equation, and a procedure is given to obtain the covariance matrix in an explicit form. Some applications of these results to random trees and adaptive designs in clinical trials are also presented.
An urn model arising from all-optical networks
Authors: Morrison John A.
Published: July 2016

An occupancy model that has arisen in the investigation of randomized distributed schedules in all-optical networks is considered. The model consists of B initially empty urns, and at stage j of the process d

j
≤ B balls are placed in distinct urns with uniform probability. Let M

i
(j) denote the number of urns containing i balls at the end of stage j. An explicit expression for the joint factorial moments of M
0(j) and M
1(j) is obtained. A multivariate generating function for the joint factorial moments of M

i
(j), 0 ≤ i ≤ I, is derived (where I is a positive integer). Finally, the case in which the d

j
, j ≥ 1, are independent, identically distributed random variables is investigated.
Limit theorems for a diffusion process with a one-sided Brownian potential
Authors: Kawazu Kiyoshi , Suzuki Yuki
Published: July 2016

We consider a diffusion process X(t) with a one-sided Brownian potential starting from the origin. The limiting behavior of the process as time goes to infinity is studied. For each t > 0, the sample space describing the random potential is divided into two parts, Ã

t
and B̃

t
, both having probability ½, in such a way that our diffusion process X(t) exhibits quite different limiting behavior depending on whether it is conditioned on Ã

t
or on B̃

t
(t → ∞). The asymptotic behavior of the maximum process of X(t) is also investigated. Our results improve those of Kawazu, Suzuki, and Tanaka (2001).
Reversibility and entropy production of inhomogeneous Markov chains
Authors: Ge Hao , Jiang Da-Quan , Qian Min
Published: July 2016

In this paper we introduce the concepts of instantaneous reversibility and instantaneous entropy production rate for inhomogeneous Markov chains with denumerable state spaces. The following statements are proved to be equivalent: the inhomogeneous Markov chain is instantaneously reversible; it is in detailed balance; its entropy production rate vanishes. In particular, for a time-periodic birth-death chain, which can be regarded as a simple version of a physical model (Brownian motors), we prove that its rotation number is 0 when it is instantaneously reversible or periodically reversible. Hence, in our model of Markov chains, the directed transport phenomenon of Brownian motors can occur only in nonequilibrium and irreversible systems.
On the total reward variance for continuous-time Markov reward chains
Authors: Van Dijk Nico M. , Sladký Karel
Published: July 2016

As an extension of the discrete-time case, this note investigates the variance of the total cumulative reward for continuous-time Markov reward chains with finite state spaces. The results correspond to discrete-time results. In particular, the variance growth rate is shown to be asymptotically linear in time. Expressions are provided to compute this growth rate.
Extinction of branching symmetric α-stable processes
Authors: Shiozawa Yuichi
Published: July 2016

We give a criterion for extinction or local extinction of branching symmetric α-stable processes in terms of the principal eigenvalue for time-changed processes of symmetric α-stable processes. Here the branching rate and the branching mechanism are spatially dependent. In particular, the branching rate is allowed to be singular with respect to the Lebesgue measure. We apply this criterion to some branching processes.
Strategic equilibrium versus global optimum for a pair of competing servers
Authors: Avi-Itzhak Benjamin , Golany Boaz , Rothblum Uriel G.
Published: July 2016

Christ and Avi-Itzhak (2002) analyzed a queueing system with two competing servers who determine their service rates so as to optimize their individual utilities. The system is formulated as a two-person game; Christ and Avi-Itzhak proved the existence of a unique Nash equilibrium which is symmetric. In this paper, we explore globally optimal solutions. We prove that the unique Nash equilibrium is generally strictly inferior to a globally optimal solution and that optimal solutions are symmetric and require the servers to adopt service rates that are smaller than those occurring in equilibrium. Furthermore, given a symmetric globally optimal solution, we show how to impose linear penalties on the service rates so that the given optimal solution becomes a unique Nash equilibrium. When service rates are not observable, we show how the same effect is achieved by imposing linear penalties on a corresponding signal.
Shapes of stationary autocovariances
Authors: Lund Robert , Zhao Ying , Kiessler Peter C.
Published: July 2016

This note introduces shape orderings for stationary time series autocorrelation and partial autocorrelation functions and explores some of their convergence rate ramifications. The shapes explored include decreasing hazard rate and new better than used, orderings that are familiar from stochastic processes settings. Time series models where these shapes arise are presented. The shapes are used to obtain explicit geometric convergence rates for mean squared errors of one-step-ahead forecasts.
Coupling of Two SIR Epidemic Models with Variable Susceptibilities and Infectivities
Authors: Neal Peter
Published: July 2016

The variable generalised stochastic epidemic model, which allows for variability in both the susceptibilities and infectivities of individuals, is analysed. A very different epidemic model which exhibits variable susceptibility and infectivity is the random-graph epidemic model. A suitable coupling of the two epidemic models is derived which enables us to show that, whilst the epidemics are very different in appearance, they have the same asymptotic final size distribution. The coupling provides a novel approach to studying random-graph epidemic models.
On Generating Functions of Waiting Times and Numbers of Occurrences of Compound Patterns in a Sequence of Multistate Trials
Authors: Inoue Kiyoshi , Aki Sigeo
Published: July 2016

In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.
Comparison Results for Branching Processes in Random Environments
Authors: Pellerey Franco
Published: July 2016

In this note we consider branching processes whose behavior depends on a dynamic random environment, in the sense that we assume that the offspring distributions of individuals are parametrized, over time, by the realizations of a process describing the environmental evolution. We study how the variability in time of the environment modifies the variability of total population by considering two branching processes of this kind (but subjected to different environments). We also provide conditions on the random environments in order to stochastically compare their marginal distributions in the increasing convex sense. Weaker conditions are also provided for comparisons at every fixed time of the expected values of the two populations.
Arbitrary Threshold Widths for Monotone, Symmetric Properties
Authors: Rossignol Raphaël
Published: July 2016

We show that there exist symmetric properties in the discrete n-cube whose threshold widths range asymptotically between 1/√n and 1/logn. These properties are built using a combination of failure sets arising in reliability theory. This combination of sets is simply called a product. Some general results on the threshold width of the product of two sets A and B in terms of the threshold locations and widths of A and B are provided.
Optimal Strategy for the Vardi Casino with Interest Payments
Authors: Grigorescu Ilie , Chen Robert , Shepp Larry
Published: July 2016

A gambler starts with fortune f < 1 and plays in a Vardi casino with infinitely many tables indexed by their odds, r ≥ 0. In addition, all tables return the same expected winnings per dollar, c < 0, and a discount factor is applied after each round. We determine the optimal probability of reaching fortune 1, as well as an optimal strategy that is different from bold play for fortunes larger than a critical value depending exclusively on c and 1 + a, the discount factor. The general result is computed explicitly for some relevant special cases. The question of whether bold play is an optimal strategy is discussed for various choices of the parameters.
A Remark on the Uniqueness of Weighted Markov Branching Processes
Authors: Chen Anyue , Pollett Phil , Li Junping , Zhang Hanjun
Published: July 2016

We present an elegant uniqueness criterion for the weighted Markov branching process in the potentially explosive case.
Distribution of the Present Value of Dividend Payments in a Lévy Risk Model
Authors: Renaud Jean-François , Zhou Xiaowen
Published: July 2016

In this short paper, we show how fluctuation identities for Lévy processes with no positive jumps yield the distribution of the present value of dividends paid until ruin in a Lévy insurance risk model with a dividend barrier.
Coupling and Ergodicity of Adaptive Markov Chain Monte Carlo Algorithms
Authors: Roberts Gareth O. , Rosenthal Jeffrey S.
Published: July 2016

We consider basic ergodicity properties of adaptive Markov chain Monte Carlo algorithms under minimal assumptions, using coupling constructions. We prove convergence in distribution and a weak law of large numbers. We also give counterexamples to demonstrate that the assumptions we make are not redundant.
Computable Bounds for the Decay Parameter of a Birth–Death Process
Authors: Sirl David , Zhang Hanjun , Pollett Phil
Published: July 2016

We present bounds on the decay parameter for absorbing birth–death processes adapted from results of Chen (2000), (2001). We address numerical issues associated with computing these bounds, and assess their accuracy for several models, including the stochastic logistic model, for which estimates of the decay parameter have been obtained previously by Nåsell (2001).
Probabilistic Approximation of a Nonlinear Parabolic Equation Occurring in Rheology
Authors: Ben Alaya Mohamed , Jourdain Benjamin
Published: July 2016

In this paper we are interested in a nonlinear parabolic evolution equation occurring in rheology. We give a probabilistic interpretation to this equation by associating a nonlinear martingale problem with it. We prove the existence of a unique solution, P, to this martingale problem. For any t, the time marginal of P at time t admits a density ρ(t,x) with respect to the Lebesgue measure, where the function ρ is the unique weak solution to the evolution equation in a well-chosen energy space. Next we introduce a simulable system of n interacting particles and prove that the empirical measure of this system converges to P as n tends to ∞. This propagation-of-chaos result ensures that the solution to the equation of interest can be approximated using a Monte Carlo method. Finally, we illustrate the convergence in some numerical experiments.
Stochastic Epidemic Models in Structured Populations Featuring Dynamic Vaccination and Isolation
Authors: Ball Frank , O'Neill Philip D. , Pike James
Published: July 2016

We consider a stochastic model for the spread of an SEIR (susceptible → exposed → infective → removed) epidemic among a population of individuals partitioned into households. The model incorporates both vaccination and isolation in response to the detection of cases. When the infectious period is exponential, we derive an explicit formula for a threshold parameter, and analytic results that enable computation of the probability of the epidemic taking off. These quantities are found to be independent of the exposure period distribution. An approximation for the expected final size of an epidemic that takes off is obtained, evaluated numerically, and found to be reasonably accurate in large populations. When the infectious period is not exponential, but has an increasing hazard rate, we obtain stochastic comparison results in the case where the exposure period is fixed. Our main result shows that as the exposure period increases, both the severity of the epidemic in a single household and the threshold parameter decrease, under certain assumptions concerning isolation. Corresponding results for infectious periods with decreasing hazard rates are also derived.
Stability of Processor Sharing Networks with Simultaneous Resource Requirements
Authors: Hansen Jennie , Reynolds Cian , Zachary Stan
Published: July 2016

We study the phenomenon of entrainment in processor sharing networks, whereby, while individual network resources have sufficient capacity to meet demand, the requirement for simultaneous availability of resources means that a network may nevertheless be unstable. We show that instability occurs through poor control, and that, for a variety of network topologies, only small modifications to controls are required in order to ensure stability. For controls which possess a natural monotonicity property, we give some new results for the classification of the corresponding Markov processes, which lead to conditions both for stability and for instability.
Generalized Pólya urn Designs with Null Balance
Authors: Baldi Antognini Alessandro , Giannerini Simone
Published: July 2016

In this paper we propose a class of sequential urn designs based on generalized Pólya urn (GPU) models for balancing the allocations of two treatments in sequential clinical trials. In particular, we consider a GPU model characterized by a 2 x 2 random addition matrix with null balance (i.e. null row sums) and replacement rule depending upon the urn composition. Under this scheme, the urn process has a Markovian structure and can be regarded as a random extension of the classical Ehrenfest model. We establish almost sure convergence and asymptotic normality for the frequency of treatment allocations and show that in some peculiar cases the asymptotic variance of the design admits a natural representation based on the set of orthogonal polynomials associated with the corresponding Markov process.
The Joint Density of the Surplus Before and After Ruin in the Sparre Andersen Model
Authors: Pitts Susan M. , Politis Konstadinos
Published: July 2016

Gerber and Shiu (1997) have studied the joint density of the time of ruin, the surplus immediately before ruin, and the deficit at ruin in the classical model of collective risk theory. More recently, their results have been generalised for risk models where the interarrival density for claims is nonexponential, but belongs to the Erlang family. Here we obtain generalisations of the Gerber-Shiu (1997) results that are valid in a general Sparre Andersen model, i.e. for any interclaim density. In particular, we obtain a generalisation of the key formula in that paper. Our results are made more concrete for the case where the distribution between claim arrivals is phase-type or the integrated tail distribution associated with the claim size distribution belongs to the class of subexponential distributions. Furthermore, we obtain conditions for finiteness of the joint moments of the surplus before ruin and the deficit at ruin in the Sparre Andersen model.
Counts of Failure Strings in Certain Bernoulli Sequences
Authors: Holst Lars
Published: July 2016

In a sequence of independent Bernoulli trials the probability for success in the kth trial is p

k
, k = 1, 2, …. The number of strings with a given number of failures between two subsequent successes is studied. Explicit expressions for distributions and moments are obtained for the case in which p

k
= a/(a + b + k − 1), a > 0, b ≥ 0. Also, the limit behaviour of the longest failure string in the first n trials is considered. For b = 0, the strings correspond to cycles in random permutations.
Precise Large Deviations for Sums of Random Variables with Consistently Varying Tails in Multi-Risk Models
Authors: Wang Shijie , Wang Wensheng
Published: July 2016

Assume that there are k types of insurance contracts in an insurance company. The ith related claims are denoted by {X

ij
, j ≥ 1}, i = 1,…,k. In this paper we investigate large deviations for both partial sums S(k; n
1,…,n

k
) = ∑
i=1

k
∑
j=1

n

i

X

ij
and random sums S(k; t) = ∑
i=1

k
∑
j=1

N

i
(t)
X

ij
, where N

i
(t), i = 1,…,k, are counting processes for the claim number. The obtained results extend some related classical results.
An Exponential Continuous-Time GARCH Process
Authors: Haug Stephan , Czado Claudia
Published: July 2016

In this paper we introduce an exponential continuous-time GARCH(p, q) process. It is defined in such a way that it is a continuous-time extension of the discrete-time EGARCH(p, q) process. We investigate stationarity, mixing, and moment properties of the new model. An instantaneous leverage effect can be shown for the exponential continuous-time GARCH(p, p) model.
Estimation for Nonnegative Lévy-Driven Ornstein-Uhlenbeck Processes
Authors: Brockwell Peter J. , Davis Richard A. , Yang Yu
Published: July 2016

Continuous-time autoregressive moving average (CARMA) processes with a nonnegative kernel and driven by a nondecreasing Lévy process constitute a very general class of stationary, nonnegative continuous-time processes. In financial econometrics a stationary Ornstein-Uhlenbeck (or CAR(1)) process, driven by a nondecreasing Lévy process, was introduced by Barndorff-Nielsen and Shephard (2001) as a model for stochastic volatility to allow for a wide variety of possible marginal distributions and the possibility of jumps. For such processes, we take advantage of the nonnegativity of the increments of the driving Lévy process to study the properties of a highly efficient estimation procedure for the parameters when observations are available of the CAR(1) process at uniformly spaced times 0,h,…,Nh. We also show how to reconstruct the background driving Lévy process from a continuously observed realization of the process and use this result to estimate the increments of the Lévy process itself when h is small. Asymptotic properties of the coefficient estimator are derived and the results illustrated using a simulated gamma-driven Ornstein-Uhlenbeck process.
Optimal Stopping with Rank-Dependent Loss
Authors: Gnedin Alexander V.
Published: July 2016

For τ, a stopping rule adapted to a sequence ofnindependent and identically distributed observations, we define the loss to be E[q(Rτ)], whereRjis the rank of thejth observation andqis a nondecreasing function of the rank. This setting covers both the best-choice problem, withq(r) =1(r> 1), and Robbins' problem, withq(r) =r. Asntends to ∞, the stopping problem acquires a limiting form which is associated with the planar Poisson process. Inspecting the limit we establish bounds on the stopping value and reveal qualitative features of the optimal rule. In particular, we show that the complete history dependence persists in the limit; thus answering a question asked by Bruss (2005) in the context of Robbins' problem.
Asymptotic Blocking Probabilities in Loss Networks with Subexponential Demands
Authors: Lu Yingdong , Radovanović Ana
Published: July 2016

The analysis of stochastic loss networks has long been of interest in computer and communications networks and is becoming important in the areas of service and information systems. In traditional settings computing the well-known Erlang formula for blocking probabilities in these systems becomes intractable for larger resource capacities. Using compound point processes to capture stochastic variability in the request process, we generalize existing models in this framework and derive simple asymptotic expressions for the blocking probabilities. In addition, we extend our model to incorporate reserving resources in advance. Although asymptotic, our experiments show an excellent match between derived formulae and simulation results even for relatively small resource capacities and relatively large values of the blocking probabilities.
Fixation in Conditional Branching Process Models in Population Genetics
Authors: Prince Thomas , Weber Neville
Published: July 2016

An alternative version of the necessary and sufficient condition for almost sure fixation in the conditional branching process model is derived. This formulation provides an insight into why the examples considered in Buckley and Seneta (1983) all have the same condition for fixation.
Sums of Dependent Nonnegative Random Variables with Subexponential Tails
Authors: Ko Bangwon , Tang Qihe
Published: July 2016

In this paper we study the asymptotic tail probabilities of sums of subexponential, nonnegative random variables, which are dependent according to certain general structures with tail independence. The results show that the subexponentiality of the summands eliminates the impact of the dependence on the tail behavior of the sums.
On the Structure and Estimation of Reflection Positive Processes
Authors: McVinish R.
Published: July 2016

The class of processes formed as the aggregation of Ornstein-Uhlenbeck processes has proved useful in modeling time series from a number of areas and includes several interesting special cases. This paper examines the second-order properties of this class. Bounds on the one-step prediction error variance are proved and consistency of the minimum contrast estimation is demonstrated.
Transient Moments of the TCP Window Size Process
Authors: Löpker Andreas H. , van Leeuwaarden Johan S. H.
Published: July 2016

The TCP window size process can be modeled as a piecewise-deterministic Markov process that increases linearly and experiences downward jumps at Poisson times. We present a transient analysis of this window size process. Our main result is the Laplace transform of the transient moments. Formulae for the integer and fractional moments are derived, as well as an explicit characterization of the speed of convergence to steady state. Central to our approach are the infinitesimal generator and Dynkin's martingale.
Pattern Markov Chains: Optimal Markov Chain Embedding Through Deterministic Finite Automata
Authors: Nuel Grégory
Published: July 2016

In the framework of patterns in random texts, the Markov chain embedding techniques consist of turning the occurrences of a pattern over an order-m Markov sequence into those of a subset of states into an order-1 Markov chain. In this paper we use the theory of language and automata to provide space-optimal Markov chain embedding using the new notion of pattern Markov chains (PMCs), and we give explicit constructive algorithms to build the PMC associated to any given pattern problem. The interest of PMCs is then illustrated through the exact computation of P-values whose complexity is discussed and compared to other classical asymptotic approximations. Finally, we consider two illustrative examples of highly degenerated pattern problems (structured motifs and PROSITE signatures), which further illustrate the usefulness of our approach.
On a Comparison Result for Markov Processes
Authors: Rüschendorf Ludger
Published: July 2016

A comparison theorem is stated for Markov processes in Polish state spaces. We consider a general class of stochastic orderings induced by a cone of real functions. The main result states that stochastic monotonicity of one process and comparability of the infinitesimal generators imply ordering of the processes. Several applications to convex type and to dependence orderings are given. In particular, Liggett's theorem on the association of Markov processes is a consequence of this comparison result.
Queues with Delays in Two-State Strategies and Lévy Input
Authors: Bekker R. , Boxma O. J. , Kella O.
Published: July 2016

We consider a reflected Lévy process without negative jumps, starting at the origin. When the reflected process first upcrosses level K, a timer is activated. After D time units, the timer expires and the Lévy exponent of the Lévy process is changed. As soon as the process hits zero again, the Lévy exponent reverses to the original function. If the process has reached the origin before the timer expires then the Lévy exponent does not change. Using martingale techniques, we analyze the steady-state distribution of the resulting process, reflected at the origin. We pay special attention to the cases of deterministic and exponential timers, and to the following three special Lévy processes: (i) a compound Poisson process plus negative drift (corresponding to an M/G/1 queue), (ii) Brownian motion, and (iii) a Lévy process that is a subordinator until the timer expires.
Weak Convergence Limits for Sojourn Times in Cyclic Queues Under Heavy Traffic Conditions
Authors: Daduna Hans , Malchin Christian , Szekli Ryszard
Published: July 2016

We consider sequences of closed cycles of exponential single-server nodes with a single bottleneck. We study the cycle time and the successive sojourn times of a customer when the population sizes go to infinity. Starting from old results on the mean cycle times under heavy traffic conditions, we prove a central limit theorem for the cycle time distribution. This result is then utilised to prove a weak convergence characteristic of the vector of a customer's successive sojourn times during a cycle for a sequence of networks with population sizes going to infinity. The limiting picture is a composition of a central limit theorem for the bottleneck node and an exponential limit for the unscaled sequences of sojourn times for the nonbottleneck nodes.
Strong Optimality of Bold Play for Discounted Dubins-Savage Gambling Problems with Time-Dependent Parameters
Authors: Yao Yi-Ching , Chen May-Ru
Published: July 2016

In the classic Dubins-Savage subfair primitive casino gambling problem, the gambler can stake any amount in his possession, winning (1 − r)/r times the stake with probability w and losing the stake with probability 1 − w, 0 ≤ w ≤ r ≤ 1. The gambler seeks to maximize the probability of reaching a fixed fortune (the goal) by gambling repeatedly with suitably chosen stakes. This problem has recently been extended in a unifying framework to account for limited playing time as well as future discounting, under which bold play is known to be optimal provided that w ≤ ½ ≤ r. This paper is concerned with a further extension of the Dubins-Savage gambling problem involving time-dependent parameters, and shows that bold play not only maximizes the probability of reaching the goal, but also stochastically minimizes the number of plays needed to reach the goal. As a result, bold play also maximizes the expected utility, where the utility at the goal is only required to be monotone decreasing with respect to the number of plays needed to reach the goal. It is further noted that bold play remains optimal even when the time-dependent parameters are random.
A Poisson Approximation for an Occupancy Problem with Collisions
Authors: Nakata Toshio
Published: July 2016

We study collision probabilities concerning the simple balls-and-bins problem developed by Wendl (2003). In this article we give the factorial moment of the number of collisions. Moreover, we obtain a Poisson approximation for the number of collisions using the Chen-Stein method.
On Negative Binomial Approximation to k-Runs
Authors: Wang Xiaoxin , Xia Aihua
Published: July 2016

The distributions of the run occurrences for a sequence of independent and identically distributed (i.i.d.) experiments are usually obtained by combinatorial methods (see Balakrishnan and Koutras (2002, Chapter 5)) and the resulting formulae are often very tedious, while the distributions for non i.i.d. experiments are generally intractable. It is therefore of practical interest to find a suitable approximate model with reasonable approximation accuracy. In this paper we demonstrate that the negative binomial distribution is the most suitable approximate model for the number ofk-runs: it outperforms the Poisson approximation, the general compound Poisson approximation as observed in Eichelsbacher and Roos (1999), and the translated Poisson approximation in Rollin (2005). In particular, its accuracy of approximation in terms of the total variation distance improves when the number of experiments increases, in the same way as the normal approximation improves in the Berry-Esseen theorem.
Growth of Integral Transforms and Extinction in Critical Galton-Watson Processes
Authors: Tokarev Daniel
Published: July 2016

The mean time to extinction of a critical Galton-Watson process with initial population size k is shown to be asymptotically equivalent to two integral transforms: one involving the kth iterate of the probability generating function and one involving the generating function itself. Relating the growth of these transforms to the regular variation of their arguments, immediately connects statements involving the regular variation of the probability generating function, its iterates at 0, the quasistationary measures, their partial sums, and the limiting distribution of the time to extinction. In the critical case of finite variance we also give the growth of the mean time to extinction, conditioned on extinction occurring by time n.
Relating Time and Customer Averages for Queues Using ‘forward’ Coupling from the Past
Authors: Peköz Erol A. , Ross Sheldon M.
Published: July 2016

We give a new method for simulating the time average steady-state distribution of a continuous-time queueing system, by extending a ‘read-once’ or ‘forward’ version of the coupling from the past (CFTP) algorithm developed for discrete-time Markov chains. We then use this to give a new proof of the ‘Poisson arrivals see time averages’ (PASTA) property, and a new proof for why renewal arrivals see either stochastically smaller or larger congestion than the time average if interarrival times are respectively new better than used in expectation (NBUE) or new worse than used in expectation (NWUE).
Controlled Heterogeneous Collection: The Role of Occupation Numbers
Authors: Gerardi A. , Tardelli P.
Published: July 2016

A controlled heterogeneous collection of identical items is presented. According to their level of wear and tear, they are divided into a finite number of classes and the partition of the collection is allowed to change over time. A suitable exchangeability assumption is made to preserve the property that the items be identical. The role of the occupation numbers is investigated and a filtering problem is set up, where the observation is the cardinality of a particular class. A control on the dynamics of the items is introduced, and the existence of an optimal control is proved. A discrete-time approximation for the separated problem, which is a finite-dimensional one, is performed. As a consequence, an approximation for the value function is given.
The Noisy Veto-Voter Model: A Recursive Distributional Equation on [0, 1]
Authors: Jacka Saul , Sheehan Marcus
Published: July 2016

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.
Random Walk Delayed on Percolation Clusters
Authors: Comets Francis , Simenhaus François
Published: July 2016

We study a continuous-time random walk on thed-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.
On Prolific Individuals in a Supercritical Continuous-State Branching Process
Authors: Bertoin Jean , Fontbona Joaquin , Martínez Servet
Published: July 2016

We describe the genealogy of individuals with infinite descent in a supercritical continuous-state branching process.
Functional Large Deviations and Moderate Deviations for Markov-Modulated Risk Models with Reinsurance
Authors: Gao Fuqing , Yan Jun
Published: July 2016

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.
Pricing of Catastrophe Insurance Options Under Immediate Loss Reestimation
Authors: Biagini Francesca , Bregman Yuliya , Meyer-Brandis Thilo
Published: July 2016

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.
Stochastic Models of Cascading Failures
Authors: Swift Andrew W.
Published: July 2016

What is a ‘cascading failure’? Whilst most people have an intuitive idea of what is meant by cascading, no definition exists within the framework of reliability theory. The aim of this paper is to explore the field of component dependence within a network structure, and to construct a definition of cascading failure.
Bin Packing with Queues
Authors: Shah Devavrat , Tsitsiklis John N.
Published: July 2016

We study the best achievable performance (in terms of the average queue size and delay) in a stochastic and dynamic version of the bin-packing problem. Items arrive to a queue according to a Poisson process with rate 2ρ, where ρ ∈ (0, 1). The item sizes are independent and identically distributed (i.i.d.) with a uniform distribution in [0, 1]. At each time unit, a single unit-size bin is available and can receive any of the queued items, as long as their total size does not exceed 1. Coffman and Stolyar (1999) and Gamarnik (2004) have established that there exist packing policies under which the average queue size is finite for every ρ ∈ (0, 1). In this paper we study the precise scaling of the average queue size, as a function of ρ, with emphasis on the critical regime where ρ approaches 1. Standard results on the probabilistic (but static) bin-packing problem can be readily applied to produce policies under which the queue size scales as O(h
2), where h = 1 / (1 - ρ), which raises the question of whether this is the best possible. We establish that the average queue size scales as Ω(hlogh), under any policy. Furthermore, we provide an easily implementable policy, which packs at most two items per bin. Under that policy, the average queue size scales as O(hlog3/2
h), which is nearly optimal. On the other hand, if we impose the additional requirement that any two items packed together must have near-complementary sizes (in a sense to be made precise), we show that the average queue size must scale as Θ(h
2).
Ranking of Weighted Majority Rules
Authors: Berend Daniel , Chernyavsky Yuri , Sapir Luba
Published: July 2016

A decision-making body may utilize a wide variety of different strategies when required to make a collective decision. In principle, we would like to use the most effective decision rule, that is, the rule yielding the highest probability of making the correct decision. However, in reality we often have to choose a decision rule out of some restricted family of rules. Therefore, it is important to be able to rank various families of rules. In this paper we consider three classes of decision rules: (i) balanced expert rules, (ii) the so-called single expert rules, and (iii) restricted majority rules. For the first two classes, we show that, as we deviate from the best rule in the family, the effectiveness of the decision rule decreases. For the last class, we obtain a very different phenomenon: any inner ranking is possible.
Run Statistics Defined on the Multicolor URN Model
Authors: Eryilmaz Serkan
Published: July 2016

Recently, Makri, Philippou and Psillakis (2007b) studied the exact distribution of success run statistics defined on an urn model. They derived the exact distributions of various success run statistics for a sequence of binary trials generated by the Pólya-Eggenberger sampling scheme. In our study we derive the joint distributions of run statistics defined on the multicolor urn model using a simple unified combinatorial approach and extend some of the results of Makri, Philippou and Psillakis (2007b). As a consequence of our results, we obtain the joint distributions of success and failure runs defined on the two-color urn model. The results enable us to compute the characteristics of particular consecutive-type systems and start-up demonstration tests.
On Nonoptimality of Bold Play for Subfair Red-And-Black with a Rational-Valued House Limit
Authors: Chen May-Ru , Chung Pei-Shou , Hsiau Shoou-Ren , Yao Yi-Ching
Published: July 2016

In the subfair red-and-black gambling problem, a gambler can stake any amount in his possession, winning an amount equal to the stake with probability w and losing the stake with probability 1 − w, where 0 < w < ½. The gambler seeks to maximize the probability of reaching a fixed fortune (to be normalized to unity) by gambling repeatedly with suitably chosen stakes. In their classic work, Dubins and Savage (1965), (1976) showed that it is optimal to play boldly. When there is a house limit of l (0 < l < ½), so that the gambler can stake no more than l, Wilkins (1972) showed that bold play remains optimal provided that 1 / l is an integer. On the other hand, building on an earlier surprising result of Heath, Pruitt and Sudderth (1972), Schweinsberg (2005) recently showed that, for all irrational 0 < l < ½ and all 0 < w < ½, bold play is not optimal for some initial fortune. The purpose of the present paper is to present several results supporting the conjecture that, for all rational l with 1 / l not an integer and all 0 < w < ½, bold play is not optimal for some initial fortune. While most of these results are based on Schweinsberg's method, in a special case where his method is shown to be inapplicable, we argue that the conjecture can be verified with the help of symbolic-computation software.
A Stochastic Volatility Alternative to SABR
Authors: Rogers L. C. G. , Veraart L. A. M.
Published: July 2016

We present two new stochastic volatility models in which option prices for European plain-vanilla options have closed-form expressions. The models are motivated by the well-known SABR model, but use modified dynamics of the underlying asset. The asset process is modelled as a product of functions of two independent stochastic processes: a Cox-Ingersoll-Ross process and a geometric Brownian motion. An application of the models to options written on foreign currencies is studied.
Mixture Representations of Residual Lifetimes of Used Systems
Authors: Navarro Jorge , Balakrishnan N. , Samaniego Francisco J.
Published: July 2016

The representation of the reliability function of the lifetime of a coherent system as a mixture of the reliability function of order statistics associated with the lifetimes of its components is a very useful tool to study the ordering and the limiting behaviour of coherent systems. In this paper, we obtain several representations of the reliability functions of residual lifetimes of used coherent systems under two particular conditions on the status of the components or the system in terms of the reliability functions of residual lifetimes of order statistics.
On Asymptotics of Exchangeable Coalescents with Multiple Collisions
Authors: Gnedin Alexander , Iksanov Alex , Möhle Martin
Published: July 2016

We study the number of collisions,Xn, of an exchangeable coalescent with multiple collisions (Λ-coalescent) which starts withnparticles and is driven by rates determined by a finite characteristic measure η(dx) =x−2Λ(dx). Via a coupling technique, we derive limiting laws ofXn, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws ofXninclude normal, stable with index 1 ≤ α < 2, and Mittag-Leffler distributions. The results apply, in particular, to the case when η is a beta(a− 2,b) distribution with parametersa> 2 andb> 0. The approach taken allows us to derive asymptotics of three other functionals of the coalescent: the absorption time, the length of an external branch chosen at random from thenexternal branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours.
Optimal Allocation of Active Spares in Series Systems and Comparison of Component and System Redundancies
Authors: Misra Neeraj , Dhariyal Ishwari D. , Gupta Nitin
Published: July 2016

We consider the problem of allocating k active spares to n components of a series system in order to optimize its lifetime. Under the hypotheses that lifetimes of n components are identically distributed with distribution function F(⋅), lifetimes of k spares are identically distributed with distribution function G(⋅), lifetimes of components and spares are independently distributed, and that ln(G(x))/ln(F(x)) is increasing in x, we show that the strategy of balanced allocation of spares optimizes the failure rate function of the system. Furthermore, under the hypotheses that lifetimes of n components are stochastically ordered, lifetimes of k spares are identically distributed, and that lifetimes of components and spares are independently distributed, we show that the strategy of balanced allocation of spares is superior to the strategy of allocating a larger number of components to stronger components. For coherent systems consisting of n identical components with n identical redundant (spare) components, we compare strategies of component and system redundancies under the criteria of reversed failure rate and likelihood ratio orderings. When spares and original components do not necessarily match in their life distributions, we provide a sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under reversed failure rate ordering. As a consequence, we show that, for r-out-of-n systems, the strategy of component redundancy is superior to the strategy of system redundancy under the criterion of reversed failure rate ordering. When spares and original components match in their life distributions, we provide a necessary and sufficient condition, on the structure of the coherent system, for the strategy of component redundancy to be superior to the strategy of system redundancy under the likelihood ratio ordering. As a consequence, we show that, for r-out-of-n systems, with spares and original components matching in their life distributions, the strategy of component redundancy is superior to the strategy of system redundancy under the likelihood ratio ordering.
State-Dependent Utility
Authors: Londoño Jaime A.
Published: July 2016

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.
An Optimal Dividends Problem with a Terminal Value for Spectrally Negative Lévy Processes with a Completely Monotone Jump Density
Authors: Loeffen R. L.
Published: July 2016

We consider a modified version of the classical optimal dividends problem of de Finetti in which the objective function is altered by adding in an extra term which takes account of the ruin time of the risk process, the latter being modeled by a spectrally negative Lévy process. We show that, with the exception of a small class, a barrier strategy forms an optimal strategy under the condition that the Lévy measure has a completely monotone density. As a prerequisite for the proof, we show that, under the aforementioned condition on the Lévy measure, theq-scale function of the spectrally negative Lévy process has a derivative which is strictly log-convex.