bluuebunny/crossref_metadata_2025 huggingface dataset:
What is the time value of a stream of investments?
Authors: Norberg Ragnar , Steffensen Mogens
Published: July 2016

The titular question is investigated for fairly general semimartingale investment and asset price processes. A discrete-time consideration suggests a stochastic differential equation and an integral expression for the time value in the continuous-time framework. It is shown that the two are equivalent if the jump part of the price process converges. The integral expression, which is the answer to the titular question, is the sum of all investments accumulated with returns on the asset (a stochastic integral) plus a term that accounts for the possible covariation between the two processes. The arbitrage-free price of the time value is the expected value of the sum (i.e. integral) of all investments discounted with the locally risk-free asset.
The effect of service time variability on maximum queue lengths in MX/G/1 queues
Authors: Koole Ger , Nuyens Misja , Righter Rhonda
Published: July 2016

We study the impact of service time distributions on the distribution of the maximum queue length during a busy period for the M
X
/G/1 queue. The maximum queue length is an important random variable to understand when designing the buffer size for finite-buffer (M/G/1/n) systems. We show the somewhat surprising result that, for three variations of the preemptive last-come–first-served discipline, the maximum queue length during a busy period is smaller when service times are more variable (in the convex sense).
Asymptotic baseline of the hazard rate function of mixtures
Authors: Li Yulin
Published: July 2016

In this article, we consider the limit behavior of the hazard rate function of mixture distributions, assuming knowledge of the behavior of each individual distribution. We show that the asymptotic baseline function of the hazard rate function is preserved under mixture.
A Central Limit Theorem for Non-Overlapping Return Times
Authors: Johnson Oliver
Published: July 2016

Define the non-overlapping return time of a block of a random process to be the number of blocks that pass by before the block in question reappears. We prove a central limit theorem based on these return times. This result has applications to entropy estimation, and to the problem of determining if digits have come from an independent, equidistributed sequence. In the case of an equidistributed sequence, we use an argument based on negative association to prove convergence under conditions weaker than those required in the general case.
An Explicit Formula for the Optimal Gain in the Full-Information Problem of Owning a Relatively Best Object
Authors: Mazalov Vladimir V. , Tamaki Mitsushi
Published: July 2016

A full-information version of the secretary problem in which the objective is to maximize the time of possession of a relatively best object is considered both in the case in which only the most recently observed object may be chosen and in the case in which any of the previously observed objects may be chosen. The main purpose of this paper is to obtain, under an optimal rule, both the asymptotic proportional durations when the number of objects tends to infinity, and the expected durations when the number of objects remains finite. The integral expressions for these asymptotic values are derived in both cases: the approximate numerical values are 0.435 in the former (no-recall) case and 0.449 in the latter (recall) case, indicating a surprisingly small difference. The expected values of the optimal stopping times are also obtained from the planar Poisson process analysis.
Gambling Teams and Waiting Times for Patterns in Two-State Markov Chains
Authors: Glaz Joseph , Kulldorff Martin , Pozdnyakov Vladimir , Steele J. Michael
Published: July 2016

Methods using gambling teams and martingales are developed and applied to find formulae for the expected value and the generating function of the waiting time to observation of an element of a finite collection of patterns in a sequence generated by a two-state Markov chain of first, or higher, order.
A Lévy Process Reflected at a Poisson Age Process
Authors: Kella Offer , Boxma Onno , Mandjes Michel
Published: July 2016

We consider a Lévy process with no negative jumps, reflected at a stochastic boundary that is a positive constant multiple of an age process associated with a Poisson process. We show that the stability condition for this process is identical to the one for the case of reflection at the origin. In particular, there exists a unique stationary distribution that is independent of initial conditions. We identify the Laplace-Stieltjes transform of the stationary distribution and observe that it satisfies a decomposition property. In fact, it is a sum of two independent random variables, one of which has the stationary distribution of the process reflected at the origin, and the other the stationary distribution of a certain clearing process. The latter is itself distributed as an infinite sum of independent random variables. Finally, we discuss the tail behavior of the stationary distribution and in particular observe that the second distribution in the decomposition always has a light tail.
Stabilization of an Overloaded Queueing Network Using Measurement-Based Admission Control
Authors: Leskelä Lasse
Published: July 2016

Admission control can be employed to avoid congestion in queueing networks subject to overload. In distributed networks, the admission decisions are often based on imperfect measurements on the network state. In this paper, we study how the lack of complete state information affects the system performance, by considering a simple network model for distributed admission control. The stability region of the network is characterized and it is shown how feedback signaling makes the system very sensitive to its parameters.
On a random-coefficient AR(1) process with heavy-tailed renewal switching coefficient and heavy-tailed noise
Authors: Leipus Remigijus , Paulauskas Vygantas , Surgailis Donatas
Published: July 2016

We discuss the limit behavior of the partial sums process of stationary solutions to the (autoregressive) AR(1) equation X

t
= a

t

X

t−1 + ε
t
with random (renewal-reward) coefficient, a

t
, taking independent, identically distributed values A

j
∈ [0,1] on consecutive intervals of a stationary renewal process with heavy-tailed interrenewal distribution, and independent, identically distributed innovations, ε
t
, belonging to the domain of attraction of an α-stable law (0 < α ≤ 2, α ≠ 1). Under suitable conditions on the tail parameter of the interrenewal distribution and the singularity parameter of the distribution of A

j
near the unit root a = 1, we show that the partial sums process of X

t
converges to a λ-stable Lévy process with index λ < α. The paper extends the result of Leipus and Surgailis (2003) from the case of finite-variance X

t
to that of infinite-variance X

t
.
The untraceable events method for absorbing processes
Authors: Nakatsuka Toshinao
Published: July 2016

In this paper we propose a new method of determining the stability of queueing systems. We attain it using the absorbing process and introduce the untraceable events method to show the existence of the absorbing process. The advantage of our method is that we are able to discuss the stability of various variables for both discrete and continuous parameters in a general framework with nonstationary input. An untraceable event has the property that the state loses the memory of its origin. In a concrete model, we use the boundedness of the state at an epoch in time with respect to the initial condition and choose the form of the untraceable event corresponding to the input distribution.
Critical scaling for the SIS stochastic epidemic
Authors: Dolgoarshinnykh R. G. , Lalley Steven P.
Published: July 2016

We exhibit a scaling law for the criticalSISstochastic epidemic. If at time 0 the population consists ofinfected andsusceptible individuals, then when the time and the number currently infected are both scaled by, the resulting process converges, asN→ ∞, to a diffusion process related to the Feller diffusion by a change of drift. As a consequence, the rescaled size of the epidemic has a limit law that coincides with that of a first passage time for the standard Ornstein-Uhlenbeck process. These results are the analogs for the SIS epidemic of results of Martin-Löf (1998) and Aldous (1997) for the simple SIR epidemic.
A class of risk processes with delayed claims: ruin probability estimates under heavy tail conditions
Authors: Ganesh Ayalvadi , Torrisi Giovanni Luca
Published: July 2016

We consider a class of risk processes with delayed claims, and we provide ruin probability estimates under heavy tail conditions on the claim size distribution.
Multitype linear fractional branching processes
Authors: Joffe A. , Letac G.
Published: July 2016

We complete a paper written by Edward Pollak in 1974 on a multitype branching process the generating functions of whose birth law are fractional linear functions with the same denominator. The main tool is a parameterization of these functions adapted using the mean matrix
M
and an element
w
of the first quadrant. We use this opportunity to give a self-contained presentation of Pollak's theory.
Some limiting properties of the bounds of the present value function of a life insurance portfolio
Authors: Zhang Yi , Lin Zhengyan , Weng Chengguo
Published: July 2016

Under certain assumptions on the dependence structure of the residual lives of the insured (i.e. their independence, positive association, or negative association), in this paper we establish some laws of large numbers for the convex upper bounds, derived by the technique of comonotonicity, of the present value function of a homogeneous portfolio composed of the whole-life insurance policies.
Asymptotic Behavior of a Generalized TCP Congestion Avoidance Algorithm
Authors: Ott Teunis J. , Swanson Jason
Published: July 2016

The transmission control protocol (TCP) is a transport protocol used in the Internet. In Ott (2005), a more general class of candidate transport protocols called ‘protocols in the TCP paradigm’ was introduced. The long-term objective of studying this class is to find protocols with promising performance characteristics. In this paper we study Markov chain models derived from protocols in the TCP paradigm. Protocols in the TCP paradigm, as TCP, protect the network from congestion by decreasing the ‘congestion window’ (i.e. the amount of data allowed to be sent but not yet acknowledged) when there is packet loss or packet marking, and increasing it when there is no loss. When loss of different packets are assumed to be independent events and the probabilitypof loss is assumed to be constant, the protocol gives rise to a Markov chain {Wn}, whereWnis the size of the congestion window after the transmission of thenth packet. For a wide class of such Markov chains, we prove weak convergence results, after appropriate rescaling of time and space, asp→ 0. The limiting processes are defined by stochastic differential equations. Depending on certain parameter values, the stochastic differential equation can define an Ornstein-Uhlenbeck process or can be driven by a Poisson process.
Discounted Optimal Stopping for Maxima of Some Jump-Diffusion Processes
Authors: Gapeev Pavel V.
Published: July 2016

In this paper we present closed form solutions of some discounted optimal stopping problems for the maximum process in a model driven by a Brownian motion and a compound Poisson process with exponential jumps. The method of proof is based on reducing the initial problems to integro-differential free-boundary problems, where the normal-reflection and smooth-fit conditions may break down and the latter then replaced by the continuous-fit condition. We show that, under certain relationships on the parameters of the model, the optimal stopping boundary can be uniquely determined as a component of the solution of a two-dimensional system of nonlinear ordinary differential equations. The obtained results can be interpreted as pricing perpetual American lookback options with fixed and floating strikes in a jump-diffusion model.
Optimal Smooth Portfolio Selection for an Insider
Authors: Hu Yaozhong , Øksendal Bernt
Published: July 2016

We study the optimal portfolio problem for an insider, in the case where the performance is measured in terms of the logarithm of the terminal wealth minus a term measuring the roughness and the growth of the portfolio. We give explicit solutions in some cases. Our method uses stochastic calculus of forward integrals.
Explicit Optimal Strategy for the Vardi Casino with Limited Playing Time
Authors: Yao Yi-Ching
Published: July 2016

The Vardi casino with parameter 0 <c< 1 consists of infinitely many tables indexed by their odds, each of which returns the same (negative) expected winnings -cper dollar. A gambler seeks to maximize the probability of reaching a fixed fortune by gambling repeatedly with suitably chosen stakes and tables (odds). The optimal strategy is derived explicitly subject to the constraint that the gambler is allowed to play only a given finite number of times. Some properties of the optimal strategy are also discussed.
A Weakest Link Marked Stopping Problem
Authors: Lippman Steven A. , Ross Sheldon M. , Seshadri Sridhar
Published: July 2016

We consider a model in which we have k items to be sold. Potential buyers make offers in a sequential fashion. Once made, the offer is either rejected or marked for acceptance. Once k items have been marked, the items are then sold to the buyers whose offers were marked, but at a price equal to the minimum of the k marked offers. Assuming that the successive offers are independent and identically distributed according to a specified distribution and that there is a fixed cost incurred whenever an offer is rejected, we determine structural results about the optimal policy, present computational approaches for finding the optimal policy, and give some heuristic policies.
Bounds for Owen's Multilinear Extension
Authors: Freixas Josep
Published: July 2016

Owen's multilinear extension (MLE) of a game is a very important tool in game theory and particularly in the field of simple games. Among other applications it serves to efficiently compute several solution concepts. In this paper we provide bounds for the MLE. Apart from its self-contained theoretical interest, the bounds offer the means in voting system studies of approximating the probability that a proposal is approved in a particular simple game having a complex component arrangement. The practical interest of the bounds is that they can be useful for simple games having a tedious MLE to evaluate exactly, but whose minimal winning coalitions and minimal blocking coalitions can be determined by inspection. Such simple games are quite numerous.
On Utility-Based Superreplication Prices of Contingent Claims with Unbounded Payoffs
Authors: Oertel Frank , Owen Mark
Published: July 2016

Consider a financial market in which an agent trades with utility-induced restrictions on wealth. For a utility function which satisfies the condition of reasonable asymptotic elasticity at -∞, we prove that the utility-based superreplication price of an unbounded (but sufficiently integrable) contingent claim is equal to the supremum of its discounted expectations under pricing measures with finite loss-entropy. For an agent whose utility function is unbounded from above, the set of pricing measures with finite loss-entropy can be slightly larger than the set of pricing measures with finite entropy. Indeed, the former set is the closure of the latter under a suitable weak topology. Central to our proof is a proof of the duality between the cone of utility-based superreplicable contingent claims and the cone generated by pricing measures with finite loss-entropy.
On a Model for the Storage of Files on a Hardware. II. Evolution of a Typical Data Block
Authors: Bansaye Vincent
Published: July 2016

We consider the generalized version in continuous time of the parking problem of Knuth introduced in Bansaye (2006). Files arrive following a Poisson point process and are stored on a hardware identified with the real line, at the right of their arrival point. Here we study the evolution of the endpoints of the data block straddling 0, which is empty at time 0 and is equal to R at a deterministic time.
A Gamma Activity Time Process with Noninteger Parameter and Self-Similar Limit
Authors: Finlay Richard , Seneta Eugene
Published: July 2016

We construct a process with gamma increments, which has a given convex autocorrelation function and asymptotically a self-similar limit. This construction validates the use of long-range dependent t and variance-gamma subordinator models for actual financial data as advocated in Heyde and Leonenko (2005) and Finlay and Seneta (2006), in that it allows for noninteger-valued model parameters to occur as found empirically by data fitting.
On a Theorem of Breiman and a Class of Random Difference Equations
Authors: Denisov Denis , Zwart Bert
Published: July 2016

We consider the tail behavior of the product of two independent nonnegative random variables X and Y. Breiman (1965) has considered this problem, assuming that X is regularly varying with index α and that E{Yα+ε} < ∞ for some ε > 0. We investigate when the condition on Y can be weakened and apply our findings to analyze a class of random difference equations.
Long-Range Dependence of Markov Chains in Discrete Time on Countable State Space
Authors: Carpio K. J. E. , Daley D. J.
Published: July 2016

When {X

n
} is an irreducible, stationary, aperiodic Markov chain on the countable state space
X
= {i, j,…}, the study of long-range dependence of any square integrable functional {Y

n
} := {y

X

n

} of the chain, for any real-valued function {y

i
: i ∈
X
}, involves in an essential manner the functions Q

ij

n
= ∑
r=1

n
(p

ij

r
− π
j
), where p

ij

r
= P{X

r
= j | X
0 = i} is the r-step transition probability for the chain and {π
i
: i ∈
X
} = P{X

n
= i} is the stationary distribution for {X

n
}. The simplest functional arises when Y

n
is the indicator sequence for visits to some particular state i, I

ni
= I
{X

n
=i} say, in which case limsup
n→∞
n
−1var(Y
1 + ∙ ∙ ∙ + Y

n
) = limsup
n→∞
n
−1 var(N

i
(0, n]) = ∞ if and only if the generic return time random variable T

ii
for the chain to return to state i starting from i has infinite second moment (here, N

i
(0, n] denotes the number of visits of X

r
to state i in the time epochs {1,…,n}). This condition is equivalent to Q

ji

n
→ ∞ for one (and then every) state j, or to E(T

jj

2) = ∞ for one (and then every) state j, and when it holds, (Q

ij

n
/ π
j
) / (Q

kk

n
/ π
k
) → 1 for n → ∞ for any triplet of states i, j
k.
Negative Association Does not Imply Log-Concavity of the Rank Sequence
Authors: Markström Klas
Published: July 2016

We present a minimum counterexample to the conjecture that a negatively associated random variable has an ultra-log-concave rank sequence. The rank sequence does not in fact even need to be unimodal.
On the Structure of a Swing Contract's Optimal Value and Optimal Strategy
Authors: Ross Sheldon M. , Zhu Zegang
Published: July 2016

Consider a sales contract, called a swing contract, between a seller and a buyer concerning some underlying commodity, with the contract specifying that during some future time interval the buyer will purchase an amount of the commodity between some specified minimum and maximum values. The purchase price and capacity at each time point is also prespecified in the contract. Assuming a random market price process and ignoring the possibility of storage, we look for the maximal expected net gain for the buyer of such a contract, and the strategy that achieves this maximal expected net gain. We study the effects that various contract constraints and market price processes have on the optimal strategy and on the contract value. We show how we can reduce the general swing contract to a multiple exercising of American (Bermudan) style options. Also, in important special cases, we give explicit expressions for the optimal contract value function and the optimal strategy.
Evaluating Scale Functions of Spectrally Negative Lévy Processes
Authors: Surya B. A.
Published: July 2016

In this paper we present a robust numerical method to compute the scale function W
(q)(x) of a general spectrally negative Lévy process (X, P). The method is based on the Esscher transform of measure Pν under which X is taken and the scale function is determined. This change of measure makes it possible for the scale function to be bounded and, hence, makes numerical computation easy, fast, and stable. Working under the new measure Pν and using the method of Abate and Whitt (1992) and Choudhury, Lucantoni, and Whitt (1994), we give a fast stable numerical algorithm for the computation of W
(q)(x).
The Necklace Process
Authors: Mallows Colin , Shepp Larry
Published: July 2016

Start with a necklace consisting of one white bead and one black bead, and add new beads one at a time by inserting each new bead between a randomly chosen adjacent pair of old beads, with the proviso that the new bead will be white if and only if both beads of the adjacent pair are black. Let W

n
denote the number of white beads when the total number of beads is n. We show that EW

n
= n/3 and, with c
2 = 2/45, that (W

n
− n/3) / c√n
is asymptotically standard normal. We find that, for all r ≥ 1 and n > 2r, the rth cumulant of the distribution of W

n
is of the form n
h

r
. We find the expected numbers of gaps of given length between white beads, and examine the asymptotics of the longest gaps.
Critical Behaviors and Critical Values of Branching Random Walks on Multigraphs
Authors: Bertacchi Daniela , Zucca Fabio
Published: July 2016

We consider weak and strong survival for branching random walks on multigraphs with bounded degree. We prove that, at the strong critical value, the process dies out locally almost surely. We relate the weak critical value to a geometric parameter of the multigraph. For a large class of multigraphs (which enlarges the class of quasi-transitive or regular graphs), we prove that, at the weak critical value, the process dies out globally almost surely. Moreover, for the same class, we prove that the existence of a pure weak phase is equivalent to nonamenability. The results are extended to branching random walks on weighted graphs.
Bounding the Size and Probability of Epidemics on Networks
Authors: Miller Joel C.
Published: July 2016

We consider an infectious disease spreading along the edges of a network which may have significant clustering. The individuals in the population have heterogeneous infectiousness and/or susceptibility. We define the out-transmissibility of a node to be the marginal probability that it would infect a randomly chosen neighbor given its infectiousness and the distribution of susceptibility. For a given distribution of out-transmissibility, we find the conditions which give the upper (or lower) bounds on the size and probability of an epidemic, under weak assumptions on the transmission properties, but very general assumptions on the network. We find similar bounds for a given distribution of in-transmissibility (the marginal probability of being infected by a neighbor). We also find conditions giving global upper bounds on the size and probability. The distributions leading to these bounds are network independent. In the special case of networks with high girth (locally tree-like), we are able to prove stronger results. In general, the probability and size of epidemics are maximal when the population is homogeneous and minimal when the variance of in- or out-transmissibility is maximal.
Laplace Transforms of Probability Distributions and Their Inversions are Easy on Logarithmic Scales
Authors: Rossberg A. G.
Published: July 2016

It is shown that, when expressing arguments in terms of their logarithms, the Laplace transform of a function is related to the antiderivative of this function by a simple convolution. This allows efficient numerical computations of moment generating functions of positive random variables and their inversion. The application of the method is straightforward, apart from the necessity to implement it using high-precision arithmetics. In numerical examples the approach is demonstrated to be particularly useful for distributions with heavy tails, such as lognormal, Weibull, or Pareto distributions, which are otherwise difficult to handle. The computational efficiency compared to other methods is demonstrated for an M/G/1 queueing problem.
Convergence of At-The-Money Implied Volatilities to the Spot Volatility
Authors: Durrleman Valdo
Published: July 2016

We study the convergence of at-the-money implied volatilities to the spot volatility in a general model with a Brownian component and a jump component of finite variation. This result is a consequence of the robustness of the Black-Scholes formula and of the central limit theorem for martingales.
Epidemic Size in the SIS Model of Endemic Infections
Authors: Kessler David A.
Published: July 2016

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.
Estimates for the Absolute Ruin Probability in the Compound Poisson Risk Model with Credit and Debit Interest
Authors: Zhu Jinxia , Yang Hailiang
Published: July 2016

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.
Assessing a Linear Nanosystem's Limiting Reliability from its Components
Authors: Ebrahimi Nader
Published: July 2016

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 in nature. We consider the linear nanosystem at a fixed moment of time, say the present moment, and we assume that the present state of the linear nanosystem depends only on the present states of its atoms.
Average-Case Analysis of Cousins in m-ary Tries
Authors: Mahmoud Hosam M. , Ward Mark Daniel
Published: July 2016

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 arem-ary, and the corresponding trees arem-ary trees. Certain analytic issues arise in them-ary case that do not have an analog in the binary case.
On the Sums of Compound Negative Binomial and Gamma Random Variables
Authors: Vellaisamy P. , Upadhye N. S.
Published: July 2016

We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability ofm-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.
A Central Limit Theorem Associated with the Transformed Two-Parameter Poisson–Dirichlet Distribution
Authors: Xu Fang
Published: July 2016

In this paper we introduce the transformed two-parameter Poisson–Dirichlet distributionon the ordered infinite simplex. Furthermore, we prove the central limit theorem related to this distribution when both the mutation rateθand the selection rateσbecome large in a specified manner. As a consequence, we find that the properly scaled homozygosities have asymptotical normal behavior. In particular, there is a certain phase transition with the limit depending on the relative strength ofσandθ.
Approximating Critical Parameters of Branching Random Walks
Authors: Bertacchi Daniela , Zucca Fabio
Published: July 2016

Given a branching random walk on a graph, we consider two kinds of truncations: either by inhibiting the reproduction outside a subset of vertices or by allowing at most m particles per vertex. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.
L Log L Criterion for a Class of Superdiffusions
Authors: Liu Rong-Li , Ren Yan-Xia , Song Renming
Published: July 2016

In Lyons, Pemantle and Peres (1995), a martingale change of measure method was developed in order to give an alternative proof of the Kesten–Stigum L log L theorem for single-type branching processes. Later, this method was extended to prove the L log L theorem for multiple- and general multiple-type branching processes in Biggins and Kyprianou (2004), Kurtz et al. (1997), and Lyons (1997). In this paper we extend this method to a class of superdiffusions and establish a Kesten–Stigum L log L type theorem for superdiffusions. One of our main tools is a spine decomposition of superdiffusions, which is a modification of the one in Englander and Kyprianou (2004).
Geometric Convergence of Genetic Algorithms Under Tempered Random Restart
Authors: Mendivil F. , Shonkwiler R. , Spruill M. C.
Published: July 2016

Geometric convergence to 0 of the probability that the goal has not been encountered by the nth generation is established for a class of genetic algorithms. These algorithms employ a quickly decreasing mutation rate and a crossover which restarts the algorithm in a controlled way depending on the current population and restricts execution of this crossover to occasions when progress of the algorithm is too slow. It is shown that without the crossover studied here, which amounts to a tempered restart of the algorithm, the asserted geometric convergence need not hold.
Systems with Failure-Dependent Lifetimes of Components
Authors: Burkschat M.
Published: July 2016

A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequentialk-out-of-nsystem, where the distribution of the remaining components' lifetimes is allowed to change after every failure of a component. In the present paper, general component lifetimes which can be influenced by failures are considered. The ordered failure times of these components can be used to extend the concept of sequential order statistics. In particular, a definition of sequential order statistics based on exchangeable components is proposed. By utilizing the system signature (cf. Samaniego (2007)), the distribution of the lifetime of a coherent system with failure-dependent exchangeable component lifetimes is shown to be given by a mixture of the distributions of sequential order statistics. Furthermore, some results on the joint distribution of sequential order statistics based on exchangeable components are given.
Space–Time Duality for Fractional Diffusion
Authors: Baeumer Boris , Meerschaert Mark M. , Nane Erkan
Published: July 2016

Zolotarev (1961) proved a duality result that relates stable densities with different indices. In this paper we show how Zolotarev's duality leads to some interesting results on fractional diffusion. Fractional diffusion equations employ fractional derivatives in place of the usual integer-order derivatives. They govern scaling limits of random walk models, with power-law jumps leading to fractional derivatives in space, and power-law waiting times between the jumps leading to fractional derivatives in time. The limit process is a stable Lévy motion that models the jumps, subordinated to an inverse stable process that models the waiting times. Using duality, we relate the density of a spectrally negative stable process with index 1<α<2 to the density of the hitting time of a stable subordinator with index 1/α, and thereby unify some recent results in the literature. These results provide a concrete interpretation of Zolotarev's duality in terms of the fractional diffusion model. They also illuminate a current controversy in hydrology, regarding the appropriate use of space- and time-fractional derivatives to model contaminant transport in river flows.
Scaling Limit for a Drainage Network Model
Authors: Coletti C. F. , Fontes L. R. G. , Dias E. S.
Published: July 2016

We consider the two-dimensional version of a drainage network model introduced in Gangopadhyay, Roy and Sarkar (2004), and show that the appropriately rescaled family of its paths converges in distribution to the Brownian web. We do so by verifying the convergence criteria proposed in Fontes, Isopi, Newman and Ravishankar (2002).
Prediction in a Poisson cluster model
Authors: Matsui Muneya , Mikosch Thomas
Published: July 2016

We consider a Poisson cluster model, motivated by insurance applications. At each claim arrival time, modeled by the point of a homogeneous Poisson process, we start a cluster process which represents the number or amount of payments triggered by the arrival of a claim in a portfolio. The cluster process is a Lévy or truncated compound Poisson process. Given the observations of the process over a finite interval, we consider the expected value of the number and amount of payments in a future time interval. We also give bounds for the error encountered in this prediction procedure.
The queue with impatience: construction of the stationary workload under FIFO
Authors: Moyal Pascal
Published: July 2016

In this paper we study the stability of queueing systems with impatient customers and a single server operating under a FIFO (first-in-first-out) discipline. We first give a sufficient condition for the existence of a stationary workload in the case of impatience until the beginning of service. We then provide a weaker condition of existence on an enriched probability space using the theory of Anantharam et al. (1997), (1999). The case of impatience until the end of service is also investigated.
Recovering a Piecewise Constant Volatility from Perpetual Put Option Prices
Authors: Lu Bing
Published: July 2016

In this paper we present a method to recover a time-homogeneous piecewise constant volatility from a finite set of perpetual put option prices. The whole calculation process of the volatility is decomposed into easy computations in many fixed disjoint intervals. In each interval, the volatility is obtained by solving a system of nonlinear equations.
Gibbs Random Fields with Unbounded Spins on Unbounded Degree Graphs
Authors: Kondratiev Yuri , Kozitsky Yuri , Pasurek Tanja
Published: July 2016

Gibbs fields are constructed and studied which correspond to systems of real-valued spins (e.g. systems of interacting anharmonic oscillators) indexed by the vertices of unbounded degree graphs of a certain type, for which the Gaussian Gibbs fields need not be existing. In these graphs, the vertex degree growth is controlled by a summability requirement formulated with the help of a generalized Randić index. In particular, it is proven that the Gibbs fields obey uniform integrability estimates, which are then used in the study of the topological properties of the set of Gibbs fields. In the second part, a class of graphs is introduced in which the mentioned summability is obtained by assuming that the vertices of large degree are located at large distances from each other. This is a stronger version of the metric property employed in Bassalygo and Dobrushin (1986).
Some New Results on the Moment Generating Function Order and Related Life Distributions
Authors: Zhang Shuhong , Li Xiaohu
Published: July 2016

In this paper we study the moment generating function order and the new better than used in the moment generating function order (NBUMG) life distributions. A closure property of this order under an independent random sum is deduced, and stochastic comparisons among the block replacement policy, the age replacement policy, the complete repair policy, and the minimal repair policy of an NBUMG component are investigated.
Shortfall Risk Approximations for American Options in the Multidimensional Black-Scholes Model
Authors: Dolinsky Yan
Published: July 2016

We show that shortfall risks of American options in a sequence of multinomial approximations of the multidimensional Black-Scholes (BS) market converge to the corresponding quantities for similar American options in the multidimensional BS market with path-dependent payoffs. In comparison to previous papers we consider the multiassets case for which we use the weak convergence approach.
Discrete Scan Statistics Generated by Exchangeable Binary Trials
Authors: Eryilmaz Serkan
Published: July 2016

Let {X

i
}
i=1

n
be a sequence of random variables with two possible outcomes, denoted 0 and 1. Define a random variable S

n,m
to be the maximum number of 1s within any m consecutive trials in {X

i
}
i=1

n
. The random variable S

n,m
is called a discrete scan statistic and has applications in many areas. In this paper we evaluate the distribution of discrete scan statistics when {X

i
}
i=1

n
consists of exchangeable binary trials. We provide simple closed-form expressions for both conditional and unconditional distributions of S

n,m
for 2m ≥ n. These results are also new for independent, identically distributed Bernoulli trials, which are a special case of exchangeable trials.
Odds Theorem with Multiple Selection Chances
Authors: Ano Katsunori , Kakinuma Hideo , Miyoshi Naoto
Published: July 2016

We study the multi-selection version of the so-called odds theorem by Bruss (2000). We observe a finite number of independent 0/1 (failure/success) random variables sequentially and want to select the last success. We derive the optimal selection rule when m (≥ 1) selection chances are given and find that the optimal rule has the form of a combination of multiple odds-sums. We provide a formula for computing the maximum probability of selecting the last success when we have m selection chances and also provide closed-form formulae for m = 2 and 3. For m = 2, we further give the bounds for the maximum probability of selecting the last success and derive its limit as the number of observations goes to ∞. An interesting implication of our result is that the limit of the maximum probability of selecting the last success for m = 2 is consistent with the corresponding limit for the classical secretary problem with two selection chances.
Optimal Time to Exchange Two Baskets
Authors: Nishide Katsumasa , Rogers L. C. G.
Published: July 2016

In this paper we present simple extensions of earlier works on the optimal time to exchange one basket of log Brownian assets for another. A superset and subset of the optimal stopping region in the case where both baskets consist of multiple assets are obtained. It is also shown that a conjecture of Hu and Øksendal (1998) is false except in the trivial case where all the assets in a basket are the same processes.
Quasistationary Distributions and Fleming-Viot Processes in Finite Spaces
Authors: Asselah Amine , Ferrari Pablo A. , Groisman Pablo
Published: July 2016

Consider a continuous-time Markov process with transition rates matrixQin the state space Λ ⋃ {0}. In the associated Fleming-Viot processNparticles evolve independently in Λ with transition rates matrixQuntil one of them attempts to jump to state 0. At this moment the particle jumps to one of the positions of the other particles, chosen uniformly at random. When Λ is finite, we show that the empirical distribution of the particles at a fixed time converges asN→ ∞ to the distribution of a single particle at the same time conditioned on not touching {0}. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process withNparticles converges asN→ ∞ to the unique quasistationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations are of order 1 /N.
Processes with Block-Associated Increments
Authors: Jakubowski Adam , Karłowska-Pik Joanna
Published: July 2016

This paper is motivated by relations between association and independence of random variables. It is well known that, for real random variables, independence implies association in the sense of Esary, Proschan and Walkup (1967), while, for random vectors, this simple relationship breaks. We modify the notion of association in such a way that any vector-valued process with independent increments also has associated increments in the new sense - association between blocks. The new notion is quite natural and admits nice characterization for some classes of processes. In particular, using the covariance interpolation formula due to Houdré, Pérez-Abreu and Surgailis (1998), we show that within the class of multidimensional Gaussian processes, block association of increments is equivalent to supermodularity (in time) of the covariance functions. We also define corresponding versions of weak association, positive association, and negative association. It turns out that the central limit theorem for weakly associated random vectors due to Burton, Dabrowski and Dehling (1986) remains valid, if the weak association is relaxed to the weak association between blocks.
Concave Renewal Functions do not Imply DFR Interrenewal Times
Authors: Yu Yaming
Published: July 2016

Brown (1980), (1981) proved that the renewal function is concave if the interrenewal distribution is DFR (decreasing failure rate), and conjectured the converse. This note settles Brown's conjecture with a class of counterexamples. We also give a short proof of Shanthikumar's (1988) result that the DFR property is closed under geometric compounding.
On the First Exit Time of a Nonnegative Markov Process Started at a Quasistationary Distribution
Authors: Pollak Moshe , Tartakovsky Alexander G.
Published: July 2016

Let {M

n
}
n≥0 be a nonnegative time-homogeneous Markov process. The quasistationary distributions referred to in this note are of the form Q

A
(x) = lim
n→∞P(M

n
≤ x | M
0 ≤ A,
M
1 ≤ A, …, M

n
≤ A). Suppose that M
0 has distribution Q

A
, and define T

A

Q

A

= min{n | M

n
> A, n ≥ 1}, the first time when M

n
exceeds A. We provide sufficient conditions for Q

A
(x) to be nonincreasing in A (for fixed x) and for T

A

Q

A

to be stochastically nondecreasing in A.
Rumor Processes on N
Authors: Junior Valdivino V. , Machado Fábio P. , Zuluaga Mauricio
Published: July 2016

We study four discrete-time stochastic systems on N, modeling processes of rumor spreading. The involved individuals can either have an active or a passive role, speaking up or asking for the rumor. The appetite for spreading or hearing the rumor is represented by a set of random variables whose distributions may depend on the individuals. Our goal is to understand - based on the distribution of the random variables - whether the probability of having an infinite set of individuals knowing the rumor is positive or not.
Optimal Control of Capital Injections by Reinsurance with a Constant Rate of Interest
Authors: Eisenberg Julia , Schmidli Hanspeter
Published: July 2016

We consider a classical risk model and its diffusion approximation, where the individual claims are reinsured by a reinsurance treaty with deductible b ∈ [0, b̃]. Here b = b̃ means ‘no reinsurance’ and b= 0 means ‘full reinsurance’. In addition, the insurer is allowed to invest in a riskless asset with some constant interest rate m > 0. The cedent can choose an adapted reinsurance strategy {b

t
}
t≥0, i.e. the parameter can be changed continuously. If the surplus process becomes negative, the cedent has to inject additional capital. Our aim is to minimise the expected discounted capital injections over all admissible reinsurance strategies. We find an explicit expression for the value function and the optimal strategy using the Hamilton-Jacobi-Bellman approach in the case of a diffusion approximation. In the case of the classical risk model, we show the existence of a ‘weak’ solution and calculate the value function numerically.
Realization of an Ergodic Markov Chain as a Random Walk Subject to a Synchronizing Road Coloring
Authors: Yano Kouji , Yasutomi Kenji
Published: July 2016

An ergodic Markov chain is proved to be the realization of a random walk in a directed graph subject to a synchronizing road coloring. The result ensures the existence of appropriate random mappings in Propp-Wilson's coupling from the past. The proof is based on the road coloring theorem. A necessary and sufficient condition for approximate preservation of entropies is also given.
A Note on a Paper by Wong and Heyde
Authors: Mijatović Aleksandar , Urusov Mikhail
Published: July 2016

In this note we re-examine the analysis of the paper ‘On the martingale property of stochastic exponentials’ by Wong and Heyde (2004). Some counterexamples are presented and alternative formulations are discussed.
An Integrated Probabilistic Model for Assessing a Nanocomponent's Reliability
Authors: Ebrahimi Nader , Yang Yarong
Published: July 2016

We construct an integrated probabilistic model to capture interactions between atoms of a nanocomponent. We then use this model to assess reliabilities of nanocomponents with different structures. Several properties of our proposed model are also described under a sparseness condition. The model is an extension of our previous model based on Markovian random field theory. The proposed integrated model is flexible in that pairwise relationship information among atoms as well as features of individual atoms can be easily incorporated. An important feature that distinguishes the integrated probabilistic model from our previous model is that the integrated approach uses all available sources of information with different weights for different types of interaction. In this paper we consider the nanocomponent at a fixed moment of time, say the present moment, and we assume that the present state of the nanocomponent depends only on the present states of its atoms.