We present a detailed analysis of random motions moving in higher spaces with a natural number of velocities. In the case of the so-called minimal random dynamics, under some broad assumptions, we give the joint distribution of the position of the motion (for both the inner part and the boundary of the support) and the number of displacements performed with each velocity. Explicit results for cyclic and complete motions are derived. We establish useful relationships between motions moving in different spaces, and we derive the form of the distribution of the movements in arbitrary dimension. Finally, we investigate further properties for stochastic motions governed by non-homogeneous Poisson processes.

We present a reflection principle for a wide class of symmetric random motions with finite velocities. We propose a deterministic argument which is then applied to trajectories of stochastic processes. In the case of symmetric correlated random walks and the symmetric telegraph process, we provide a probabilistic result recalling the classical reflection principle for Brownian motion, but where the initial velocity has a crucial role. In the case of the telegraph process we also present some consequences which lead to further reflection-type characteristics of the motion.

We prove that for a discrete determinantal process the BK inequality occurs for increasing events generated by simple points. We also give some elementary but nonetheless appealing relationships between a discrete determinantal process and the well-known CS decomposition.

We study the integrated telegraph process $X_t$ under the assumption of general distribution for the random times between consecutive reversals of direction. Specifically, $X_t$ represents the position, at time t, of a particle moving U time units upwards with velocity c and D time units downwards with velocity $-c$ . The latter motions are repeated cyclically, according to independent alternating renewals. Explicit expressions for the probability law of $X_t$ are given in the following cases: (i) (U, D) gamma-distributed; (ii) U exponentially distributed and D gamma-distributed. For certain values of the parameters involved, the probability law of $X_t$ is provided in a closed form. Some expressions for the moment generating function of $X_t$ and its Laplace transform are also obtained. The latter allows us to prove the existence of a Kac-type condition under which the probability density function of the integrated telegraph process, with identically distributed gamma intertimes, converges to that of the standard Brownian motion.

Finally, we consider the square of $X_t$ and disclose its distribution function, specifying the expression for some choices of the distribution of (U, D).

We propose a wide generalization of known results related to the telegraph process. Functionals of the simple telegraph process on a straight line and their generalizations on an arbitrary state space are studied.

We study the joint degree counts in linear preferential attachment random graphs and find a simple representation for the limit distribution in infinite sequence space. We show weak convergence with respect to the p-norm topology for appropriate p and also provide optimal rates of convergence of the finite-dimensional distributions. The results hold for models with any general initial seed graph and any fixed number of initial outgoing edges per vertex; we generate nontree graphs using both a lumping and a sequential rule. Convergence of the order statistics and optimal rates of convergence to the maximum of the degrees is also established.

The explosion probability before time t of a branching diffusion satisfies a nonlinear parabolic partial differential equation. This equation, along with the natural boundary and initial conditions, has only the trivial solution, i.e. explosion in finite time does not occur, provided the creation rate does not grow faster than the square power at ∞.

In this paper the optimal dividend (subject to transaction costs) and reinsurance (with two reinsurers) problem is studied in the limit diffusion setting. It is assumed that transaction costs and taxes are required when dividends occur, and that the premiums charged by two reinsurers are calculated according to the exponential premium principle with different parameters, which makes the stochastic control problem nonlinear. The objective of the insurer is to determine the optimal reinsurance and dividend policy so as to maximize the expected discounted dividends until ruin. The problem is formulated as a mixed classical-impulse stochastic control problem. Explicit expressions for the value function and the corresponding optimal strategy are obtained. Finally, a numerical example is presented to illustrate the impact of the parameters associated with the two reinsurers' premium principle on the optimal reinsurance strategy.

We study the one-dimensional random motion X = X(t), t ≥ 0, which takes two different velocities with two different alternating intensities. The closed-form formulae for the density functions of X and for the moments of any order, as well as the distributions of the first passage times, are obtained. The limit behaviour of the moments is analysed under nonstandard Kac's scaling.

The yeast Saccharomyces cerevisiae has emerged as an ideal model system to study the dynamics of prion proteins which are responsible for a number of fatal neurodegenerative diseases in humans. Within an infected cell, prion proteins aggregate in complexes which may increase in size or be fragmented and are transmitted upon cell division. Recent work in yeast suggests that only aggregates below a critical size are transmitted efficiently. We formulate a continuous-time branching process model of a yeast colony under conditions of prion curing. We generalize previous approaches by providing an explicit formula approximating prion loss as influenced by both aggregate growth and size-dependent transmission.

In this paper we study the fractional moments of the stationary solution to the stochastic recurrence equation Xt = AtXt−1 + Bt, t ∈ Z, where ((At, Bt))t∈Z is an independent and identically distributed bivariate sequence. We derive recursive formulae for the fractional moments E|X0|p, p ∈ R. Special attention is given to the case when Bt has an Erlang distribution. We provide various approximations to the moments E|X0|p and show their performance in a small numerical study.

Using a characterizing equation for the beta distribution, Stein's method is applied to obtain bounds of the optimal order for the Wasserstein distance between the distribution of the scaled number of white balls drawn from a Pólya-Eggenberger urn and its limiting beta distribution. The bound is computed by making a direct comparison between characterizing operators of the target and the beta distribution, the former derived by extending Stein's density approach to discrete distributions. In addition, refinements are given to Döbler's (2012) result for the arcsine approximation for the fraction of time a simple random walk of even length spends positive, and so also to the distributions of its last return time to 0 and its first visit to its terminal point, by supplying explicit constants to the present Wasserstein bound and also demonstrating that its rate is of the optimal order.

The asymptotic variance of the sample mean of a homogeneous Poisson marked point process has been studied in the literature, but confusion has arisen as to the correct expression due to some technical intricacies. This note sets the record straight with regards to the variance of the sample mean. In addition, a central limit theorem in the general d-dimensional case is also established.

An infinite sequence (Y1, Y2,…) of independent Bernoulli random variables with P(Yi = 1) = a / (a + b + i - 1), i = 1, 2,…, where a > 0 and b ≥ 0, will be called a Bern(a, b) sequence. Consider the counts Z1, Z2, Z3,… of occurrences of patterns or strings of the form {11}, {101}, {1001},…, respectively, in this sequence. The joint distribution of the counts Z1, Z2,… in the infinite Bern(a, b) sequence has been studied extensively. The counts from the initial finite sequence (Y1, Y2,…, Yn) have been studied by Holst (2007), (2008b), who obtained the joint factorial moments for Bern(a, 0) and the factorial moments of Z1, the count of the string {1, 1}, for a general Bern(a, b). We consider stopping the Bernoulli sequence at a random time and describe the joint distribution of counts, which extends Holst's results. We show that the joint distribution of counts from a class of randomly stopped Bernoulli sequences possesses the mixture of independent Poissons property: there is a random vector conditioned on which the counts are independent Poissons. To obtain these results, we extend the conditional marked Poisson process technique introduced in Huffer, Sethuraman and Sethuraman (2009). Our results avoid previous combinatorial and induction methods which generally only yield factorial moments.

Some major companies have the policy of annually giving numerical scores to their employees according to their performance, firing those whose performance scores are below a given percentile of the scores of all employees, and then recruiting new employees to replace those who were fired. We introduce a probabilistic model to describe how this practice affects the quality of employee performance as measured over time by the annual scores. Let n be the number of years that the policy has been in effect, and let Fn(x) be the distribution function of the evaluation scores in year n. We show, under certain technical assumptions, that the sequence (Fn(x)) satisfies a particular nonlinear difference equation, and furnish estimates of the solution of the equation and expressions for the quantiles of Fn. The mathematical tools that are used include convex functions, difference equations, and extreme value theory for independent and identically distributed random variables.

Suppose that I1, I2,… is a sequence of independent Bernoulli random variables with E(In) = λ/(λ + n − 1), n = 1, 2,…. If λ is a positive integer k, {In}n≥1 can be interpreted as a k-record process of a sequence of independent and identically distributed random variables with a common continuous distribution. When In−1In = 1, we say that a consecutive k-record occurs at time n. It is known that the total number of consecutive k-records is Poisson distributed with mean k. In fact, for general is Poisson distributed with mean λ. In this paper, we want to find an optimal stopping time τλ which maximizes the probability of stopping at the last n such that In−1In = 1. We prove that τλ is of threshold type, i.e. there exists a tλ ∈ ℕ such that τλ = min{n | n ≥ tλ, In−1In = 1}. We show that tλ is increasing in λ and derive an explicit expression for tλ. We also compute the maximum probability Qλ of stopping at the last consecutive record and study the asymptotic behavior of Qλ as λ → ∞.

We introduce and motivate the study of (n + 1) × r arrays X with Bernoulli entries Xk,j and independently distributed rows. We study the distribution of which denotes the number of consecutive pairs of successes (or runs of length 2) when reading the array down the columns and across the rows. With the case r = 1 having been studied by several authors, and permitting some initial inferences for the general case r > 1, we examine various distributional properties and representations of Sn for the case r = 2, and, using a more explicit analysis, the case of multinomial and identically distributed rows. Applications are also given in cases where the array X arises from a Pólya sampling scheme.

We introduce a stochastic process that describes a finite-velocity damped motion on the real line. Differently from the telegraph process, the random times between consecutive velocity changes have exponential distribution with linearly increasing parameters. We obtain the probability law of the motion, which admits a logistic stationary limit in a special case. Various results on the distributions of the maximum of the process and of the first passage time through a constant boundary are also given.

We consider the operation of an insurer with a large initial surplus x>0. The insurer's surplus process S(t) (with S(0)=x) evolves in the range S(t)≥ 0 as a generalized renewal process with positive mean drift and with jumps at time epochs T1,T2,…. At the time Tη(x) when the process S(t) first becomes negative, the insurer's ruin (in the ‘classical’ sense) occurs, but the insurer can borrow money via a line of credit. After this moment the process S(t) behaves as a solution to a certain stochastic differential equation which, in general, depends on the indebtedness, -S(t). This behavior of S(t) lasts until the time θ(x,y) at which the indebtedness reaches some ‘critical’ level y>0. At this moment the line of credit will be closed and the insurer's absolute ruin occurs with deficit -S(θ(x,y)). We find the asymptotics of the absolute ruin probability and the limiting distributions of η(x), θ(x,y), and -S(θ(x,y)) as x → ∞, assuming that the claim distribution is regularly varying. The second-order approximation to the absolute ruin probability is also obtained. The abovementioned results are obtained by using limiting theorems for the joint distribution of η(x) and -S(Tη(x)).

In an infinite sequence of independent Bernoulli trials with success probabilities pk=a/(a+b +k-1) for k=1,2,3,…, let Nr be the number of r≥2 consecutive successes. Expressions for the first two moments of Nr are derived. Asymptotics of the probability of no occurrence of r consecutive successes for large r are obtained. Using an embedding in a marked Poisson process, it is indicated how the distribution of Nr can be calculated for small r.