We consider a class of correlated Bernoulli variables, which have the following form: for some 0 < p < 1,

$$\begin{align}{P(X_{j+1}=1 \vert {\cal F}_{j})= (1-\theta_j)p+\theta_jS_j/j,}\end{align}$$

$S_n=\sum _{j=1}^nX_j$

${\cal F}_n=\sigma \{X_1,\ldots , X_n\}$

The transient crystal size distribution (CSD) in a continuous mixed suspension, mixed product removal crystallizer has been modeled through a stochastic approach. Effects of the seed size distribution and size-dependent growth rate on both the transient and steady-state CSD have been investigated. It has been found that the seed size distribution causes a maximum in the steady-state CSD and the size-dependent growth rate results in an upward curvature at the lower end in a semilogarithmic plot of steady-state CSD against crystal size. Under appropriate conditions, the mean CSD of the present stochastic model reduces to the results predicted by the corresponding deterministic population balance model.

Zhu and He [(2018). A new closed-form formula for pricing European options under a skew Brownian motion. The European Journal of Finance 24(12): 1063–1074] provided an innovative closed-form solution by replacing the standard Brownian motion in the Black–Scholes framework using a particular skew Brownian motion. Their formula involves numerically integrating the product of the Guassian density and corresponding distribution function. Being different from their pricing formula, we derive a much simpler formula that only involves the Gaussian distribution function and Owen's $T$ function.

This paper investigates continuous-time optimal portfolio and consumption problems under loss aversion in an infinite time horizon. The investor's goal is to choose the optimal portfolio and consumption policies to maximize total discounted S-shaped utility from consumption. The problems are solved under two different situations respectively for the reference level: exogenous or endogenous. For the case of exogenous reference level, which is independent of the consumption policy, the optimal consumption policy and wealth process are obtained through the martingale method and replicating technique. For the case of endogenous reference level, which is related to the past actual consumption, the optimization problem with stochastic reference level is first transformed into an equivalent optimization problem with zero reference point, the corresponding relationship between them is proved, and then the relevant optimal consumption policy and wealth process are also obtained. When the investment opportunity sets are constants, the closed-form solutions of the portfolio and consumption policies are derived under two different situations respectively.

This paper studies the optimal allocation policy of a coherent system with independent heterogeneous components and dependent subsystems, the systems are assumed to consist of two groups of components whose lifetimes follow proportional hazard (PH) or proportional reversed hazard (PRH) models. We investigate the optimal allocation strategy by finding out the number $k$ of components coming from Group A in the up-series system. First, some sufficient conditions are provided in the sense of the usual stochastic order to compare the lifetimes of two-parallel–series systems with dependent subsystems, and we obtain the hazard rate and reversed hazard rate orders when two subsystems have independent lifetimes. Second, similar results are also obtained for two-series–parallel systems under certain conditions. Finally, we generalize the corresponding results to parallel–series and series–parallel systems with multiple subsystems in the viewpoint of the minimal path and the minimal cut sets, respectively. Some numerical examples are presented to illustrate the theoretical findings.

This paper describes a procedure for computing tightest possible best-case and worst-case bounds on the coefficient of variation of a discrete, bounded random variable when lower and upper bounds are available for its unknown probability mass function. An example from the application of the Monte Carlo method to the estimation of network reliability illustrates the procedure and, in particular, reveals considerable tightening in the worst-case bound when compared to the trivial worst-case bound based exclusively on range.

A new class of weighted generalized gamma distribution (WGGD) and related distributions are presented. Theoretical properties of the generalized gamma model, WGGD including the hazard function, reverse hazard function, moments, coefficient of variation, coefficient of skewness, coefficient of kurtosis, and entropy measures are derived. The results presented here generalizes the generalized gamma distribution and includes several distributions as special cases. The special cases include generalized gamma, weighted gamma, weighted exponential, weighted Weibull, weighted Rayleigh distributions, and their underlying or parent distributions.

We prove that no stochastic domination exists between the effective resistance of a spherically symmetric random tree and that of a branching process in a varying environments tree if they grow according to the same law of distribution.

This article presents two methods for predicting weather-related overhead distribution feeder failures. The first model is based on linear regression, which uses a regression function to determine the correlation between the weather factors and overhead feeder failures. The second method is based on a one-layer Bayesian network, which uses conditional probabilities to model the correlation. Both methods are discussed and followed by tests to assess their performance. The results obtained using these methods are discussed and compared.

Consider a multidimensional risk model, in which an insurer simultaneously confronts m (m ≥ 2) types of claims sharing a common non-stationary and non-renewal arrival process. Assuming that the claims arrival process satisfies a large deviation principle and the claim-size distributions are heavy-tailed, asymptotic estimates for two common types of ruin probabilities for this multidimensional risk model are obtained. As applications, we give two examples of the non-stationary point process: a Hawkes process and a Cox process with shot noise intensity, and asymptotic ruin probabilities are obtained for these two examples.

Traveling times in a FIFO-stochastic event graph are compared in increasing convex ordering for different arrival processes. As a special case, a stochastic lower bound is obtained for the sojourn time in a tandem network of FIFO queues with a Markov arrival process. A counterexample shows that the extended Ross conjecture is not true for discrete-time arrival processes.

We consider a growth collapse model in a random environment for which the input rates might depend on the state of an underlying irreducible Markov chain and at state change epochs there is a possible downward jump to a level that is a random fraction of the level just before the jump. The distributions of these jumps are allowed to depend on both the originating and target states. Under a very weak assumption we develop an explicit formula for the conditional moments (of all orders) of the time stationary distribution. We then consider special cases and show how to use this result to study a growth collapse process in which the times between collapses have a phase-type distribution.

This paper deals with the multivariate tail conditional expectation (MTCE) for generalized skew-elliptical distributions. We present tail conditional expectation for univariate generalized skew-elliptical distributions and MTCE for generalized skew-elliptical distributions. There are many special cases for generalized skew-elliptical distributions, such as generalized skew-normal, generalized skew Student-t, generalized skew-logistic and generalized skew-Laplace distributions.

The general problem under investigation is to understand how the complexity of a system which has been adapted to its random environment affects the level of randomness of its output (which is a function of its random input). In this paper, we consider a specific instance of this problem in which a deterministic finite-state decision system operates in a random environment that is modeled by a binary Markov chain. The system interacts with it by trying to match states of inactivity (represented by 0). Matching means that the system selects the (t + 1)th bit from the Markov chain whenever it predicts at time t that the environment will take a 0 value. The actual value at time t + 1 may be 0 or 1 thus the selected sequence of bits (which forms the system's output) may have both binary values. To try to predict well, the system's decision function is inferred based on a sample of the random environment.

We are interested in assessing how non-random the output sequence may be. To do that, we apply the adapted system on a second random sample of the environment and derive an upper bound on the deviation between the average number of 1 bit in the output sequence and the probability of a 1. The bound shows that the complexity of the system has a direct effect on this deviation and hence on how non-random the output sequence may be. The bound takes the form of $O(\sqrt {(2^k/n} ))$ where 2k is the complexity of the system and n is the length of the second sample.

We consider a discounted cost Markov decision process with a constraint. Relating this to a vector-valued Markov decision process, we prove that there exists a constrained optimal randomized semistationary policy if there exists at least one policy satisfying a constraint. Moreover, we present an algorithm by which we can find the constrained optimal randomized semistationary policy, or we can discover that there exist no policies satisfying a given constraint.

We analyze the optimal preemptive sequencing of n jobs on M + 1 parallel identical machines to minimize expected total flowtime. The running times of the jobs are independent samples from the distribution Pr(X = H) = p, Pr(X = H + T) = 1 − p, where H, T are random variables of general distribution. Preemption of a job is allowed when H is completed. This problem does not have a simple optimal solution. We show that the scheme of shortest expected remaining processing time first (SERPT) is close to optimal in two senses. The expected flowtime under SERPT and under the optimal policy differ by no more than a constant, independent of the number of jobs, and the expected number of optimal decisions that are not according to SERPT is bounded by a constant, independent of the number of jobs.

Techniques of exploratory data analysis are used in the construction of simple approximations for the percentage points of the chi-square distribution. The approximations are direct functions of the degrees of freedom and the tail probability. They are more accurate than Hoaglin's approximations.

Markov modulated fluid models are studied in this paper. When the input of the fluid model is represented by one Markov chain, two approaches are given that result in asymptotic expressions for the overflow probability. Both approaches are based on large deviations theories. The equivalence of the expressions is proved. When the input is represented by N similar Markov chains, a reduction property is derived.

We consider an inventory system with compound Poisson demands replenished by discrete production of units on a single-server facility. This facility may start a vacation at any production completion epoch; at the completion of a vacation the inventory level is inspected to decide whether or not to resume production. Unit production and vacation times are independent and identically distributed with general distributions. Under an (s, S) policy, production is terminated when the inventory level reaches a critical level S, and production is resumed when the inventory level, upon completion of a vacation, is at or below a prespecified value s. In this paper we prove that an (s, S) policy is optimal among all possible policies. We also derive a highly efficient algorithm for the determination and evaluation of an optimal (s, S) policy.

Most of the multivariate counting processes studied in the literature are regular processes, which implies, ignoring the types of the events, the non-occurrence of multiple events. However, in practice, several different types of events may occur simultaneously. In this paper, a new class of multivariate counting processes which allow simultaneous occurrences of multiple types of events is suggested and its stochastic properties are studied. For the modeling of such kind of process, we rely on the tool of superposition of seed counting processes. It will be shown that the stochastic properties of the proposed class of multivariate counting processes are explicitly expressed. Furthermore, the marginal processes are also explicitly obtained. We analyze the multivariate dependence structure of the proposed class of counting processes.