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Table Of Integrals Series And Products

Tables of integrals

The Theory of Stochastic Processes. By D. R. Cox and H. D. Miller. Pp. x, 398. 70s. (Methuen)

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

The basic jump-telegraph process with exponentially distributed interarrival times deserves interest in various applied fields such as financial modelling and queueing theory. Aiming to propose a more general setting, we analyse such a stochastic process when the interarrival times separating consecutive velocity changes (and jumps) have generalized Mittag-Leffler distributions, and constitute the random times of a fractional alternating Poisson process. By means of renewal theory-based issues we obtain the forward and backward transition densities of the motion in series form, and prove their uniform convergence. Specific attention is then given to the case of jumps with constant size, for which we also obtain the mean of the process. Finally, we investigate the first-passage time of the process through a constant positive boundary, providing its formal distribution and suitable lower bounds.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/D7166DDFD95CAE4502F5BF04816EB661/S0021900218000086a.pdf/div-class-title-on-a-jump-telegraph-process-driven-by-an-alternating-fractional-poisson-process-div.pdf

On a jump-telegraph process driven by an alternating fractional Poisson process

Dual usual stochastic ordering and dual increasing linear ordering are introduced as the dual of two well-known stochastic orderings in order to provide characterization of various ageing notions useful to describe ageing properties of devices with unknown age. The “harmonic used better than aged in expectation” ageing notion is then given, and various of its properties as well as an application to Poisson shock models are discussed

Dual stochastic orderings describing ageing properties of devices of unknown age

Motivated by applications in mathematical biology concerning randomly alternatingmotion ofmicro-organisms, we analyze a generalized integrated telegraph process. The random times between consecutive velocity reversals are gamma-distributed, and perform an alternating renewal process. We obtain the probability law and the mean of the process.

Random motion with gamma-distributed alternating velocities in biological modeling

We consider a "length-biased" shift-dependent information measure, related to the differential entropy in which higher weight is assigned to large values of observed random variables This allows us to introduce the notions of "weighted residual entropy" and "weighted past entropy", that are suitable to describe dynamic information of random lifetimes, in analogy with the entropies of residual and past lifetimes introduced in [9] and [6], respectively The obtained results include their behaviors under monotonic transformations

On Weighted Residual and Past Entropies

We consider a continuous-time Ehrenfest model defined over the integers from -N to N, and subject to catastrophes occurring at constant rate. The effect of each catastrophe instantaneously resets the process to state 0. We investigate both the transient and steady-state probabilities of the above model. Further, the first passage time through state 0 is discussed. We perform a jump-diffusion approximation of the above model, which leads to the Ornstein-Uhlenbeck process with catastrophes. The underlying jump-diffusion process is finally studied, with special attention to the symmetric case arising when the Ehrenfest model has equal upward and downward transition rates.

Antonio Di Crescenzo

Papers

On a jump-telegraph process driven by an alternating fractional Poisson process

Dual stochastic orderings describing ageing properties of devices of unknown age

Random motion with gamma-distributed alternating velocities in biological modeling

On Weighted Residual and Past Entropies

A continuous-time Ehrenfest model with catastrophes and its jump-diffusion approximation