Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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Table Of Integrals Series And Products

Fractional Differential Equations

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

A branching process of particles moving at finite velocity over the geodesic lines of the hyperbolic space (Poincare half-plane and Poincare disk) is examined. Each particle can split into two particles only once at Poisson spaced times and deviates orthogonally when splitted. At time t, after N(t) Poisson events, there are N(t)+1 particles moving along different geodesic lines. We are able to obtain the exact expression of the mean hyperbolic distance of the center of mass of the cloud of particles. We derive such mean hyperbolic distance from two different and independent ways and we study the behavior of the relevant expression as t increases and for different values of the parameters c (hyperbolic velocity of motion) and λ (rate of reproduction). The mean hyperbolic distance of each moving particle is also examined and a useful representation, as the distance of a randomly stopped particle moving over the main geodesic line, is presented.

Cascades of Particles Moving at Finite Velocity in Hyperbolic Spaces

This paper deals with shot noise models driven by various point processes. By means of an approximating step response function the distribution of the Poisson shot noise is obtained. The exponential response function case is studied as well, writing down explicit univariate and bivariate distributions. In this case also the crossing problem is analyzed and upcrossing and downcrossing rates are evaluated The last part of the paper is concerned with bivariate shot noise fields in circular regions of the plane, both at inner and boundary points.

Probability Distributions and Level Crossings of Shot Noise Models

We present and analyze the nonlinear classical pure birth process &#x1d4a9;(t), t > 0, and the fractional pure birth process &#x1d4a9;ν(t), t > 0, subordinated to various random times. We derive the state probability distribution , k ≥ 1 and, in some cases, we also present the corresponding governing differential equation. Various types of compositions of the fractional pure birth process &#x1d4a9;ν(t) have been examined in the second part of the paper. In particular, the processes &#x1d4a9;ν(T t ), &#x1d4a9;ν(&#x1d4ae;α(t)), &#x1d4a9;ν(T 2ν(t)), have been analyzed, where T 2ν(t), t > 0, is a process related to fractional diffusion equations. As a byproduct of our analysis, some formulae relating Mittag–Leffler functions are obtained.

Randomly Stopped Nonlinear Fractional Birth Processes

We consider the stochastic process describing the position of a particle whose velocity changes in sign and magnitude at the occurrences of two independent Poisson processes We decompose the probability law of the process into n components, which jointly yield a solution of a system of n telegraph equations. In the particular case n = 2, the fourth-order equation governing the probability law is presented and its explicit expression is obtained when the Poisson rates are suitably connected

Enzo Orsingher

Papers

Cascades of Particles Moving at Finite Velocity in Hyperbolic Spaces

Probability Distributions and Level Crossings of Shot Noise Models

Randomly Stopped Nonlinear Fractional Birth Processes

On a 2n-valued telegraph signal and the related integrated process

Fractional spherical random fields