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

Fractional equations governing the distribution of reflecting drifted Brownian motions are presented. The equations are expressed in terms of tempered Riemann--Liouville type derivatives. For these operators a Marchaud-type form is obtained and a Riesz tempered fractional derivative is examined, together with its Fourier transform.

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Drifted Brownian motions governed by fractional tempered derivatives

Semi-Markov processes are a generalization of Markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution. This paper provides an integro-differential form of the Kolmogorov's backward equations for a large class of homogeneous semi-Markov processes, having the form of a Volterra integro-differential equation. An equivalent evolutionary (differential) form of the equations is also provided. Weak limits of semi-Markov processes are also considered and their corresponding integro-differential Kolmogorov's equations are identified.

On semi-Markov processes and their Kolmogorov's integro-differential equations

We study the scaling limit of random fields which are the solutions of a nonlinear partial differential equation known as the Burgers equation, under stochastic initial condition. These are assumed to be a Gaussian process with long-range dependence. We present some results on the rate of convergence to the normal law.

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On the rate of convergence to the normal law for solutions of the burgers equation with singular initial data

We consider here point processes $N^f(t)$, $t>0$, with independent increments and integer-valued jumps whose distribution is expressed in terms of Bern\v{s}tein functions $f$ with L\'evy measure $\nu$. We obtain the general expression of the probability generating functions $G^f$ of $N^f$, the equations governing the state probabilities $p_k^f$ of $N^f$, and their corresponding explicit forms. We also give the distribution of the first-passage times $T_k^f$ of $N^f$, and the related governing equation. We study in detail the cases of the fractional Poisson process, the relativistic Poisson process and the Gamma Poisson process whose state probabilities have the form of a negative binomial. The distribution of the times $\tau_j^{l_j}$ of jumps with height $l_j$ ($\sum_{j=1}^rl_j = k$) under the condition $N(t) = k$ for all these special processes is investigated in detail.

Counting processes with Bern\v{s}tein intertimes and random jumps

In this paper a random motion on the surface of the 3-sphere whose probability law is a solution of the telegraph equation in spherical coordinates is presented. The connection of equations governing the random motion with Maxwell equations is examined together with some qualitative features of its sample paths. Finally Brownian motion on the 3-sphere is derived as the limiting process of a random walk with latitude-changing probabilities.

Enzo Orsingher

Papers

Drifted Brownian motions governed by fractional tempered derivatives

On semi-Markov processes and their Kolmogorov's integro-differential equations

On the rate of convergence to the normal law for solutions of the burgers equation with singular initial data

Counting processes with Bern\v{s}tein intertimes and random jumps

Stochastic motions on the 3-sphere governed by wave and heat equations