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

Convergence of Probability Measures

Thank you very much for downloading table of integrals series and products. Maybe you have knowledge that, people have look hundreds times for their chosen books like this table of integrals series and products, but end up in harmful downloads. Rather than reading a good book with a cup of coffee in the afternoon, instead they cope with some harmful virus inside their laptop. table of integrals series and products is available in our book collection an online access to it is set as public so you can get it instantly. Our book servers saves in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Merely said, the table of integrals series and products is universally compatible with any devices to read.

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

In this paper we consider the relation between random sums and compositions of different processes. In particular, for independent Poisson processes $N_\alpha(t)$, $N_\beta(t)$, $t>0$, we show that $N_\alpha(N_\beta(t)) \overset{\text{d}}{=} \sum_{j=1}^{N_\beta(t)} X_j$, where the $X_j$s are Poisson random variables. We present a series of similar cases, the most general of which is the one in which the outer process is Poisson and the inner one is a nonlinear fractional birth process. We highlight generalisations of these results where the external process is infinitely divisible. A section of the paper concerns compositions of the form $N_\alpha(\tau_k^\nu)$, $\nu \in (0,1]$, where $\tau_k^\nu$ is the inverse of the fractional Poisson process, and we show how these compositions can be represented as random sums. Furthermore we study compositions of the form $\Theta(N(t))$, $t>0$, which can be represented as random products. The last section is devoted to studying continued fractions of Cauchy random variables with a Poisson number of levels. We evaluate the exact distribution and derive the scale parameter in terms of ratios of Fibonacci numbers.

Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

We consider random flights in $$\mathbb {R}^d$$
 reflecting on the surface of a sphere $$\mathbb {S}^{d-1}_R,$$
 with center at the origin and with radius R,  where reflection is performed by means of circular inversion. Random flights studied in this paper are motions where the orientation of the deviations are uniformly distributed on the unit-radius sphere $$\mathbb {S}^{d-1}_1$$
 . We obtain the explicit probability distributions of the position of the moving particle when the number of changes of direction is fixed and equal to $$n\ge 1$$
 . We show that these distributions involve functions which are solutions of the Euler–Poisson–Darboux equation. The unconditional probability distributions of the reflecting random flights are obtained by suitably randomizing n by means of a fractional-type Poisson process. Random flights reflecting on hyperplanes according to the optical reflection form are considered and the related distributional properties derived.

Reflecting Random Flights

In this paper, we study persistent piecewise linear multidimensional random motions. Their velocities, switching at Poisson times, are uniformly distributed on a sphere. The changes of direction are accompanied with subsequent jumps of random length and of uniformly distributed orientation.

In this paper, we obtain some useful properties and formulae of distributions of these processes. In particular, we get these distributions in the cases of jumps with Gaussian and exponential distributions of jump magnitudes.

Planar piecewise linear random motions with jumps

How the sojourn time distributions of Brownian motion are affected by different forms of conditioning

In this paper we introduce new distributions which are solutions of higher-order Laplace equations. It is proved that their densities can be obtained by folding and symmetrizing Cauchy distributions. Another class of probability laws related to higher-order Laplace equations is obtained by composing pseudo-processes with positively skewed stable distributions which produce asymmetric Cauchy densities in the odd-order case. Special attention is devoted to the third-order Laplace equation where the connection between the Cauchy distribution and the Airy functions is obtained and analyzed.

https://www.dss.uniroma1.it/it/system/files/pubblicazioni/28-RT_3_2012_orsingher.pdf

Enzo Orsingher

Papers

Compositions, Random Sums and Continued Random Fractions of Poisson and Fractional Poisson Processes

Reflecting Random Flights

Planar piecewise linear random motions with jumps

How the sojourn time distributions of Brownian motion are affected by different forms of conditioning

Higher-Order Laplace Equations and Hyper-Cauchy Distributions