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

Integral transforms (Laplace, Fourier and Mellin) are introduced with their properties, the relationship between these transforms was discussed and some important applications in physics and engineering were given. ااااااا دقل مت ضارعتسإ ةساردو ل ةيلماكتلا تليوحتلا لك ، سلبل تلوحت نم روف ي ر نيليمو عم ةشقانم كلذكو ،اهنم لك صاوخ و صئاصخ ةقلعلا ةشقانم مت هذه نيب طبرلاو و ،تليوحتلا مت ميدقت تاقيبطتلا ضعب تليوحتلا هذهل ةمهملا يف تلاجم ءايزيفلا ةسدنهلاو.

Integral transforms and their applications

The fractional Poisson process is a renewal process with Mittag-Leffler waiting times. Its distributions solve
a time-fractional analogue of the Kolmogorov forward equation for a Poisson process. This paper shows that a
traditional Poisson process, with the time variable replaced by an independent inverse stable subordinator, is also a
 fractional Poisson process. This result unifies the two main approaches in the stochastic theory of time-fractional
 diffusion equations. The equivalence extends to a broad class of renewal processes that include models for tempered
 fractional diffusion, and distributed-order (e.g., ultraslow) fractional diffusion. The paper also {discusses the relation between} the fractional Poisson process and Brownian time.

The Fractional Poisson Process and the Inverse Stable Subordinator

The Sample Analysis at Mars (SAM) instrument on board the Mars Science Laboratory Curiosity rover is designed to conduct inorganic and organic chemical analyses of the atmosphere and the surface regolith and rocks to help evaluate the past and present habitability potential of Mars at Gale Crater. Central to this task is the development of an inventory of any organic molecules present to elucidate processes associated with their origin, diagenesis, concentration, and long-term preservation. This will guide the future search for biosignatures. Here we report the definitive identification of chlorobenzene (150–300 parts per billion by weight (ppbw)) and C2 to C4 dichloroalkanes (up to 70 ppbw) with the SAM gas chromatograph mass spectrometer (GCMS) and detection of chlorobenzene in the direct evolved gas analysis (EGA) mode, in multiple portions of the fines from the Cumberland drill hole in the Sheepbed mudstone at Yellowknife Bay. When combined with GCMS and EGA data from multiple scooped and drilled samples, blank runs, and supporting laboratory analog studies, the elevated levels of chlorobenzene and the dichloroalkanes cannot be solely explained by instrument background sources known to be present in SAM. We conclude that these chlorinated hydrocarbons are the reaction products of Martian chlorine and organic carbon derived from Martian sources (e.g., igneous, hydrothermal, atmospheric, or biological) or exogenous sources such as meteorites, comets, or interplanetary dust particles.

Organic molecules in the Sheepbed Mudstone, Gale Crater, Mars

In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson proce...

/pdf/tempered-fractional-poisson-processes-and-fractional-2rz18gie5z.pdf

Tempered fractional Poisson processes and fractional equations with Z-transform

In this article, we introduce the Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular, we discuss the space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Skellam process by independent stable subordinator and tempered stable subordinator, respectively. We derive the marginal probabilities, Levy measures, governing difference-differential equations of the introduced processes. Our results generalize the Skellam process and running average of Poisson process in several directions.

https://www.mdpi.com/1099-4300/22/11/1193/pdf

Skellam type processes of order K and beyond

In this article, we introduce mixtures of tempered stable subordinators (TSS). These mixtures define a class of subordinators which generalize tempered stable subordinators. The main properties like probability density function (pdf), Levy density, moments, governing Fokker-Planck-Kolmogorov (FPK) type equations, asymptotic form of potential density and asymptotic form of the renewal function for the corresponding inverse subordinator are discussed. We also generalize these results to n-th order mixtures of TSS.

Mixtures of Tempered Stable Subordinators

In this article, we derive the state probabilities of different type of space- and time-fractional Poisson processes using z-transform. We work on tempered versions of time-fractional Poisson process and space-fractional Poisson processes. We also introduce Gegenbauer type fractional differential equations and their solutions using z-transform. Our results generalize and complement the result available on fractional Poisson processes in several directions.

Tempered Fractional Poisson Processes and Fractional Equations with Z-Transform

\n Daily precipitation extremes are crucial in the hydrological design of major water control structures and are expected to show a changing tendency over time due to climate change. The magnitude and frequency of extreme precipitation can be assessed by studying the upper tail behavior of probability distributions of daily precipitation. Depending on the tail behavior, the distributions can be classified into two categories: heavy-tailed and light-tailed distributions. Heavier tails indicate more frequent occurrences of extreme precipitation events. In this paper, we have analyzed the temporal change in the tail behavior of daily precipitation over India from pre- to post-1970 time periods as per the global climatic shift. A modified Probability Ratio Mean Square Error norm is used to identify the best-fit distribution to the tails of daily precipitation among four theoretical distributions (e.g., Pareto-type II, Lognormal, Weibull, and Gamma distributions). The results indicate that the Lognormal distribution, which is a heavy-tailed distribution, fits the tails of daily precipitation for the majority of the grids. It is inferred from the study that there is an increase in the heaviness of tails of daily precipitation data over India from pre- to post-1970 time periods.

/pdf/assessment-of-temporal-change-in-the-tails-of-probability-39feumhnpb.pdf

Neha Gupta

Papers

Tempered fractional Poisson processes and fractional equations with Z-transform

Skellam type processes of order K and beyond

Mixtures of Tempered Stable Subordinators

Tempered Fractional Poisson Processes and Fractional Equations with Z-Transform

Assessment of temporal change in the tails of probability distribution of daily precipitation over India due to climatic shift in the 1970s