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 introduce Skellam process of order k and its running average. We also discuss the time-changed Skellam process of order k. In particular we discuss space-fractional Skellam process and tempered space-fractional Skellam process via time changes in Poisson 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 Skellam process and running average of Poisson process in several directions.

Skellam Type Processes of Order K and Beyond

Reliable estimation of extreme precipitation is of utmost importance to ensure the structural safety of major civil engineering infrastructures (e.g., urban storm drainage network, spillways of major water control structures, etc.). Conventional practice for design of those infrastructures includes determination of magnitude and frequency of extreme precipitation events. The extreme events usually lie in the tail part of the probability distribution of daily precipitation data. Based on the nature of the tails, different distributions are classified into two categories: heavy-tailed and light-tailed distributions. The tails of heavy-tailed distributions tend to approach zero less rapidly when compared to those of light-tailed distribution (which are having exponential tails). Heavier tails imply more frequency of extreme precipitation events as compared to lighter tails. There is a dearth of attempts to assess the tail behavior of probability distributions of daily precipitation data over India. In this paper, we compare the tails of empirical distributions of daily precipitation data and tails of fitted theoretical distributions (e.g., pareto, lognormal, Weibull and gamma distributions). Gridded precipitation data prepared by the India Meteorological Department (IMD) having a resolution of 0.25° was used for the analysis. The results indicated that heavy-tailed distributions describe the observed precipitation extremes more effectively than the light-tailed distributions. This result shows that the light-tailed distributions, which are widely adopted for the determination of extreme precipitation events, are inadequate to capture the tail behavior of precipitation data accurately. This study reveals the importance of considering heavy-tailed distributions for reliable estimation of extreme precipitation events required for the design of major civil engineering infrastructure. The results from the study can find use in the design of urban storm drainage network, spillways of major water control structures.

Assessment of Tail Behavior of Probability Distributions of Daily Precipitation Data Over India

The Lower Santa Cruz River Basin Study (LSCRB Study) is a collaborative effort of regional and statewide water management stakeholders working with the US Bureau of Reclamation under the auspices of the 2009 SECURE Water Act. The impacts of climate change, land use, and population growth on projected water supply in the LSCRB were evaluated to (1) identify projected water supply and demand imbalances and (2) develop adaptation strategies to proactively respond over the next 40 years. A multi-step hydroclimate modeling and risk assessment process was conducted to assess a range of futures in terms of temperature, precipitation, runoff, soil moisture, and evapotranspiration, with a particular focus on implications for ecosystem health. Key hydroclimate modeling process decisions were informed by ongoing multi-stakeholder engagement. To incorporate the region’s highly variable precipitation pattern, the study used a numerical “weather generator” to develop ensembles of precipitation and temperature time series for input to surface hydrology modeling efforts. Hydroclimate modeling outcomes consistently included increasing temperatures, and generated information related to precipitation responses (season length and timing, precipitation amount) considered useful for evaluating potential ecosystem impacts. A range of risks was identified using the hydroclimate modeling outputs that allowed for development of potential adaptation strategies.

https://www.mdpi.com/2073-4441/15/10/1884/pdf?version=1684230478

Stakeholder-Informed Hydroclimate Scenario Modeling in the Lower Santa Cruz River Basin for Water Resource Management

In this article, the infinite series form of the probability densities of tempered stable and inverse tempered stable subordinators are obtained using Mellin transform. Further, the densities of the products and quotients of stable and inverse stable subordinators are worked out. The asymptotic behaviours of these densities are obtained as $x \rightarrow 0^+$. Similar results for tempered and inverse tempered stable subordinators are discussed. Our results provide alternative methods to find the densities of these subordinators and complement the results available in literature.

Neha Gupta

Papers

Skellam Type Processes of Order K and Beyond

Assessment of Tail Behavior of Probability Distributions of Daily Precipitation Data Over India

Stakeholder-Informed Hydroclimate Scenario Modeling in the Lower Santa Cruz River Basin for Water Resource Management

Densities of Inverse Tempered Stable Subordinators and Related Processes With Mellin Transforrm

Investigating the tail behaviour and associated risk with daily discharges in South Indian Rivers