Within the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson ...distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD’s) to allow both overdispersion and underdispersion. Similarly to Kemp’s generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD’s related to a generalization of Mittag–Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.
We construct admissible circulant Laplacian matrix functions as generators for strictly increasing random walks on the integer line. These Laplacian matrix functions refer to a certain class of ...Bernstein functions. The approach has connections with biased walks on digraphs. Within this framework, we introduce a space-time generalization of the Poisson process as a strictly increasing walk with discrete Mittag-Leffler jumps time-changed with an independent (continuous-time) fractional Poisson process. We call this process ‘space-time Mittag-Leffler process’. We derive explicit formulae for the state probabilities which solve a Cauchy problem with a Kolmogorov-Feller (forward) difference-differential equation of general fractional type. We analyze a “well-scaled” diffusion limit and obtain a Cauchy problem with a space-time convolution equation involving Mittag-Leffler densities. We deduce in this limit the ‘state density kernel’ solving this Cauchy problem. It turns out that the diffusion limit exhibits connections to Prabhakar general fractional calculus. We also analyze in this way a generalization of the space-time Mittag-Leffler process. The approach of constructing good Laplacian generator functions has a large potential in applications of space-time generalizations of the Poisson process and in the field of continuous-time random walks on digraphs.
On a fractional linear birth—death process ORSINGHER, ENZO; POLITO, FEDERICO
Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability,
02/2011, Letnik:
17, Številka:
1
Journal Article
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In this paper, we introduce and examine a fractional linear birth—death process N v (t), t > 0, whose fractionality is obtained by replacing the time derivative with a fractional derivative in the ...system of difference-differential equations governing the state probabilities $p_{k}^{v}(t)$ , t > 0, k ≥ 0. We present a subordination relationship connecting N v (t), t > 0, with the classical birth—death process N(t), t > 0, by means of the time process T 2v (t), t > 0, whose distribution is related to a time-fractional diffusion equation. We obtain explicit formulas for the extinction probability $p_{0}^{v}(t)$ and the state probabilities $p_{k}^{v}(t)$ , t > 0, k ≥ 1, in the three relevant cases λ > μ, λ < μ, λ = μ (where λ and μ are, respectively, the birth and death rates) and discuss their behaviour in specific situations. We highlight the connection of the fractional linear birth—death process with the fractional pure birth process. Finally, the mean values 𝔼N v (t) and 𭕍arN v (t) are derived and analyzed.
In a recent work we introduced a semi-Markovian discrete-time generalization of the telegraph process. We referred to this random walk as the ‘squirrel random walk’ (SRW). The SRW is a discrete-time ...random walk on the one-dimensional infinite lattice where the step direction is reversed at arrival times of a discrete-time renewal process and remains unchanged at uneventful time instants. We first recall general notions of the SRW. The main subject of the paper is the study of the SRW where the step direction switches at the arrival times of a generalization of the Sibuya discrete-time renewal process (GSP) which only recently appeared in the literature. The waiting time density of the GSP, the ‘generalized Sibuya distribution’ (GSD), is such that the moments are finite up to a certain order r≤m−1 (m≥1) and diverging for orders r≥m capturing all behaviors from broad to narrow and containing the standard Sibuya distribution as a special case (m=1). We also derive some new representations for the generating functions related to the GSD. We show that the generalized Sibuya SRW exhibits several regimes of anomalous diffusion depending on the lowest order m of diverging GSD moment. The generalized Sibuya SRW opens various new directions in anomalous physics.
We present a generalization of Hilfer derivatives in which Riemann–Liouville integrals are replaced by more general Prabhakar integrals. We analyze and discuss its properties. Furthermore, we show ...some applications of these generalized Hilfer–Prabhakar derivatives in classical equations of mathematical physics such as the heat and the free electron laser equations, and in difference–differential equations governing the dynamics of generalized renewal stochastic processes.
The space-fractional Poisson process Orsingher, Enzo; Polito, Federico
Statistics & probability letters,
April 2012, 2012-4-00, Letnik:
82, Številka:
4
Journal Article
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In this paper, we introduce the space-fractional Poisson process whose state probabilities pkα(t), t≥0, α∈(0,1, are governed by the equations (d/dt)pkα(t)=−λα(1−B)αpkα(t), where (1−B)α is the ...fractional difference operator found in the time series analysis. We explicitly obtain the distributions pkα(t), the probability generating functions Gα(u,t), which are also expressed as distributions of the minimum of i.i.d. uniform random variables. The comparison with the time-fractional Poisson process is investigated and finally, we arrive at the more general space–time-fractional Poisson process of which we give the explicit distribution.
•The class of regularly-varying random vectors is a possible choice to obtain input-output consistency.•The presence of dependency in the pooled input is necessary to obtain such consistency.•The ...proved consistency property regards the tail asymptotics of the ISIs probability distribution.
Interspike intervals describe the output of neurons. Signal transmission in a neuronal network implies that the output of some neurons becomes the input of others. The output should reproduce the main features of the input to avoid a distortion when it becomes the input of other neurons, that is input and output should exhibit some sort of consistency. In this paper, we consider the question: how should we mathematically characterize the input in order to get a consistent output? Here we interpret the consistency by requiring the reproducibility of the input tail behaviour of the interspike intervals distributions in the output. Our answer refers to a system of interconnected neurons with stochastic perfect integrate and fire units. In particular, we show that the class of regularly-varying vectors is a possible choice to obtain such consistency. Some further necessary technical hypotheses are added.
On some operators involving Hadamard derivatives Garra, Roberto; Polito, Federico
Integral transforms and special functions,
10/1/2013, 2013-10-00, 20131001, Letnik:
24, Številka:
10
Journal Article
Recenzirano
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In this paper, we introduce a novel Mittag-Leffler-type function and study its properties in relation to some integro-differential operators involving Hadamard fractional derivatives or ...hyper-Bessel-type operators. We discuss then the utility of these results to solve some integro-differential equations involving these operators by means of operational methods. We show the advantage of our approach through some examples. Among these, an application to a modified Lamb-Bateman integral equation is presented.
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