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Table Of Integrals Series And Products

Tables of integrals

(1987). Applied Probability and Queues. Journal of the Operational Research Society: Vol. 38, No. 11, pp. 1095-1096.

https://link.springer.com/content/pdf/10.1057%2Fjors.1987.184.pdf

Applied Probability and Queues

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An Introduction To The Theory Of Point Processes

Calderas are collapse structures related to the emptying of magmatic reservoirs, often associated with large eruptions from long-lived magmatic systems. Understanding how magma is transferred from a magma reservoir to the surface before eruptions is a major challenge. Here we exploit the historical, archaeological and geological record of Campi Flegrei caldera to estimate the surface deformation preceding the Monte Nuovo eruption and investigate the shallow magma transfer. Our data suggest a progressive magma accumulation from ~1251 to 1536 in a 4.6 ± 0.9 km deep source below the caldera centre, and its transfer, between 1536 and 1538, to a 3.8 ± 0.6 km deep magmatic source ~4 km NW of the caldera centre, below Monte Nuovo; this peripheral source fed the eruption through a shallower source, 0.4 ± 0.3 km deep. This is the first reconstruction of pre-eruptive magma transfer at Campi Flegrei and corroborates the existence of a stationary oblate source, below the caldera centre, that has been feeding lateral eruptions for the last ~5 ka. Our results suggest: 1) repeated emplacement of magma through intrusions below the caldera centre; 2) occasional lateral transfer of magma feeding non-central eruptions within the caldera. Comparison with historical unrest at calderas worldwide suggests that this behavior is common.

Magma transfer at Campi Flegrei caldera (Italy) before the 1538 AD eruption

We analyze the one-dimensional telegraph random process confined by two boundaries, 0 and $H>0$. The process experiences hard reflection at the boundaries (with random switching to full absorption). Namely, when the process hits the origin (the threshold $H$) it is either absorbed, with probability $\alpha$, or reflected upwards (downwards), with probability $1-\alpha$, for $0<\alpha<1$. We provide various results on the expected values of the renewal cycles and of the absorption time. The adopted approach is based on the analysis of the first-crossing times of a suitable compound Poisson process through linear boundaries. Our analysis includes also some comparisons between suitable stopping times of the considered telegraph process and of the corresponding diffusion process obtained under the classical Kac's scaling conditions.

Some results on the telegraph process confined by two non-standard boundaries

A review of a stochastic neuronal model with jumps driven by Poisson process

A classical assumption in the theory of degradation models is that the quality of a device is described by a non-negative diffusion process {X(t), t ≥ 0} starting at X(0) = x0 > 0. Assuming that large values of X(t) correspond to a better quality, the level X(t) = 0 stands for zero quality, i.e. for the lack of functionality of the device. Consequently, the first-passage time T of {X(t)} through state zero gives the failure time of the device. We shall denote its probability density function (p.d.f.) by

Some results on degradation processes with repairs and catastrophes

For nonnegative random variables with finite means we introduce an analogous of the equilibrium residual-lifetime distribution based on the quantile function. This allows to construct new distributions with support (0,1), and to obtain a new quantile-based version of the probabilistic generalization of Taylor's theorem. Similarly, for pairs of stochastically ordered random variables we come to a new quantile-based form of the probabilistic mean value theorem. The latter involves a distribution that generalizes the Lorenz curve. We investigate the special case of proportional quantile functions and apply the given results to various models based on classes of distributions and measures of risk theory. Motivated by some stochastic comparisons, we also introduce the `expected reversed proportional shortfall order', and a new characterization of random lifetimes involving the reversed hazard rate function.

A quantile-based probabilistic mean value theorem

We consider an extended birth-death-immigration process defined on a lattice formed by the integers of $d$ semiaxes joined at the origin. When the process reaches the origin, then it may jumps toward any semiaxis with the same rate. The dynamics on each ray evolves according to a one-dimensional linear birth-death process with immigration. We investigate the transient and asymptotic behavior of the process via its probability generating function. The stationary distribution, when existing, is a zero-modified negative binomial distribution. We also study a diffusive approximation of the process, which involves a diffusion process with linear drift and infinitesimal variance on each ray. It possesses a gamma-type transient density admitting a stationary limit. As a byproduct of our study, we obtain a closed form of the number of permutations with a fixed number of components, and a new series form of the polylogarithm function expressed in terms of the Gauss hypergeometric function.

Barbara Martinucci

Papers

Some results on the telegraph process confined by two non-standard boundaries

A review of a stochastic neuronal model with jumps driven by Poisson process

Some results on degradation processes with repairs and catastrophes

A quantile-based probabilistic mean value theorem

A linear birth-death process on a star graph and its diffusion approximation