On certain self-decomposable distributions

Some probabilistic aspects of the number variance statistic are investigated. Infinite systems of independent Brownian motions and symmetric α-stable processes are used to construct new examples of processes which exhibit both divergent and saturating number variance behaviour. We derive a general expression for the number variance for the spatial particle configurations arising from these systems and this enables us to deduce various limiting distribution results for the fluctuations of the associated counting functions. In particular, knowledge of the number variance allows us to introduce and characterize a novel family of centered, long memory Gaussian processes. We obtain fractional Brownian motion as a weak limit of these constructed processes.2000 MSC Primary 60G52, 60G15; Secondary 60F17, 15A52.

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“…By [22], if X is a symmetric α-stable random variable with 0 < α ≤ 2 and scale σ, then for −1 < δ < α we have…”

Section: The Brownian and Symmetric α-Stable Casesmentioning

confidence: 99%

Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes.

Hambly

Jones

2007

Electron. J. Probab.

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“…with obvious notation and hypothesis; whereas (5), which has been discussed in particular in Shanbhag and Sreehari [15,16], exhibits the self-decomposability of log e. More precisely, one has the identity log e L = α log e + α log 1/τ α ,…”

Section: Examplesmentioning

confidence: 99%

On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable

Bertoin

Yor

2001

Electron. Commun. Probab.

103

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“…As an illustration, the particular case of α = 0.9756 and the corresponding t * α = 0.1 is marked on the graph. Figure 2(b) displays the values of φ(t) as were determined by (12), with α = 0.9756 and its minimum value at t * α = 0.1. …”

Section: Example 1 the Gamma Distribution With Shape Parameter Tmentioning

confidence: 99%

A Numerical Study of Small Parameter Behavior of Some Families of Distributions

Bar‐Lev

Boukai

Kleiner

2016

Communications in Statistics - Simulation and Computation

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Limiting distributions play an important role in approximating the exact distributions, especially when they have a rather cumbersome analytic form, or simply when they do not have a closed from. The question that naturally arises is how good the approximation is. In this paper we propose a procedure for the numerical assessment of the 'goodness' of some easyto-calculate limiting distributions, originally proposed in Bar-Lev and Enis (1987), in various cases of the underlying distributions, some of which are inherently computationally challenging. The details of the procedure are provided in three examples. The first example, deals with the gamma distributions; the second deals with Bessel distributions related to a symmetric random walk, and the third example deals with positive stable distributions. The details of two additional variations of these examples are also discussed. These examples illustrate the ease with which the limiting approximations could be applied in the various cases, well-demonstrating their computational simplicity and attractiveness.

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On certain self-decomposable distributions

Cited by 84 publications

References 5 publications

Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes.

Number variance from a probabilistic perspective: infinite systems of independent Brownian motions and symmetric alpha stable processes.

On Subordinators, Self-Similar Markov Processes and Some Factorizations of the Exponential Variable

A Numerical Study of Small Parameter Behavior of Some Families of Distributions

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