Stochastic processes relating to particles distributed in a continuous infinity of states

1950 ◽  
Vol 46 (4) ◽  
pp. 595-602 ◽  
Author(s):  
Alladi Ramakrishnan
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Stochastic Processes ◽  
Stochastic Variable ◽  
Stochastic Problems ◽  
Image Position ◽  
Number Of Particles

Many stochastic problems arise in physics where we have to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and this distribution varies with another parameter t (which may be continuous or discrete; if t represents time or thickness it is of course continuous). This variation occurs because of transitions characteristic of the stochastic process under consideration. If the E-space were discrete and the states represented by E1, E2, …, then it would be possible to define a functionrepresenting the probability that there are ν1 particles in E1, ν2 particles in E2, …, at t. The variation of π with t is governed by the transitions defined for the process; ν1, ν2, … are thus stochastic variables, and it is possible to study the moments or the distribution function of the sum of such stochastic variableswith the help of the π function which yields also the correlation between the stochastic variables νi.

scholarly journals A Note on the Most “Dangerous” and Skewest Class of Distributions

1963 ◽  
Vol 2 (3) ◽  
pp. 387-390 ◽  
Author(s):  
Gunnar Benktander
Keyword(s):  
Distribution Function ◽  
Theoretical Models ◽  
Stochastic Variable ◽  
Classical Definition ◽  
Image Position ◽  
Third Moment

In the classical definition skewness is departure from symmetry. It was therefore natural to measure skewness by using a normalized third moment μ3/σ3. This condensed measure, however, is not refined enough to be used as an operational instrument for studying various functions which might be used to describe actual claim distributions. This is true especially when the interest is concentrated towards the higher values of the variate.In their paper (1) Benktander-Segerdahl have suggested that the average excess claim m(x) as a function of the priority x should be used to reveal the characteristics of the tail of the distribution where P(x) = 1 — H(x) denotes the distribution function.This statistic is very apt when comparing actual claim distributions with possible theoretical models. It is also useful when classifying these models.If, however, emphasis mainly is laid on classifying distributions according to their skewness, another statistic might be preferable. Let μ(x)dx denote the probability that a stochastic variable which is known to be at least equal to x, does not exceed x+dx. In other words, μ(x)dx represents the probability that a claim or the corresponding stochastic variable, which, when observed from the bottom, is “alive” at x, “dies” in the interval (x, x + dx). The lower this claims rate of mortality, the skewer and more dangerous is the claim distribution.

scholarly journals A Double-indexed Functional Hill Process and Applications

2016 ◽  
Vol 8 (4) ◽  
pp. 144
Author(s):  
Modou Ngom ◽  
Gane Samb Lo
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Asymptotic Behaviour ◽  
Stochastic Processes ◽  
Order Statistics ◽  
Finite Dimension ◽  
Domain Of Attraction ◽  
Extreme Value ◽  
Extreme Value Index

<div>Let $X_{1,n} \leq .... \leq X_{n,n}$ be the order statistics associated with a sample $X_{1}, ...., X_{n}$ whose pertaining distribution function (\textit{df}) is $F$. We are concerned with the functional asymptotic behaviour of the sequence of stochastic processes</div><div> </div><div>\begin{equation}<br />T_{n}(f,s)=\sum_{j=1}^{j=k}f(j)\left( \log X_{n-j+1,n}-\log<br />X_{n-j,n}\right)^{s} ,  \label{fme}<br />\end{equation}</div><div> </div><div>indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N}%^{\ast}\longmapsto \mathbb{R}_{+}$ and $s \in ]0,+\infty[$ and where $k=k(n)$ satisfies</div><div> </div><div>\begin{equation*}<br />1\leq k\leq n,k/n\rightarrow 0\text{ as }n\rightarrow \infty .<br />\end{equation*}</div><div> </div><div>We show that this is a stochastic process whose margins generate estimators of the extreme value index when $F$ is in the extreme domain of attraction. We focus in this paper on its finite-dimension asymptotic law and provide a class of new estimators of the extreme value index whose performances are compared to analogous ones. The results are next particularized for one explicit class $\mathcal{F}$.</div>

On a three-state sojourn time problem

1971 ◽  
Vol 8 (04) ◽  
pp. 716-723 ◽  
Author(s):  
A. E. Gibson ◽  
B. W. Conolly
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Sojourn Time ◽  
Random Variable ◽  
The Real ◽  
Time Problem ◽  
Arbitrary Space ◽  
Image Position

Consider the real-valued stochastic process {S(t), 0 ≦ t &lt; ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT is a random variable having a differentiable distribution function which is independent of the time t 0 at which the visit to state T begins.

A note on Janossy's mathematical model of a nucleon cascade

1952 ◽  
Vol 48 (3) ◽  
pp. 451-456 ◽  
Author(s):  
Alladi Ramakrishnan
Keyword(s):  
Mathematical Model ◽  
Specific Problem ◽  
Close Relation ◽  
Diffusion Equations ◽  
Density Functions ◽  
Stochastic Variable ◽  
Product Density ◽  
Stochastic Problems ◽  
Product Densities ◽  
Number Of Particles

In a contribution (3) to these Proceedings the author considered stochastic problems in physics where one had to deal with a stochastic variable representing the number of particles distributed in a continuous infinity of states characterized by a parameter E, and where the distribution varied with another parameter t which might be continuous or discrete (if t represents time or thickness, it is of course continuous). The author introduced the concept of product densities and derived some general results relating to the functions representing these densities. Recently, Janossy(2), using a certain mathematical model for a nuclear cascade, introduced certain functions which bear a close relation to the product-density functions. The object of this note is to establish a complete correspondence between these two sets of functions and apply them to the specific problem of the development of a nucleon cascade. The diffusion equations involving product densities can be derived from the diffusion equations involving Janossy's functions.

On a three-state sojourn time problem

1971 ◽  
Vol 8 (4) ◽  
pp. 716-723 ◽  
Author(s):  
A. E. Gibson ◽  
B. W. Conolly
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Sojourn Time ◽  
Random Variable ◽  
The Real ◽  
Time Problem ◽  
Arbitrary Space ◽  
Image Position

Consider the real-valued stochastic process {S(t), 0 ≦ t < ∞} which assumes values in an arbitrary space X. For a given subset T ⊂ X we define which represents the length in time of a visit to state T. We shall restrict ourselves to processes such that τT is a random variable having a differentiable distribution function which is independent of the time t0 at which the visit to state T begins.

scholarly journals On renewal theory, counter problems, and quasi-Poisson processes

1957 ◽  
Vol 53 (1) ◽  
pp. 175-193 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Stochastic Process ◽  
Distribution Function ◽  
Renewal Theory ◽  
Special Property ◽  
Poisson Processes ◽  
Partial Sums ◽  
Renewal Processes ◽  
Negative Exponential Distribution ◽  
Negative Exponential ◽  
Image Position

The power and appropriateness of renewal theory as a tool for the solution of general problems concerning counters has been amply demonstrated by Feller (7), who considered a variety of counter problems and reduced them to special renewal processes. The use of what may be called renewal-type arguments had certainly been made by authors other than Feller (e.g. in § 3 of Domb (3)), but it was only in (7) that the simplicity of the renewal approach to counter problems was recognized and systematically applied. More recently, Hammersley (8) was concerned with the generalization of a counter problem previously studied by Domb (2). This problem may be introduced, mathematically, as follows. Let {xi}, {yi} be two independent sequences of independent non-negative random variables which are non-zero with probability one (i.e. two independent renewal processes). The {xi}, are distributed in a negative-exponential distribution with mean λ-1, and we write Eλ for their distribution function and say ≡ {xi} is a Poisson process to imply this special property of ; the {yi} have a distribution function ‡ B(x) with mean b1 ≤ ∞. Form the partial sums and define ni to be the greatest integer k such that Xk ≥ t, taking X0 0 and nt = 0 if x1 > t. Then define the stochastic processHammersley'sx counter problem concerns the stochastic process

A computation of the Littlewood exponent of stochastic processes

1988 ◽  
Vol 103 (2) ◽  
pp. 367-370 ◽  
Author(s):  
Ron C. Blei ◽  
J.-P. Kahane
Keyword(s):  
Stochastic Process ◽  
Stochastic Processes ◽  
Total Variation ◽  
Probability Space ◽  
Complex Measure ◽  
Bona Fide ◽  
Image Position

A stochastic process X = {X(t): t ∈ [0, 1]} on a probability space (Ω, , ℙ) is said to have finite expectation if the function defined on the measureable rectangles in Ω × [0, 1] byfor A ∈ and (s, t) ⊂ [0, 1] gives rise to a complex measure in each of its two coordinates (see [1], definition 1·1). Equivalently, X has finite expectation ifis finite. The function defined by (1), effectively a generalization of the Doléans measure (see e.g. [4] pp. 33–35), is extendible to a bona fide complex measure on Ω × [0, 1] if and only if its ‘total variation’

Kinetic theory

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Distribution Function ◽  
Kinetic Theory ◽  
Liouville Equation ◽  
Vlasov Equation ◽  
Particle Distribution ◽  
Equations Of Motion ◽  
Poisson Brackets ◽  
Hamiltonian Form ◽  
First Order ◽  
Number Of Particles

This chapter covers the equations governing the evolution of particle distribution and relates the macroscopic thermodynamical quantities to the distribution function. The motion of N particles is governed by 6N equations of motion of first order in time, written in either Hamiltonian form or in terms of Poisson brackets. Thus, as this chapter shows, as the number of particles grows it becomes necessary to resort to a statistical description. The chapter first introduces the Liouville equation, which states the conservation of the probability density, before turning to the Boltzmann–Vlasov equation. Finally, it discusses the Jeans equations, which are the equations obtained by taking various averages over velocities.

scholarly journals Stochastic Processes with a Particular Type of Variograms

2007 ◽  
Vol 2007 ◽  
pp. 1-5 ◽  
Author(s):  
Chunsheng Ma
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
Stochastic Processes ◽  
Series Expansion ◽  
Real Line ◽  
Random Fields ◽  
The Other ◽  
Simple Construction ◽  
The Real ◽  
Line Form

This paper is concerned with a class of stochastic processes or random fields with second-order increments, whose variograms have a particular form, among which stochastic processes having orthogonal increments on the real line form an important subclass. A natural issue, how big this subclass is, has not been explicitly addressed in the literature. As a solution, this paper characterizes a stochastic process having orthogonal increments on the real line in terms of its variogram or its construction. Our findings are a little bit surprising: this subclass is big in terms of the variogram, and on the other hand, it is relatively “small” according to a simple construction. In particular, every such process with Gaussian increments can be simply constructed from Brownian motion. Using the characterizations we obtain a series expansion of the stochastic process with orthogonal increments.

Central Limit Theorems for Interchangeable Processes

1958 ◽  
Vol 10 ◽  
pp. 222-229 ◽  
Author(s):  
J. R. Blum ◽  
H. Chernoff ◽  
M. Rosenblatt ◽  
H. Teicher
Keyword(s):  
Stochastic Process ◽  
Probability Measure ◽  
Central Limit ◽  
Joint Distribution ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Probability Measures ◽  
Image Position ◽  
Positive Integers

Let {Xn} (n = 1, 2 , …) be a stochastic process. The random variables comprising it or the process itself will be said to be interchangeable if, for any choice of distinct positive integers i 1, i 2, H 3 … , ik, the joint distribution of depends merely on k and is independent of the integers i 1, i 2, … , i k. It was shown by De Finetti (3) that the probability measure for any interchangeable process is a mixture of probability measures of processes each consisting of independent and identically distributed random variables.

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