On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

1979 ◽  
Vol 11 (4) ◽  
pp. 750-783 ◽  
Author(s):  
Wim Vervaat
Keyword(s):  
Difference Equation ◽  
Poisson Process ◽  
Existence Of Solutions ◽  
Borel Function ◽  
Random Variables ◽  
Random Variable ◽  
Stochastic Difference Equation ◽  
Infinitely Divisible ◽  
Uniform Random Variable

The present paper considers the stochastic difference equation Yn = AnYn-1 + Bn with i.i.d. random pairs (An, Bn) and obtains conditions under which Yn converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A1A2 ··· An-1Bn. A second subject is the series ∑ Cnf(Tn) with (Cn) a sequence of i.i.d. random variables, (Tn) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case An and Cn are independent, Bn = AnCn, An = U1/αn with Un a uniform random variable, f(x) = e−x/α.

On a stochastic difference equation and a representation of non–negative infinitely divisible random variables

1979 ◽  
Vol 11 (04) ◽  
pp. 750-783 ◽  
Author(s):  
Wim Vervaat
Keyword(s):  
Difference Equation ◽  
Poisson Process ◽  
Existence Of Solutions ◽  
Borel Function ◽  
Random Variables ◽  
Random Variable ◽  
Stochastic Difference Equation ◽  
Infinitely Divisible ◽  
Uniform Random Variable

The present paper considers the stochastic difference equation Y n = A n Y n-1 + B n with i.i.d. random pairs (A n , B n ) and obtains conditions under which Y n converges in distribution. This convergence is related to the existence of solutions of and (A, B) independent, and the convergence w.p. 1 of ∑ A 1 A 2 ··· A n-1 B n . A second subject is the series ∑ C n f(T n ) with (C n ) a sequence of i.i.d. random variables, (T n ) the sequence of points of a Poisson process and f a Borel function on (0, ∞). The resulting random variable turns out to be infinitely divisible, and its Lévy–Hinčin representation is obtained. The two subjects coincide in case A n and C n are independent, B n = A n C n , A n = U 1/α n with U n a uniform random variable, f(x) = e −x/α.

A Stochastic Difference Equation with Stationary Noise on Groups

2012 ◽  
Vol 64 (5) ◽  
pp. 1075-1089 ◽  
Author(s):  
Robinson Edward Raja Chandiraraj
Keyword(s):  
Difference Equation ◽  
Existence Of Solutions ◽  
Compact Subgroup ◽  
Random Variables ◽  
Necessary And Sufficient Condition ◽  
Coset Space ◽  
Dimensional Torus ◽  
Stochastic Difference Equation ◽  
One Dimensional ◽  
Necessary And Sufficient

AbstractWe consider the stochastic difference equation on a locally compact group G, where is an automorphism of G, ξk are given G-valued random variables and ηk are unknown G-valued random variables. This equation was considered by Tsirelson and Yor on a one-dimensional torus. We consider the case when ξk have a common law μ and prove that if G is a distal group and is a distal automorphism of G and if the equation has a solution, then extremal solutions of the equation are in one-to-one correspondence with points on the coset space K\G for some compact subgroup K of G such that μ is supported on for any z in the support of μ. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

scholarly journals On Exact Sampling of Nonnegative Infinitely Divisible Random Variables

2012 ◽  
Vol 44 (3) ◽  
pp. 842-873 ◽  
Author(s):  
Zhiyi Chi
Keyword(s):  
Basic Result ◽  
Random Variables ◽  
Random Variable ◽  
Fixed Number ◽  
Rejection Sampling ◽  
Infinitely Divisible ◽  
Exact Sampling ◽  
Special Cases ◽  
Wide Range ◽  
Lévy Density

Nonnegative infinitely divisible (i.d.) random variables form an important class of random variables. However, when this type of random variable is specified via Lévy densities that have infinite integrals on (0, ∞), except for some special cases, exact sampling is unknown. We present a method that can sample a rather wide range of such i.d. random variables. A basic result is that, for any nonnegative i.d. random variable X with its Lévy density explicitly specified, if its distribution conditional on X ≤ r can be sampled exactly, where r > 0 is any fixed number, then X can be sampled exactly using rejection sampling, without knowing the explicit expression of the density of X. We show that variations of the result can be used to sample various nonnegative i.d. random variables.

scholarly journals Fractional negative binomial and Pólya processes

2018 ◽  
Vol 38 (1) ◽  
pp. 77-101
Author(s):  
Palaniappan Vellai Samy ◽  
Aditya Maheshwari
Keyword(s):  
Poisson Process ◽  
Negative Binomial ◽  
Random Variable ◽  
Infinitely Divisible ◽  
One Dimensional ◽  
Fractional Poisson Process ◽  
Negative Binomial Process ◽  
The One ◽  
Definition Of ◽  
Binomial Process

In this paper, we define a fractional negative binomial process FNBP by replacing the Poisson process by a fractional Poisson process FPP in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

scholarly journals On Tails of Perpetuities

2010 ◽  
Vol 47 (4) ◽  
pp. 1191-1194 ◽  
Author(s):  
Paweł Hitczenko
Keyword(s):  
Lower Bound ◽  
Difference Equation ◽  
Upper Bound ◽  
Random Variable ◽  
Stochastic Difference Equation

We establish an upper bound on the tails of a random variable that arises as a solution of a stochastic difference equation. In the nonnegative case our bound is similar to a lower bound obtained in Goldie and Grübel (1996).

MAKING SIMULATIONS MORE EFFICIENT WHEN ANALYZING POISSON ARRIVAL SYSTEMS AND MEANS OF MONOTONE FUNCTIONS

2006 ◽  
Vol 20 (2) ◽  
pp. 251-256
Author(s):  
Sheldon M. Ross
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Random Variable ◽  
Expected Value ◽  
Stratified Sampling ◽  
Arrival Process ◽  
Monotone Functions ◽  
New Approach ◽  
Poisson Arrival ◽  
Increasing Function

For a system in which arrivals occur according to a Poisson process, we give a new approach for using simulation to estimate the expected value of a random variable that is independent of the arrival process after some specified time t. We also give a new approach for using simulation to estimate the expected value of an increasing function of independent uniform random variables. Stratified sampling is a key technique in both cases.

Some further results in infinite divisibility

1977 ◽  
Vol 82 (2) ◽  
pp. 289-295 ◽  
Author(s):  
D. N. Shanbhag ◽  
D. Pestana ◽  
M. Sreehari
Keyword(s):  
Positive Integer ◽  
Random Variables ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stable Laws ◽  
Infinitely Divisible ◽  
Stable Random Variable ◽  
Stable Random Variables ◽  
Image Position

Goldie (2), Steutel (8, 9), Kelker (4), Keilson and Steutel (3) and several others have studied the mixtures of certain distributions which are infinitely divisible. Recently Shanbhag and Sreehari (7) have proved that if Z is exponential with unit parameter and for 0 < α < 1, if Yx is a positive stable random variable with , t ≥ 0 and independent of Z, then for every 0 < α < 1Using this result, they have obtained several interesting results concerning stable random variables including some extensions of the results of the above authors. More recently, Williams (11) has used the same approach to show that if , where n is a positive integer ≥ 2, then is distributed as the product of n − 1 independent gamma random variables with index parameters α, 2α, …, (n − 1) α. Prior to these investigations, Zolotarev (12) had studied the problems of M-divisibility of stable laws.

scholarly journals On Some Compound Random Variables Motivated by Bulk Queues

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
Romeo Meštrović
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Mass Function ◽  
Probability Mass Function ◽  
Time Of Arrival ◽  
Uniform Random Variable ◽  
Wide Range ◽  
Bulk Queue ◽  
Probability Mass ◽  
Number Of Customers

We consider the distribution of the number of customers that arrive in an arbitrary bulk arrival queue system. Under certain conditions on the distributions of the time of arrival of an arriving group (Y(t)) and its size (X) with respect to the considered bulk queue, we derive a general expression for the probability mass function of the random variableQ(t)which expresses the number of customers that arrive in this bulk queue during any considered periodt. Notice thatQ(t)can be considered as a well-known compound random variable. Using this expression, without the use of generating function, we establish the expressions for probability mass function for some compound distributionsQ(t)concerning certain pairs(Y(t),X)of discrete random variables which play an important role in application of batch arrival queues which have a wide range of applications in different forms of transportation. In particular, we consider the cases whenY(t)and/orXare some of the following distributions: Poisson, shifted-Poisson, geometrical, or uniform random variable.

scholarly journals On Tails of Perpetuities

2010 ◽  
Vol 47 (04) ◽  
pp. 1191-1194 ◽  
Author(s):  
Paweł Hitczenko
Keyword(s):  
Lower Bound ◽  
Difference Equation ◽  
Upper Bound ◽  
Random Variable ◽  
Stochastic Difference Equation

We establish an upper bound on the tails of a random variable that arises as a solution of a stochastic difference equation. In the nonnegative case our bound is similar to a lower bound obtained in Goldie and Grübel (1996).

scholarly journals On Exact Sampling of Nonnegative Infinitely Divisible Random Variables

2012 ◽  
Vol 44 (03) ◽  
pp. 842-873 ◽  
Author(s):  
Zhiyi Chi
Keyword(s):  
Basic Result ◽  
Random Variables ◽  
Random Variable ◽  
Fixed Number ◽  
Rejection Sampling ◽  
Infinitely Divisible ◽  
Exact Sampling ◽  
Special Cases ◽  
Wide Range ◽  
Lévy Density

Nonnegative infinitely divisible (i.d.) random variables form an important class of random variables. However, when this type of random variable is specified via Lévy densities that have infinite integrals on (0, ∞), except for some special cases, exact sampling is unknown. We present a method that can sample a rather wide range of such i.d. random variables. A basic result is that, for any nonnegative i.d. random variableXwith its Lévy density explicitly specified, if its distributionconditionalonX≤rcan be sampled exactly, wherer&gt; 0 is any fixed number, thenXcan be sampled exactly using rejection sampling, without knowing the explicit expression of the density ofX. We show that variations of the result can be used to sample various nonnegative i.d. random variables.

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