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 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> 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.

Subsequence principles for vector-valued random variables

1979 ◽  
Vol 86 (2) ◽  
pp. 301-312 ◽  
Author(s):  
D. J. H. Garling
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Principal Result ◽  
Moment Condition ◽  
Special Cases ◽  
Cesaro Averages ◽  
Image Position ◽  
Vector Valued ◽  
Subsequence Principle ◽  
Independent Identically Distributed

1. Introduction. Révész(8) has shown that if (fn) is a sequence of random variables, bounded in L2, there exists a subsequence (fnk) and a random variable f in L2 such that converges almost surely whenever . Komlós(5) has shown that if (fn) is a sequence of random variables, bounded in L1, then there is a subsequence (A*) with the property that the Cesàro averages of any subsequence converge almost surely. Subsequently Chatterji(2) showed that if (fn) is bounded in LP (where 0 < p ≤ 2) then there is a subsequence (gk) = (fnk) and f in Lp such thatalmost surely for every sub-subsequence. All of these results are examples of subsequence principles: a sequence of random variables, satisfying an appropriate moment condition, has a subsequence which satisfies some property enjoyed by sequences of independent identically distributed random variables. Recently Aldous(1), using tightness arguments, has shown that for a general class of properties such a subsequence principle holds: in particular, the results listed above are all special cases of Aldous' principal result.

Stochastic flowshop no-wait scheduling

1985 ◽  
Vol 22 (1) ◽  
pp. 240-246 ◽  
Author(s):  
E. Frostig ◽  
I. Adiri
Keyword(s):  
Independent Random Variable ◽  
Processing Time ◽  
Random Variables ◽  
Random Variable ◽  
Schedule Length ◽  
Special Cases ◽  
No Wait ◽  
Completion Times ◽  
Sum Of Completion Times

This paper deals with special cases of stochastic flowshop, no-wait, scheduling. n jobs have to be processed by m machines . The processing time of job Ji on machine Mj is an independent random variable Ti. It is possible to sequence the jobs so that , . At time 0 the realizations of the random variables Ti, (i are known. For m (m ≧ 2) machines it is proved that a special SEPT–LEPT sequence minimizes the expected schedule length; for two (m = 2) machines it is proved that the SEPT sequence minimizes the expected sum of completion times.

scholarly journals A Note on Goldbach Partitions of Large Even Integers

10.37236/4542 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Ljuben Mutafchiev
Keyword(s):  
Limit Theorem ◽  
Tauberian Theorem ◽  
Random Variables ◽  
Random Variable ◽  
Fixed Number ◽  
Asymptotic Approach ◽  
Partitions Of Integers

Let $\Sigma_{2n}$ be the set of all partitions of the even integers from the interval $(4,2n], n>2,$ into two odd prime parts. We show that $|\Sigma_{2n}| \sim 2n^2/\log^2{n}$ as $n\to\infty$. We also assume that a partition is selected uniformly at random from the set $\Sigma_{2n}$. Let $2X_n\in (4,2n]$ be the size of this partition. We prove a limit theorem which establishes that $X_n/n$ converges weakly to the maximum of two random variables which are independent copies of a uniformly distributed random variable in the interval $(0,1)$. Our method of proof is based on a classical Tauberian theorem due to Hardy, Littlewood and Karamata. We also show that the same asymptotic approach can be applied to partitions of integers into an arbitrary and fixed number of odd prime parts.

scholarly journals A numerical study of entropy and residual entropy estimators based on smooth density estimators for non-negative random variables

2021 ◽  
Vol 54 (2) ◽  
pp. 99-121
Author(s):  
Yogendra P. Chaubey ◽  
Nhat Linh Vu
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Numerical Study ◽  
Random Variables ◽  
Random Variable ◽  
Residual Entropy ◽  
Density Estimator ◽  
Wide Range ◽  
Smooth Density

In this paper, we are interested in estimating the entropy of a non-negative random variable. Since the underlying probability density function is unknown, we propose the use of the Poisson smoothed histogram density estimator to estimate the entropy. To study the per- formance of our estimator, we run simulations on a wide range of densities and compare our entropy estimators with the existing estimators based on different approaches such as spacing estimators. Furthermore, we extend our study to residual entropy estimators which is the entropy of a random variable given that it has been survived up to time $t$.

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.

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.

scholarly journals THE QUANTUM DECOMPOSITION OF RANDOM VARIABLES WITHOUT MOMENTS

2013 ◽  
Vol 16 (02) ◽  
pp. 1350012 ◽  
Author(s):  
LUIGI ACCARDI ◽  
HABIB REBEI ◽  
ANIS RIAHI
Keyword(s):  
Hilbert Spaces ◽  
Random Variables ◽  
Random Variable ◽  
Quantum Probability ◽  
Fock Spaces ◽  
Infinitely Divisible ◽  
New Type ◽  
First Moment ◽  
Finite Moments

The quantum decomposition of a classical random variable is one of the deep results of quantum probability: it shows that any classical random variable or stochastic process has a built-in non-commutative structure which is intrinsic and canonical, and not artificially put by hands. Up to now the technique to deduce the quantum decomposition has been based on the theory of interacting Fock spaces and on Jacobi's tri-diagonal relation for orthogonal polynomials. Therefore it requires the existence of moments of any order and cannot be applied to random variables without this property. The problem to find an analogue of the quantum decomposition for random variables without finite moments of any order remained open for about fifteen years and nobody had any idea of how such a decomposition could look like. In the present paper we prove that any infinitely divisible random variable has a quantum decomposition canonically associated to its Lévy–Khintchin triple. The analytical formulation of this result is based on Kolmogorov representation of these triples in terms of 1–cocycles (helices) in Hilbert spaces and on the Araki–Woods–Parthasarathy–Schmidt characterization of these representation in terms of Fock spaces. It distinguishes three classes of random variables: (i) with finite second moment; (ii) with finite first moment only; (iii) without any moment. The third class involves a new type of renormalization based on the associated Lévy–Khinchin function.

Stochastic flowshop no-wait scheduling

1985 ◽  
Vol 22 (01) ◽  
pp. 240-246
Author(s):  
E. Frostig ◽  
I. Adiri
Keyword(s):  
Independent Random Variable ◽  
Processing Time ◽  
Random Variables ◽  
Random Variable ◽  
Schedule Length ◽  
Special Cases ◽  
No Wait ◽  
Completion Times ◽  
Sum Of Completion Times

This paper deals with special cases of stochastic flowshop, no-wait, scheduling. n jobs have to be processed by m machines . The processing time of job Ji on machine Mj is an independent random variable Ti . It is possible to sequence the jobs so that , . At time 0 the realizations of the random variables Ti , ( i are known. For m (m ≧ 2) machines it is proved that a special SEPT–LEPT sequence minimizes the expected schedule length; for two (m = 2) machines it is proved that the SEPT sequence minimizes the expected sum of completion times.

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