Uniform saddlepoint approximations

The validity of the saddlepoint expansion evaluated at the point y is considered in the limit y tending to ∞. This is done for the expansions of the density and of the tail probability of the mean of n i.i.d. random variables and also for the expansion of the tail probability of a compound Poisson sum , where N is a Poisson random variable. We consider both general conditions that ensure the validity of the expansions and study the four classes of densities for X 1 introduced in Daniels (1954).

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On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

Journal of Applied Probability ◽

10.1239/jap/1253279848 ◽

2009 ◽

Vol 46 (3) ◽

pp. 721-731 ◽

Cited By ~ 10

Author(s):

Shibin Zhang ◽

Xinsheng Zhang

Keyword(s):

Stable Distribution ◽

Stochastic Integral ◽

Random Variables ◽

Transition Density ◽

Random Variable ◽

Independent Random Variables ◽

Poisson Random Variable ◽

Compound Poisson ◽

Tempered Stable Distribution ◽

Transition Distribution

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is a C∞-function.

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On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

Journal of Applied Probability ◽

10.1017/s0021900200005842 ◽

2009 ◽

Vol 46 (03) ◽

pp. 721-731 ◽

Cited By ~ 2

Author(s):

Shibin Zhang ◽

Xinsheng Zhang

Keyword(s):

Stable Distribution ◽

Stochastic Integral ◽

Random Variables ◽

Transition Density ◽

Random Variable ◽

Independent Random Variables ◽

Poisson Random Variable ◽

Compound Poisson ◽

Tempered Stable Distribution ◽

Transition Distribution

In this paper, a stochastic integral of Ornstein–Uhlenbeck type is represented to be the sum of two independent random variables: one has a tempered stable distribution and the other has a compound Poisson distribution. In distribution, the compound Poisson random variable is equal to the sum of a Poisson-distributed number of positive random variables, which are independent and identically distributed and have a common specified density function. Based on the representation of the stochastic integral, we prove that the transition distribution of the tempered stable Ornstein–Uhlenbeck process is self-decomposable and that the transition density is aC∞-function.

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EXPANSIONS FOR MOMENTS OF COMPOUND POISSON DISTRIBUTIONS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964813000053 ◽

2013 ◽

Vol 27 (3) ◽

pp. 319-331 ◽

Cited By ~ 2

Author(s):

S. Nadarajah ◽

C.S. Withers ◽

S.A.A. Bakar

Keyword(s):

Random Sample ◽

Recursion Formula ◽

Random Variable ◽

Bell Polynomials ◽

Poisson Random Variable ◽

Compound Poisson ◽

Poisson Distributions ◽

Poisson Variable ◽

Panjer Recursion ◽

The Mean

Expansions for moments of $\overline{X}$, the mean of a random sample of size n, are given for both the univariate and multivariate cases. The coefficients of these expansions are simply Bell polynomials. An application is given for the compound Poisson variable SN, where $S_{n} = n \overline{X}$ and N is a Poisson random variable independent of X1, X2, …, yielding expansions that are computationally more efficient than the Panjer recursion formula and Grubbström and Tang's formula.

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Fully degenerate Bell polynomials associated with degenerate Poisson random variables

Open Mathematics ◽

10.1515/math-2021-0022 ◽

2021 ◽

Vol 19 (1) ◽

pp. 284-296

Author(s):

Hye Kyung Kim

Keyword(s):

Random Variables ◽

Random Variable ◽

Combinatorial Identities ◽

Bell Polynomials ◽

Poisson Random Variable ◽

Central Moments ◽

Special Polynomials ◽

Special Numbers And Polynomials ◽

New Type ◽

Two Parameters

Abstract Many mathematicians have studied degenerate versions of quite a few special polynomials and numbers since Carlitz’s work (Utilitas Math. 15 (1979), 51–88). Recently, Kim et al. studied the degenerate gamma random variables, discrete degenerate random variables and two-variable degenerate Bell polynomials associated with Poisson degenerate central moments, etc. This paper is divided into two parts. In the first part, we introduce a new type of degenerate Bell polynomials associated with degenerate Poisson random variables with parameter α > 0 \alpha \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the fully degenerate Bell polynomials. We derive some combinatorial identities for the fully degenerate Bell polynomials related to the n n th moment of the degenerate Poisson random variable, special numbers and polynomials. In the second part, we consider the fully degenerate Bell polynomials associated with degenerate Poisson random variables with two parameters α > 0 \alpha \gt 0 and β > 0 \beta \hspace{-0.15em}\gt \hspace{-0.15em}0 , called the two-variable fully degenerate Bell polynomials. We show their connection with the degenerate Poisson central moments, special numbers and polynomials.

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Poisson approximation for a sum of dependent indicators: an alternative approach

Advances in Applied Probability ◽

10.1017/s0001867800011782 ◽

2002 ◽

Vol 34 (03) ◽

pp. 609-625 ◽

Cited By ~ 2

Author(s):

N. Papadatos ◽

V. Papathanasiou

Keyword(s):

Total Variation ◽

Upper Bound ◽

Random Variables ◽

Random Variable ◽

Poisson Approximation ◽

Total Variation Distance ◽

Poisson Random Variable ◽

Birthday Problem ◽

Alternative Approach ◽

Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

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Poisson approximation for a sum of dependent indicators: an alternative approach

Advances in Applied Probability ◽

10.1239/aap/1033662168 ◽

2002 ◽

Vol 34 (3) ◽

pp. 609-625 ◽

Cited By ~ 2

Author(s):

N. Papadatos ◽

V. Papathanasiou

Keyword(s):

Total Variation ◽

Upper Bound ◽

Random Variables ◽

Random Variable ◽

Poisson Approximation ◽

Total Variation Distance ◽

Poisson Random Variable ◽

Birthday Problem ◽

Alternative Approach ◽

Variation Distance

The random variables X1, X2, …, Xn are said to be totally negatively dependent (TND) if and only if the random variables Xi and ∑j≠iXj are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X1, x2, …, Xn with P[Xi = 1] = pi = 1 - P[Xi = 0], an upper bound for the total variation distance between ∑ni=1Xi and a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

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Poisson approximation for some statistics based on exchangeable trials

Advances in Applied Probability ◽

10.2307/1426620 ◽

1983 ◽

Vol 15 (3) ◽

pp. 585-600 ◽

Cited By ~ 48

Author(s):

A. D. Barbour ◽

G. K. Eagleson

Keyword(s):

Total Variation ◽

Limit Theorems ◽

Random Variables ◽

Random Variable ◽

Poisson Approximation ◽

Total Variation Distance ◽

Poisson Random Variable ◽

Variation Distance ◽

Image Position ◽

Poisson Approximations

Stein's (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables.Let be exchangeable random elements of a space and, for I a k-subset of , let XI be a 0–1 function. The statistics studied here are of the form where N is some collection of k -subsets of .An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.

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The Largest Missing Value in a Sample of Geometric Random Variables

Combinatorics Probability Computing ◽

10.1017/s096354831400011x ◽

2014 ◽

Vol 23 (5) ◽

pp. 670-685 ◽

Cited By ~ 1

Author(s):

MARGARET ARCHIBALD ◽

ARNOLD KNOPFMACHER

Keyword(s):

Random Variables ◽

Random Variable ◽

Missing Value ◽

Asymptotic Expressions ◽

Mean And Variance ◽

The Mean ◽

Special Case

We consider samples of n geometric random variables with parameter 0 < p < 1, and study the largest missing value, that is, the highest value of such a random variable, less than the maximum, that does not appear in the sample. Asymptotic expressions for the mean and variance for this quantity are presented. We also consider samples with the property that the largest missing value and the largest value which does appear differ by exactly one, and call this the LMV property. We find the probability that a sample of n variables has the LMV property, as well as the mean for the average largest value in samples with this property. The simpler special case of p = 1/2 has previously been studied, and verifying that the results of the present paper coincide with those previously found for p = 1/2 leads to some interesting identities.

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A Compound Poisson Approximation Inequality

Journal of Applied Probability ◽

10.1017/s0021900200001534 ◽

2006 ◽

Vol 43 (01) ◽

pp. 282-288

Author(s):

Erol A. Peköz

Keyword(s):

System Reliability ◽

Random Variable ◽

Poisson Approximation ◽

Poisson Random Variable ◽

Sequence Pattern ◽

Compound Poisson ◽

Compound Poisson Approximation ◽

Numerical Example

We give conditions under which the number of events which occur in a sequence ofm-dependent events is stochastically smaller than a suitably defined compound Poisson random variable. The results are applied to counts of sequence pattern appearances and to system reliability. We also provide a numerical example.

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