On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

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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Uniform saddlepoint approximations

Advances in Applied Probability ◽

10.2307/1427038 ◽

1988 ◽

Vol 20 (3) ◽

pp. 622-634 ◽

Cited By ~ 31

Author(s):

J. L. Jensen

Keyword(s):

Random Variables ◽

Random Variable ◽

Tail Probability ◽

Poisson Random Variable ◽

Compound Poisson ◽

Saddlepoint Approximations ◽

The Mean ◽

General Conditions

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 X1 introduced in Daniels (1954).

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Uniform saddlepoint approximations

Advances in Applied Probability ◽

10.1017/s0001867800018176 ◽

1988 ◽

Vol 20 (03) ◽

pp. 622-634 ◽

Cited By ~ 2

Author(s):

J. L. Jensen

Keyword(s):

Random Variables ◽

Random Variable ◽

Tail Probability ◽

Poisson Random Variable ◽

Compound Poisson ◽

Saddlepoint Approximations ◽

The Mean ◽

General Conditions

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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A unification of some approaches to Poisson approximation

Journal of Applied Probability ◽

10.1017/s0021900200039152 ◽

1990 ◽

Vol 27 (03) ◽

pp. 611-621 ◽

Cited By ~ 3

Author(s):

Hans-Jürgen Witte

Keyword(s):

Complex Analysis ◽

Random Variables ◽

Random Variable ◽

Poisson Approximation ◽

Upper Bounds ◽

Secretary Problem ◽

Independent Random Variables ◽

Poisson Random Variable ◽

Semigroup Approach ◽

Metric Distances

Let Sn be a sum of independent random variables. For the approximation of Sn by a Poisson random variable Y with the same mean, the complex analysis approaches based on generating functions and the semigroup approach are presented in a unified setting which permits us to refine Kerstan's complex analysis approach obtaining considerably sharper upper bounds for some metric distances of Sn and Y. These results are applied to some special Sn counting the records of an i.i.d. sequence of random variables which is important to various applied problems, for instance the secretary problem.

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A unification of some approaches to Poisson approximation

Journal of Applied Probability ◽

10.2307/3214545 ◽

1990 ◽

Vol 27 (3) ◽

pp. 611-621 ◽

Cited By ~ 26

Author(s):

Hans-Jürgen Witte

Keyword(s):

Complex Analysis ◽

Random Variables ◽

Random Variable ◽

Poisson Approximation ◽

Upper Bounds ◽

Secretary Problem ◽

Independent Random Variables ◽

Poisson Random Variable ◽

Semigroup Approach ◽

Metric Distances

Let Sn be a sum of independent random variables. For the approximation of Sn by a Poisson random variable Y with the same mean, the complex analysis approaches based on generating functions and the semigroup approach are presented in a unified setting which permits us to refine Kerstan's complex analysis approach obtaining considerably sharper upper bounds for some metric distances of Sn and Y. These results are applied to some special Sn counting the records of an i.i.d. sequence of random variables which is important to various applied problems, for instance the secretary problem.

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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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Approximation of sums by compound Poisson distributions with respect to stop-loss distances

Advances in Applied Probability ◽

10.2307/1427540 ◽

1990 ◽

Vol 22 (2) ◽

pp. 350-374 ◽

Cited By ~ 31

Author(s):

S. T. Rachev ◽

L. Rüschendorf

Keyword(s):

Random Variables ◽

Independent Random Variables ◽

Compound Poisson ◽

Poisson Distributions ◽

Stop Loss ◽

Scale Transformations

The approximation of sums of independent random variables by compound Poisson distributions with respect to stop-loss distances is investigated. These distances are motivated by risk-theoretic considerations. In contrast to the usual construction of approximating compound Poisson distributions, the method suggested in this paper is to fit several moments. For two moments, this can be achieved by scale transformations. It is shown that the new approximations are more stable and improve the usual approximations by accompanying laws in examples where the probability 1 – pi that the ith summand is zero is not too large.

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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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Convergence rates in the law of large numbers. II

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100043103 ◽

1968 ◽

Vol 64 (2) ◽

pp. 485-488 ◽

Cited By ~ 4

Author(s):

V. K. Rohatgi

Keyword(s):

Convergence Rates ◽

Random Variables ◽

Almost Sure Convergence ◽

Random Variable ◽

Rates Of Convergence ◽

Independent Random Variables ◽

Large Numbers ◽

Present Context ◽

Image Position ◽

Uniformly Bounded

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense thatSet qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence ofto zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

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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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