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

1990 ◽  
Vol 22 (02) ◽  
pp. 350-374 ◽  
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.

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

1990 ◽  
Vol 22 (2) ◽  
pp. 350-374 ◽  
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.

Compound Poisson approximations for sums of discrete nonlattice variables

2003 ◽  
Vol 35 (1) ◽  
pp. 228-250 ◽  
Author(s):  
V. Čekanavičius ◽  
Y. H. Wang
Keyword(s):  
Normal Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Compound Poisson ◽  
Poisson Distributions ◽  
Set Of Points ◽  
Poisson Measures ◽  
Poisson Approximations

Sums of independent random variables concentrated on the same finite discrete, not necessarily lattice, set of points are approximated by compound Poisson distributions and signed compound Poisson measures. Such approximations can be more accurate than the normal distribution. Short asymptotic expansions are constructed.

Compound Poisson approximations for sums of discrete nonlattice variables

2003 ◽  
Vol 35 (01) ◽  
pp. 228-250 ◽  
Author(s):  
V. Čekanavičius ◽  
Y. H. Wang
Keyword(s):  
Normal Distribution ◽  
Asymptotic Expansions ◽  
Random Variables ◽  
Independent Random Variables ◽  
Compound Poisson ◽  
Poisson Distributions ◽  
Set Of Points ◽  
Poisson Measures ◽  
Poisson Approximations

Sums of independent random variables concentrated on the same finite discrete, not necessarily lattice, set of points are approximated by compound Poisson distributions and signed compound Poisson measures. Such approximations can be more accurate than the normal distribution. Short asymptotic expansions are constructed.

Asymptotic distributions of weighted compound Poisson bridges

1992 ◽  
Vol 112 (3) ◽  
pp. 613-629 ◽  
Author(s):  
Barbara Szyszkowicz
Keyword(s):  
Poisson Process ◽  
Random Variables ◽  
Asymptotic Distributions ◽  
Independent Random Variables ◽  
Intensity Parameter ◽  
Compound Poisson ◽  
Image Position

Let S(N(t)) be defined bywhere {N(t), t ≥ 0} is a Poisson process with intensity parameter 1/μ > 0 and {Xi i ≥ 1} is a family of independent random variables which are also independent of {N(t), t ≥ 0}.

scholarly journals On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

2009 ◽  
Vol 46 (3) ◽  
pp. 721-731 ◽  
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.

Another characteristic property of the Poisson distribution

1970 ◽  
Vol 68 (1) ◽  
pp. 167-169 ◽  
Author(s):  
D. N. Shanbhag
Keyword(s):  
Poisson Distribution ◽  
Conditional Distribution ◽  
Characteristic Property ◽  
Random Variables ◽  
Independent Random Variables ◽  
Poisson Distributions

1. Introduction: In (4) Moran considers two independent random variables X and Y taking non-negative integral values to give a characterization of the Poisson distribution. He establishes that the conditional distribution of X, given the total X + Y, is binomial for all given values of X + Y and there exists at least one i so that P(x = i) > 0, P( Y = i) > 0 if and only if X and Y have Poisson distributions. A slightly improved version of this result is given by Chatterji (1). For a comprehensive bibliography on the Poisson distribution the reader is referred to (3).

Infinitely divisible approximations for discrete nonlattice variables

2003 ◽  
Vol 35 (04) ◽  
pp. 982-1006
Author(s):  
V. Čekanavičius
Keyword(s):  
Asymptotic Expansions ◽  
Random Variables ◽  
Second Order ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Compound Poisson ◽  
Set Of Points ◽  
Poisson Measures

Sums of independent random variables concentrated on discrete, not necessarily lattice, set of points are approximated by infinitely divisible distributions and signed compound Poisson measures. A version of Kolmogorov's first uniform theorem is proved. Second-order asymptotic expansions are constructed for distributions with pseudo-lattice supports.

Infinitely divisible approximations for discrete nonlattice variables

2003 ◽  
Vol 35 (4) ◽  
pp. 982-1006 ◽  
Author(s):  
V. Čekanavičius
Keyword(s):  
Asymptotic Expansions ◽  
Random Variables ◽  
Second Order ◽  
Independent Random Variables ◽  
Infinitely Divisible Distributions ◽  
Infinitely Divisible ◽  
Compound Poisson ◽  
Set Of Points ◽  
Poisson Measures

Sums of independent random variables concentrated on discrete, not necessarily lattice, set of points are approximated by infinitely divisible distributions and signed compound Poisson measures. A version of Kolmogorov's first uniform theorem is proved. Second-order asymptotic expansions are constructed for distributions with pseudo-lattice supports.

scholarly journals On the Transition Law of Tempered Stable Ornstein–Uhlenbeck Processes

2009 ◽  
Vol 46 (03) ◽  
pp. 721-731 ◽  
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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