Compound Poisson approximations for sums of discrete nonlattice variables

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.

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Infinitely divisible approximations for discrete nonlattice variables

Advances in Applied Probability ◽

10.1017/s0001867800012702 ◽

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.

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Infinitely divisible approximations for discrete nonlattice variables

Advances in Applied Probability ◽

10.1239/aap/1067436331 ◽

2003 ◽

Vol 35 (4) ◽

pp. 982-1006 ◽

Cited By ~ 11

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.

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Asymptotic Expansions in the Exponent: a Compound Poisson Approach

Advances in Applied Probability ◽

10.1017/s0001867800028044 ◽

1997 ◽

Vol 29 (02) ◽

pp. 374-387 ◽

Cited By ~ 1

Author(s):

V. Čekanavičius

Keyword(s):

Asymptotic Expansion ◽

Large Deviations ◽

Poisson Distribution ◽

Asymptotic Expansions ◽

Compound Poisson ◽

Poisson Measures ◽

Poisson Approximations

The accuracy of the Normal or Poisson approximations can be significantly improved by adding part of an asymptotic expansion in the exponent. The signed-compound-Poisson measures obtained in this manner can be of the same structure as the Poisson distribution. For large deviations we prove that signed-compound-Poisson measures enlarge the zone of equivalence for tails.

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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.

Download Full-text

Asymptotic Expansions in the Exponent: a Compound Poisson Approach

Advances in Applied Probability ◽

10.2307/1428008 ◽

1997 ◽

Vol 29 (2) ◽

pp. 374-387 ◽

Cited By ~ 16

Author(s):

V. Čekanavičius

Keyword(s):

Asymptotic Expansion ◽

Large Deviations ◽

Poisson Distribution ◽

Asymptotic Expansions ◽

Compound Poisson ◽

Poisson Measures ◽

Poisson Approximations

The accuracy of the Normal or Poisson approximations can be significantly improved by adding part of an asymptotic expansion in the exponent. The signed-compound-Poisson measures obtained in this manner can be of the same structure as the Poisson distribution. For large deviations we prove that signed-compound-Poisson measures enlarge the zone of equivalence for tails.

Download Full-text

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

Advances in Applied Probability ◽

10.1017/s0001867800019613 ◽

1990 ◽

Vol 22 (02) ◽

pp. 350-374 ◽

Cited By ~ 5

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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Asymptotic expansions for distribution of sums quasi-lattice random variables

Lietuvos matematikos rinkinys ◽

10.15388/lmr.2011.tt03 ◽

2011 ◽

Vol 52 ◽

pp. 359-364

Author(s):

Algimantas Bikelis ◽

Kazimieras Padvelskis ◽

Pranas Vaitkus

Keyword(s):

Asymptotic Expansion ◽

Asymptotic Expansions ◽

Random Variables ◽

Independent Random Variables

Althoug Chebyshev [3] and Edeworth [5] had conceived of the formal expansions for distribution of sums of independent random variables, but only in Cramer’s work [4] was laid a proper foundation of this problem. In the case when random variables are lattice Esseen get the asymptotic expansion in a new diﬀerent form. Here we extend this problem for quasi-lattice random variables.

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