Asymptotic Expansions in the Exponent: a Compound Poisson Approach

1997 ◽  
Vol 29 (02) ◽  
pp. 374-387 ◽  
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.

Asymptotic Expansions in the Exponent: a Compound Poisson Approach

1997 ◽  
Vol 29 (2) ◽  
pp. 374-387 ◽  
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.

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.

scholarly journals Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain

2007 ◽  
Vol 39 (01) ◽  
pp. 128-140 ◽  
Author(s):  
Etienne Roquain ◽  
Sophie Schbath
Keyword(s):  
Markov Chain ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Rare Event ◽  
Poisson Approximation ◽  
Rare Word ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Method ◽  
Poisson Approximations

We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.

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 Improved compound Poisson approximation for the number of occurrences of any rare word family in a stationary markov chain

2007 ◽  
Vol 39 (1) ◽  
pp. 128-140 ◽  
Author(s):  
Etienne Roquain ◽  
Sophie Schbath
Keyword(s):  
Markov Chain ◽  
Poisson Distribution ◽  
Approximation Error ◽  
Rare Event ◽  
Poisson Approximation ◽  
Rare Word ◽  
Compound Poisson ◽  
Compound Poisson Approximation ◽  
Stein Method ◽  
Poisson Approximations

We derive a new compound Poisson distribution with explicit parameters to approximate the number of overlapping occurrences of any set of words in a Markovian sequence. Using the Chen-Stein method, we provide a bound for the approximation error. This error converges to 0 under the rare event condition, even for overlapping families, which improves previous results. As a consequence, we also propose Poisson approximations for the declumped count and the number of competing renewals.

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