Poisson Approximations for the Number of kl-Scans

2017 ◽  
pp. 1-8
Author(s):  
Anant Godbole ◽  
Katherine Grzesik ◽  
Heather Shappell
Keyword(s):  
Poisson Approximations

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.

The normal and Poisson approximations.

1963 ◽  
pp. 126-150
Author(s):  
Thomas E. Kurtz
Keyword(s):  
Poisson Approximations

scholarly journals Poisson approximations for telecommunications networks

1994 ◽  
Vol 36 (1) ◽  
pp. 132-132
Author(s):  
T. C. Brown ◽  
P. K. Pollett
Keyword(s):  
Conditional Intensity ◽  
Telecommunications Networks ◽  
Image Position ◽  
Poisson Approximations

Professor Andrew Barbour (Institut für Angewandte Mathematik, Universität Zürich) has pointed out to us that the conditional intensity specified on Page 356 of our paper is incorrect and, consequently, so too is the bound (14) and the expression on Page 358 for the variance of the conditional intensity, given by (15). This variance should be 0 if ω = 1, where recall that ω = ω (r) is uniquely determined by rω = k, while if ω > 1, the variance is given by

Poisson approximations for runs and patterns of rare events

1991 ◽  
Vol 23 (4) ◽  
pp. 851-865 ◽  
Author(s):  
Anant P. Godbole
Keyword(s):  
Success Probability ◽  
Standard Condition ◽  
Random Variable ◽  
Poisson Approximation ◽  
Bernoulli Trials ◽  
Success Runs ◽  
Poisson Random Variable ◽  
Runs And Patterns ◽  
Reliability Systems ◽  
Poisson Approximations

Consider a sequence of Bernoulli trials with success probability p, and let Nn,k denote the number of success runs of length among the first n trials. The Stein–Chen method is employed to obtain a total variation upper bound for the rate of convergence of Nn,k to a Poisson random variable under the standard condition npk→λ. This bound is of the same order, O(p), as the best known for the case k = 1, i.e. for the classical binomial-Poisson approximation. Analogous results are obtained for occurrences of word patterns, where, depending on the nature of the word, the corresponding rate is at most O(pk–m) for some m = 0, 2, ···, k – 1. The technique is adapted for use with two-state Markov chains. Applications to reliability systems and tests for randomness are discussed.

Some distributional approximations in Markovian queueing networks

1982 ◽  
Vol 14 (03) ◽  
pp. 654-671 ◽  
Author(s):  
T. C. Brown ◽  
P. K. Pollett
Keyword(s):  
Queueing Networks ◽  
Poisson Processes ◽  
Single Class ◽  
Open Network ◽  
State Dependent ◽  
Service Rates ◽  
Distributional Approximations ◽  
Closed Network ◽  
Poisson Approximations ◽  
Markovian Queueing Networks

We consider single-class Markovian queueing networks with state-dependent service rates (the immigration processes of Whittle (1968)). The distance of customer flows from Poisson processes is estimated in both the open and closed cases. The bounds on distances lead to simple criteria for good Poisson approximations. Using the bounds, we give an asymptotic, closed network version of the ‘loop criterion' of Melamed (1979) for an open network. Approximation of two or more flows by independent Poisson processes is also studied.

Compound Poisson approximations for word patterns under Markovian hypotheses

1995 ◽  
Vol 32 (04) ◽  
pp. 877-892 ◽  
Author(s):  
Mark X. Geske ◽  
Anant P. Godbole ◽  
Andrew A. Schaffner ◽  
Allison M. Skolnick ◽  
Garrick L. Wallstrom
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Compound Poisson ◽  
Letter Word ◽  
Stationary Markov Chain ◽  
Letter Alphabet ◽  
Poisson Approximations

Consider a stationary Markov chainwith state space consisting of theξ-letter alphabet set Λ= {a1, a2, ···,aξ}.We study the variablesM=M(n, k) andN=N(n, k),defined, respectively, as the number of overlapping and non-overlapping occurrences of a fixed periodick-letter word, and use the Stein–Chen method to obtain compound Poisson approximations for their distribution.

Improving Poisson Approximations

1994 ◽  
Vol 8 (4) ◽  
pp. 449-462 ◽  
Author(s):  
Erol A. Peköz ◽  
Sheldon M. Ross
Keyword(s):  
Poisson Distribution ◽  
Error Bounds ◽  
Random Variables ◽  
Urn Models ◽  
Matching Problem ◽  
Bernoulli Convolutions ◽  
Poisson Approximations ◽  
Better Than

Let X1,…, Xn, be indicator random variables, and set We present a method for estimating the distribution of W in settings where W has an approximately Poisson distribution. Our method is shown to yield estimates significantly better than straight Poisson estimates when applied to Bernoulli convolutions, urn models, the circular k of n: F system, and a matching problem. Error bounds are given.

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.

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