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(3) The random variables the variances $\text{Var}(X_n) < c (E X_n)^2$ for $n$ sufficiently large, with some constant $c \in (0,1)$. The random variable \( X \) associated with a Poisson process is discrete and therefore the Poisson distribution is discrete. https://projecteuclid.org/euclid.aop/1176996766, © The question is the following: can we assure that the sequence does NOT converge in distribution to a Poisson? Poisson probability distribution is used in situations where events occur randomly and independently a number of times on average during an interval of time or space. 0000002102 00000 n
The Poisson Process is the model we use for describing randomly occurring events and by itself, isn’t that useful. Poisson Probability distribution Examples and Questions. 0000010785 00000 n
MOD-POISSON CONVERGENCE IN PROBABILITY AND NUMBER THEORY E. KOWALSKI AND A. NIKEGHBALI Abstract. 0000008135 00000 n
MathJax reference. 159 30
Probab., Volume 2, Number 1 (1974), 178-180. 0000002571 00000 n
Simons and Johnson (1971) showed that the convergence is actually much stronger than in the usual sense. A Compound Poisson Convergence Theorem Wang, Y. H., Annals of Probability, 1991; Invariance, Minimax Sequential Estimation, and Continuous Time Processes Kiefer, J., Annals of Mathematical Statistics, 1957; Right Haar Measure for Convergence in Probability to Quasi Posterior Distributions Stone, M., Annals of Mathematical Statistics, 1965; Limiting Behavior of Posterior Distributions when … It is well known that under certain conditions, binomial distributions converge to Poisson distributions. Ann. 0
To learn more, see our tips on writing great answers. 2020 Use MathJax to format equations. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. For a positive result, you need some extra condition. <]>>
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In this note the author shows that the result of Simons and Johnson is also true for Poisson binomial distributions which include binomial distributions as special cases. [�u����yJ8�r����u����8C䳤����uYܛg>o^����E]�rw>��p��P�K���=T�-�W����z����V�_e�Ǟ MathOverflow is a question and answer site for professional mathematicians. 0000002469 00000 n
Thanks for contributing an answer to MathOverflow! By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Let $\{X_n\}_{n\geq 0}$ a sequence of random variables which we know that $\mathbb{E}[X_n]$ tends to infinity. 0000008539 00000 n
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site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. It's easy to construct a sequence $Y_n$ with $Y_n \to 0$ a.s. but $E Y_n \to +\infty$. rev 2020.11.24.38066, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us, “Question closed” notifications experiment results and graduation, MAINTENANCE WARNING: Possible downtime early morning Dec 2/4/9 UTC (8:30PM…, Convergence of moments implies convergence to normal distribution, Limit of pushforward measures of random variables is “represented” by a random variable, Weak convergence of random variables in $L^2$ and vague convergence, Non-normality of limit of random variables, moment sequence which does not define a random variable vs convergence in distribution, Weak convergence to a “multi-Bernoulli” distribution. 0000003316 00000 n
Mod-Poisson Convergence 3553 Then the following hold: (1) The re-scaled variables ZN/λN converge in probability to 1, that is, for any ε>0, lim N→∞ P N Z λN −1 >ε = 0. %PDF-1.5
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We have encountered the following problem that we think that should be true. Building on earlier work introducing the notion of \mod-Gaussian" convergence of sequences of random variables, which arises naturally in Random Matrix Theory and number theory, we discuss the analogue notion of \mod-Poisson" convergence. x�b```���,A ���,Ё��r��5���=�S���NJm2��#���F\����z�>����U��=�}�. 0000014906 00000 n
Now let $X$ be a fixed Poisson random variable and $X_n = X + Y_n$. 0000007263 00000 n
Project Euclid, 60F05: Central limit and other weak theorems, 62E99: None of the above, but in this section, Invariance, Minimax Sequential Estimation, and Continuous Time Processes, Right Haar Measure for Convergence in Probability to Quasi Posterior Distributions, Limiting Behavior of Posterior Distributions when the Model is Incorrect, Consistency of maximum likelihood estimators in general random effects models for binary data, Non-Parametric Empirical Bayes Procedures, Note on Uniformly Best Unbiased Estimates, Stationary Waiting-Time Distributions for Single-Server Queues.
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