367:. The unscaled sum of iid variables doesn't converge to anything at all! Of course, if you take a distribution with support on the non-negative integers, and multiply it by a non-integer scalar, the resulting distribution will not be supported by the non-negative integers. But there is an alternate re-scaling operation which does yield an integer-supported distribution, and that has a central limit theorem that converges to a Poisson distribution (or more generally, a discrete-stable distribution, if the finite variance assumption is dropped). --
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Poisson distributed, and approaches a normal distribution in the limit, in a certain sense. Perhaps we should mention the discrete limit theorems are special cases of the continuous ones? Some care is needed to indicate which limits we are taking, and what kind of convergence. Random sums of
Poisson
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RVs can themselves approach non-Poisson Stable RVs in the limit, according to the linked reference from Lee, for example. Maybe here's not the case to explain that? In whcih case we should remove the section entirely again; currently it's ambiguous what sort of limits we are discussing etc. --
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That is not the case. Under the
Central Limit Theorem, the normal distribution is also the limit a discrete distribution with finite variance. The CLT is not dependent on discrete vs. continuous. A binomial and even Poisson itself tends to the normal.
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The discrete-stable distributions are a class of probability distributions with the property that the sum of several random variables from such a distribution is distributed according to the same family....
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The most well-known discrete stable distribution is the
Poisson distribution which is a special case as the only discrete-stable distribution for which the mean and all higher-order moments are finite.
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Something must be missing from the description of limit theorems, since the sum of many discrete iid random variables converges to a normal distribution, which is not a discrete-stable distribution.
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So now I understand better what the article is getting it. The deleted section was not wrong, but there was very important context missing from the article, hopefully I will get around to fixing it.
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89:. The project works to allow users to contribute quality articles and media files to the encyclopedia and track their progress as they are developed. To participate, please visit the
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Am I misunderstanding "stable" in this context? Or is there supposed to be a difference between "discrete stable" and "discrete-stable"? Or am I missing something else? Help? --
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Some of this confusion is perhaps due to the fact that in the limit one can approach, in distribution, both continuous and discrete RVs; Indeed a sum of independent
Poisson RVs
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I removed the section on limit theorems. I am not finding any support for discrete stable distributions satisfying this property except for lattice random variables. In fact
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323:(stable distribution with index of stability Ī± = Ī³ and with skewness parameter Ī² = 1), whose Laplace transform is given by exp(āĪ»tĪ³ )." (emphasis added)
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explicitly states that at least some discrete stable distributions belong "to the domain of normal attraction of the absolutely
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The difference of two independent
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Briefly: it is usually left unsaid, but actually, when we invoke the central limit theorem what we really mean is a sum
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and has finite variance, the limit is the normal distribution as you point out. In the case that the distribution is
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I have tried to make it more explicit that in the case when the distribution is
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As I indicate above, there is important context missing from the article. --
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has finite mean, the limit is the
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This article was accepted on 19 March 2015 by reviewer
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Generating probability
Generating Function definition
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