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On The Convolution Property of a Heavy Tailed Stable Distribution |
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PP: 173-182 |
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Author(s) |
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Subhash C. Bagui,
K. L. Mehra,
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Abstract |
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| In this Article, we present an explicit direct proof of the convolution property for a heavy tailed stable distribution. The
distribution arises and is of interest in a variety of the contexts in many disciplines: in probability and statistics, in electrical engineering,
computer vision, image and signal processing and in many physical and economic processes.We shall refer to this as L`evy’s distribution
in the sequel. The particular convolution property for the distribution, which entails its stability, shows that the sample mean based
on a random sample of n observations from this distribution has the same distribution as that of n times a single observation. The
sample mean, thus, is more variable than a single observation and increases by an order of n as the sample size n increases. The
central limit theorem, evidently, does not hold for this distribution. We also give an alternative proof for the above property based on
Laplace transforms. These proofs do not seem to be available in standard text books. The only proofs available use advanced arguments
involving the Brownian motion process. In addition, for better understanding of L`evy’s and other stable distributions, some contextually
relevant basic properties of stable distributions are also discussed and elaborated on. Stable distributions are the limiting distributions,
under appropriate conditions, of normed sums of independent random variables. Their study should be of interest per se. These proofs
in their detailed presentation along with an introductory discussion of stable distributions should help to fill up a notable gap in the
available text-book literature. The article should be of interest from a pedagogical standpoint for seniors, first year graduate students
and beginning researchers in statistics and probability. |
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