Limit of Power Sum Distribution

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Bill August - 24 Jul 2009 12:04 GMT

When we take the "mathematic" mean of n iid rvs Xi, when n approximate infinity, the CLT shows that no matter what the distribution of Xi is, the distribution is approximate to the normal distribution with mean u and variance sigam/sqrt(n).
I am curies that what if we use other kind of "mean method". For example "Power Sum":
S = 10log10(1/n * sum(10^X_i))
Is it ever possible to predict the distribution trend when n becomes infinity?
Any comment will be very appreciated.

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Herman Rubin - 24 Jul 2009 18:36 GMT

>When we take the "mathematic" mean of n iid rvs Xi, when n approximate infinity, the CLT shows that no matter what the distribution of Xi is, the distribution is approximate to the normal distribution with mean u and variance sigam/sqrt(n).
>I am curies that what if we use other kind of "mean method". For example "Power Sum":
>S = 10log10(1/n * sum(10^X_i))
>Is it ever possible to predict the distribution trend when n becomes infinity?
>Any comment will be very appreciated.

Assuming the variance exists, the central limit theorem holds
for any function of independent observations. If X_i are iid,
so are 10^X_i.

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This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

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Bill August - 27 Jul 2009 15:19 GMT

> In article
> <6939582.32058.1248433494683.JavaMail.jakarta@nitrogen
[quoted text clipped - 16 lines]
> are iid,
> so are 10^X_i.

Thank you very much for your reply.
Yes, if we assume that X_i are iid then 10^X_i also should be iid.
And by CLT, 1/n * sum(10^X_i) is approximating Gaussian.
However, we still do not know the trend of the S which takes the logarithm of 1/n * sum(10^X_i).
I noticed that from the paper "A Normal Limit Theorem for Power Sums of Independent Random Variables" published in 1967 even before CLT (1995) that both "1/n * sum(10^X_i)" and its logarithm approximate Gaussian. However the paper only tested it by Monte-carlo simulation without any prove.
I am curious to know if theirs any study on this topic.
Many thanks again.

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Math Forum / Mathematics / Statistics / July 2009

Limit of Power Sum Distribution