I am new to math research: what’s the significance of this? Should the references be included in the paper by professor Tao? Is the paper by prof. Tao completely derivative or not (is there an original contribution)?

]]>Hence, where . That is, can be seen as a product of independent sums of iid random variables. Central limit theorem for such products (and its connection to Wishart determinants) has been discussed in G. Rempala, J. Wesolowski “Asymptotics for products of independent sums with an application to Wishart determinants”, Stat. Probab. Lett. 74 (2005) 129–138.

In particular, in this paper, it has been shown that for any sequence of iid positive square integrable random variables with finite absolute moment of order

tends in distribution to , where , and is the coefficient of variation. In this case we have so that , , . From this, clearly the result in (3) follows. The assumption on was later removed in K. Kosinski “Asymptotics for sums of a function of normalized independent sums, Stat. Probab. Lett. 79 (2009) 415–419, in the sense that a different normalization than is needed in the case .

Various other generalizations have been considered for asymptotics for products of (not necessarily independent) sums of random variables.

Yes, one needs to truncate and renormalise the integral in order to eliminate the type divergence. This is discussed in detail in the paper, but I am omitting it in this post as it is a minor technical issue rather than a principal difficulty in the argument.

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