The expectation of the Vandermonde product squared for uniform random variables
Section snippets
Introduction and preliminaries
Zvonkin [11, page 303] noted that for being independent identically distributed normal random variables with mean 0 and variance 1, the following identity holds: See [2] for a bijective proof of this result. However, this raises the question for which other probability distributions can we evaluate the expected value of the Vandermonde product squared?
In this paper we answer this question for independent random variables, each uniformly distributed on the
Rectangular standard Young tableaux
For a subset X of Euclidean space, let denote the set of all real-valued square integrable functions on X. Recall that is endowed with an inner product defined by .
Let be the m-dimensional simplex given by Note that is a point and we identify with the real numbers. For a non-negative integer m define the -function on which indicates if the two vectors interlace, that is,
Concluding remarks
Is there an explicit expression for composing operators of type A and B when they are interspersed? Such an expression is . If so, what integral identities would come out from the hook formula?
It is interesting to note that the hook formula and even the Young–Frobenius formula have probabilistic proofs; see the two papers by Greene, Nijenhuis and Wilf [7], [8].
Acknowledgment
The author thanks Theodore Ehrenborg for comments on an earlier draft of this paper. This work was supported by a grant from the Simons Foundation (#429370, Richard Ehrenborg).
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