The expectation of the Vandermonde product squared for uniform random variables

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Abstract

Let X1,X2,,Xn be independent random variables, each uniformly distributed on the interval (0,1). We compute the expectation of (X1X2Xn)k1i<jn(XjXi)2. The result relies on operators, the extended Vandermonde determinant and the hook formula for standard Young tableaux.

Section snippets

Introduction and preliminaries

Zvonkin [11, page 303] noted that for X1,X2,,Xn being independent identically distributed normal random variables with mean 0 and variance 1, the following identity holds:E[1i<jn(XjXi)2]=0!1!n!. 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 L2(X) denote the set of all real-valued square integrable functions on X. Recall that L2(X) is endowed with an inner product (,) defined by (f,g)=Xfgdx.

Let Um be the m-dimensional simplex given byUm={(x1,x2,,xm)Rm:0x1x2xm1}. Note that U0 is a point and we identify L2(U0) with the real numbers. For a non-negative integer m define the 0,1-function χm on Um×Um+1 which indicates if the two vectors interlace, that is,χm(x,y)={1 if y1x1y2xmym

Concluding remarks

Is there an explicit expression for composing operators of type A and B when they are interspersed? Such an expression is B3B3A2B2A1B1B1A0(1). 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 λn(fλ)2=n! 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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