My Open Problems Page

On this page I hope to place a list of mathematical problems and half baked ideas in case anyone else would like to tell me the solutions. At the moment the list is essentially empty out of emptiness.

1.

COnvergence of moment generating functions in the permutation central limit theorem. Suppose that for each n we have vectors $a,b\in R^n$ (dependence on n suppressed) with ${\bf 1}^T a = {\bf 1}^T b = 0$ and a^T a = b^T b = n (?)> Let ${\bf P}$ be a random permutation matrix (equal to one of the n!possible $n \times n$ permutation matrices with probability 1/n!). Then

$$T_n = a^t {\bf P} b$$

has mean 0, variance ? and ?

This proves that the characteristic function of T_nconverges to the normal characteristic function:

$$\frac{1}{n!} \sum_P e^{ita^T P b} \to e^{-t^2/2}$$

for each fixed t. I would like to find conditions on a and b under which the moment generating function converges at least for t in a neighbourhood of 0; that is, I want conditions under which

$$\frac{1}{n!} \sum_P e^{ta^T P b} \to e^{t^2/2}$$

2.