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 laziness.

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

T_{n} = a^{t}Pb

has mean 0, variance 1 and converges in law to N(0, 1).

This proves that the characteristic function of T_{n} converges to the normal characteristic function:

\frac{1}{n!} ∑_{P} e^{ita^{T} P b} → e^{−t^{2}/2}

for each fixed t (the sum is over permutation matrices P). 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!} ∑_{P} e^{ta^{T} P b} → e^{t^{2}/2}

2. Suppose ϵ_{1}, …, ϵ_{n} are iid N(0, 1). The empirical process

W_{n}(·) = \frac{1}{\sqrt{n}} ∑ [1(Φ(ϵ_{i}) ≤ ·) − ·]

converges weakly to a Brownian Bridge. The process

Z_{n}(·) = \frac{W_{n}(·)}{ϕ(Φ^{−1}(·))}

converges weakly in C[ϵ, 1 − ϵ] to a mean 0 Gaussian process Z with covariance function

ρ(s, t) = \frac{min(s, t) − st}{J(s)J(t)}

where the notation is

J(t) = ϕ(Φ^{−1}(t))
ϕ = Φ′

and Φ is the standard normal cumulative. The covariance kernel ρ is square integrable over the unit square and has spectrum {1, 1/2, 1/3, … }. The kernel has, however, infinite trace:

∫_{0}^{1} ρ(t, t)dt = ∞

DeWet and Venter (?) established that if ϵ_{(1)} < ⋯ < ϵ_{(n)} are the order statistics then there are constants a_{n} → ∞ such that

∑ (ϵ_{(i)} − Φ^{−1}(i/(n + 1)))^{2} − a_{n}

converges in distribution to

∫_{0}^{1} [Z^{2}(t) − ρ(t, t)] dt = ∑ \frac{U_{i}}{i}

where the U_{i} are iid χ_{1}^{2} variates.

Now suppose that observations Y_{i} satisfy the linear model

Y_{i} = x_{i}^{T} β + σϵ_{i}

for some known p dimensional row vectors x_{i}^{T}, an unknown p-vector β of parameters and an unknown standard deviation σ. If we fit the model by least squares as usual and compute the usual residuals

ϵ̂_{i} = \frac{Y_{i} − x_{i}^{T} β̂}{σ̂}

then the empirical process of these fitted residuals analogous to Z_{n} is

Ẑ_{n}(t) = \frac{∑ [1(Φ(ϵ̂_{i}) ≤ t) − t]}{\sqrt{n}J(t)}

This process converges weakly (again in C[ϵ, 1 − ϵ] for each positive ϵ) to hatZ a mean 0 Gaussian process with covariance function

ρ̂(s, t) = \frac{min(s, t) − st − J(s)J(t) − Φ^{−1}(s)J(s)Φ^{−1}(t)J(t)}{J(s)J(t)}

This covariance function has (over the unit square) spectrum {1/3, 1/4, 1/5, … }.