An estimate about multiple stochastic integrals with respect to a normalized empirical measure

Let a sequence of iid. random variables ξ 1 , …, ξ n be given on a measurable space ( X,X ) with distribution µ together with a function f ( x1 , …, xk ) on the product space ( Xk , Xk ). Let µ n denote the empirical measure defined by these random variables and consider the random integral \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$J_{n,k} (f) = \frac{{n^{{k \mathord{\left/ {\vphantom {k 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{k!}}\smallint 'f(u_1 ,...,u_k )(\mu _n (du_1 ) - \mu (du_1 ))...(\mu _n (du_k ) - \mu (du_k )),$$ \end{document} where prime means that the diagonals are omitted from the domain of integration. A good bound is given on the probability P (| Jn,k ( f )| > x ) for all x > 0 which is similar to the estimate in the analogous problem we obtain by considering the Gaussian (multiple) Wiener-Itô integral of the function f . The proof is based on an adaptation of some methods of the theory of Wiener-Itô integrals. In particular, a sort of diagram formula is proved for the random integrals Jn,k ( f ) together with some of its important properties, a result which may be interesting in itself. The relation of this paper to some results about U -statistics is also discussed.

Isoperimetric bounds for higher eigenvalue ratios for the n-dimensional fixed membrane problem

SynopsisWe give several results which extend our recent proof of the Payne-Pólya–Weinberger conjecture to ratios of higher eigenvalues. In particular, we show that for a bounded domain Ω⊂ℝn the eigenvalues of its Dirichlet Laplacian obey where λm denotes the mth eigenvalue and jp,k denotes the kth positive zero of the Bessel function Jp(x). Certain extensions of this result are given, the most general being the bound where k≧2 and l(m) denotes the number of nodal domains of an mth eigenfunction. Our results imply certain further conjectures of Payne, Pólya, and Weinberger concerning λ3/λ2 and λ4/λ3. In addition, we find a resonably good bound on λ4/λ1. We also briefly discuss extensions to Schrödinger operators and other elliptic eigenvalue problems.

A New Bound for Nil U-Rings

A U-ring is a ring in which every subring is a meta ideal. A meta ideal of a ring R is a subring I of R which lies in a chain of subrings,with the properties:(1) Iλ is an ideal of Iλ+1 for all λ < β;(2) If α is a limit ordinal number, then Iα = ∪λ<αIλ.Freidman [3] proved that every nil U-ring is a locally nilpotent ring. Since there are many locally nilpotent rings which are not U-rings, the class of locally nilpotent rings is not a very good bound for the class of nil U-rings. This paper establishes a new bound for nil U-rings based on a property of the multiplicative semigroup of the ring.