Tail behaviour of multiple random integrals and U-statistics

Estimation of Multiple Random Integrals and U-statistics

Here I give a short survey of some problems on multiple random integrals and U -statistics together with some other questions which occur during their study in a natural way. I write down the main results, discuss their background together with some pictures and mathematical ideas which explain them better. 2000 Math. Subj. Class. 60E15.

Péter Major On the estimation of multiple random integrals and U - statistics – Lecture

un 2 00 5 Moment Inequalities for U - statistics Rados law Adamczak

We present moment inequalities for completely degenerate Banach space valued (generalized) U-statistics of arbitrary order. The estimates involve suprema of empirical processes, which in the real valued case can be replaced by simpler norms of the kernel matrix (i.e. norms of some multilinear operators associated with the kernel matrix). As a corollary we derive tail inequalities for U-statisti...

On the Estimation of Multiple Random Integrals and U-statistics

Fix a positive integer n, consider n independent and identically distributed random variables ξ1, . . . , ξn on a measurable space (X,X ) with some distribution μ and take their empirical distribution μn together with its normalization √ n(μn−μ). Beside this, take a function f(x1, . . . , xk) of k variables on the k-fold product (X k,X ) of the space (X,X ), introduce the k-th power of the norm...

Approximation Theorems for Random Permanents and Associated Stochastic Processes

The limit theorems for certain stochastic processes generated by permanents of random matrices of independent columns with exchangeable components are established. The results are based on the martingale decomposition of a random permanent function similar to the one known for U -statistics and on relating the components of this decomposition to some multiple stochastic integrals.