In the first part of the paper we study stochastic integrals of a nonrandom function with respect to a nonorthogonal Hilbert noise defined on a semiring of subsets of an arbitrary nonempty set.

In the second part we apply this construction to study limit behavior of canonical (i.e., degenerate) Von Mises statistics based on weakly dependent stationary observations.

In this paper we give simple sufficient conditions for linear type processes with short memory that imply the invariance principle. Various examples including projective criterion are considered as applications. In particular, we treat the weak invariance principle for partial sums of linear processes with short memory. We prove that whenever the partial sums of innovations satisfy the $L_{p}$--invariance principle, then so does the partial sums of its corresponding linear process.

Let $(S_0,S_1,\dots)$ be a supermartingale relative to a nondecreasing sequence of $\sigma$-algebras $H_{\le0},H_{\le1},\dots$, with $S_0\le0$ almost surely (a.s.) and differences $X_i:=S_i-S_{i-1}$. Suppose that $X_i\le d$ and $\Var(X_i|H_{\le i-1})\le \si_i^2$ a.s.\ for every $i=1,2,\dots$, where $d>0$ and $\si_i>0$ are non-random constants. Let $T_n:=Z_1+\dots+Z_n$, where $Z_1,\dots,Z_n$ are i.i.d.\ r.v.'s each taking on only two values, one of which is $d$, and satisfying the conditions $\E Z_i=0$ and $\Var Z_i=\si^2:=\frac1n(\si_1^2+\dots+\si_n^2)$. Then, based on a comparison inequality between generalized moments of $S_n$ and $T_n$ for a rich class of generalized moment functions, the tail comparison inequality

$$ \PP(S_n\ge y) \le c\, \PP^{\lin,\lc}(T_n\ge y+\tfrac h2)\quad\forall y\in\R $$

is obtained, where $c:=e^2/2=3.694\dots$, $h:=d+\si^2/d$, and the function $y\mapsto\PP^{\lin,\lc}(T_n\ge y)$ is the least log-concave majorant of the linear interpolation of the tail function $y\mapsto\PP(T_n\ge y)$ over the lattice of all points of the form $nd+kh$ ($k\in\Z$). An explicit formula for $\PP^{\lin,\lc}(T_n\ge y+\tfrac h2)$ is given. Another, similar bound is given under somewhat different conditions. It is shown that these bounds improve significantly upon known bounds.

We obtain an almost sure bound for oscillation rates of empirical distribution functions for stationary causal processes. For short-range dependent processes, the oscillation rate is shown to be optimal in the sense that it is as sharp as the one obtained under independence. The dependence conditions are expressed in terms of physical dependence measures which are directly related to the data-generating mechanism of the underlying processes and thus are easy to work with.

For $\gamma>-\frac{1}{2}$, we provide the Karhunen-Loève expansion of the weighted mean-centered Wiener process, defined by \[W_{\gamma}(t)=\frac{1}{\sqrt{1+2\gamma}}\Big\{W\big(t^{1+2\gamma}\big) -\int_{0}^1W\big(u^{1+2\gamma}\big)du\Big\},\] for $t\in(0,1]$. We show that the orthogonal functions in these expansions have simple expressions in term of Bessel functions. Moreover, we obtain that the $L^2[0,1]$ norm of $W_{\gamma}$ is identical in distribution with the $L^2[0,1]$ norm of the weighted Brownian bridge $t^{\gamma}B(t)$.

In this paper the whole family of fractional Brownian motions is constructed as a single Gaussian field indexed by time and the Hurst index simultaneously. The field has a simple covariance structure and it is related to two generalizations of fractional Brownian motion known as multifractional Brownian motions. A mistake common to the existing literature regarding multifractional Brownian motions is pointed out and corrected. The Gaussian field, due to inherited ``duality'', reveals a new way of constructing martingales associated with the odd and even part of a fractional Brownian motion and therefore of the fractional Brownian motion. The existence of those martingales and their stochastic representations is the first step to the study of natural wavelet expansions associated to those processes in the spirit of our earlier work on a construction of natural wavelets associated to Gaussian-Markov processes.

Estimation methods for the Lévy density of a Lévy process are developed under mild qualitative assumptions. A classical model selection approach made up of two steps is studied. The first step consists in the selection of a good estimator, from an approximating (finite-dimensional) linear model $\calS$ for the true Lévy density. The second is a data-driven selection of a linear model $\calS$, among a given collection $\{\calS_{m}\}_{m\in\calM}$, that approximately realizes the best trade-off between the error of estimation within $\calS$ and the error incurred when approximating the true Lévy density by the linear model $\calS$. Using recent concentration inequalities for functionals of Poisson integrals, a bound for the risk of estimation is obtained. As a byproduct, oracle inequalities and long-run asymptotics for spline estimators are derived. Even though the resulting underlying statistics are based on continuous time observations of the process, approximations based on high-frequency discrete-data can be easily devised.

In this paper the multi-dimensional random walk models governed by distributed fractional order differential equations and multi-term fractional order differential equations are constructed. The scaling limits of these random walks to a diffusion process in the sense of distributions is proved.

Let $\{u_t(x), t \ge 0,\, x \in \R \}$ be a random string taking values in $\R^d$, specified by the following stochastic partial differential equation (Funaki, 1983):

\frac{\partial u_t(x)}{\partial t}=\frac{{\partial}^2u_t(x)}{\partial x^2}+\dot{W},

here $\dot{W}(x,t)$ is an $\R^d$-valued space-time white noise.

Mueller and Tribe (2002) have proved necessary and sufficient conditions for the $\R^d$-valued process $\{u_t(x):t \ge 0,\,x \in \R\}$ to hit points and to have double points. In this paper, we continue their research by determining the Hausdorff and packing dimensions of the level sets and the sets of double times of the random string process $\{u_t(x):t \ge 0,\,x \in \R\}$. We also consider the Hausdorff and packing dimensions of the range and graph of the string.

This note deals with a problem of the probabilistic Ramsey theory in functional analysis. Given a linear operator $T$ on a Hilbert space with an orthogonal basis, we define the isomorphic structure $\Sigma(T)$ as the family of all subsets of the basis so that $T$ restricted to their span is a nice isomorphism. Our main result is a dimension-free optimal estimate of the size of $\Sigma(T)$. It improves and extends in several ways the principle of restricted invertibility due to Bourgain and Tzafriri. With an appropriate notion of randomness, we obtain a randomized principle of restricted invertibility.

We demonstrate the strength of a coupling derived from a Gaussian approximation of Zaitsev (1987a) by revisiting two strong approximation results for the empirical process of Dudley and Philipp (1983), and using the coupling to derive extended and refined versions of them.

We show that a modified Empirical process converges to the limiting Gaussian process whenever the limit is continuous. The modification depends on the properties of the limit via Talagrand's characterization of the continuity of Gaussian processes.

We give several conditions for pregaussianity of norm balls of Besov spaces defined over $\mathbb{R}^{d}$ by exploiting results in Haroske and Triebel (2005). Furthermore, complementing sufficient conditions in Nickl and Pötscher (2005), we give necessary conditions on the parameters of the Besov space to obtain the Donsker property of such balls. For certain parameter combinations Besov balls are shown to be pregaussian but not Donsker.

We find the Bahadur slope of the Lilliefors and Cramér–von Mises tests of normality.

Assumptions on a likelihood function, including a local Glivenko-Cantelli condition, imply the existence of M-estimators converging to an M-functional. Scatter matrix-valued estimators, defined on all empirical measures on $\RR^d$ for $d\geq 2$, and equivariant under all, including singular, affine transformations, are shown to be constants times the sample covariance matrix. So, if weakly continuous, they must be identically 0. Results are stated on existence and differentiability of location and scatter functionals, defined on a weakly dense, weakly open set of laws, via elliptically symmetric t distributions on $\RR^d$, following up on work of Kent, Tyler, and Dümbgen.

Almost sure bounds are established on the uniform error of smoothing spline estimators in nonparametric regression with random designs. Some results of Einmahl and Mason (2005) are used to derive uniform error bounds for the approximation of the spline smoother by an “equivalent” reproducing kernel regression estimator, as well as for proving uniform error bounds on the reproducing kernel regression estimator itself, uniformly in the smoothing parameter over a wide range. This admits data-driven choices of the smoothing parameter.

Let ${M}$ be a compact Riemannian submanifold of ${{\bf R}^m}$ of dimension $\scriptstyle{d}$ and let ${X_1,\dots, X_n}$ be a sample of i.i.d. points in ${M}$ with uniform distribution. We study the random operators

$${\Delta_{h_n,n}f(p):={1\over {nh_n^{d+2}}}\sum_{i=1}^n K({p-X_i\over h_n})(f(X_i)-f(p)),\ p\in M}$$

where ${K(u):={1\over (4\pi)^{d/2}}e^{-\|u\|^2/4}}$ is the Gaussian kernel and $ {h_n\to 0}$ as ${n\to\infty.}$ Such operators can be viewed as graph laplacians (for a weighted graph with vertices at data points) and they have been used in the machine learning literature to approximate the Laplace-Beltrami operator of ${M,}$ ${\Delta_M f}$ (divided by the Riemannian volume of the manifold). We prove several results on a.s. and distributional convergence of the deviations ${\Delta_{h_n,n}f(p)-{1\over |\mu|}\Delta_Mf(p)}$ for smooth functions ${f}$ both pointwise and uniformly in ${f}$ and ${p}$ (here ${|\mu|=\mu(M)}$ and ${\mu}$ is the Riemannian volume measure). In particular, we show that for any class ${{\cal F}}$ of three times differentiable functions on ${M}$ with uniformly bounded derivatives

$${ \sup_{p\in M}\sup_{f\in {\cal F}}\Big|\Delta_{h_n,p}f(p)-{1\over|\mu|}\Delta_Mf(p)\Big|=O\Big(\sqrt{\log(1/h_n)\over nh_n^{d+2}}\Big)\ \ {\rm a.s.}}$$

as soon as

$${nh_n^{d+2}/\log h_n^{-1}\to \infty\ \ {\rm and}\ \nh^{d+4}_n/\log h_n^{-1}\to 0,}$$

and also prove asymptotic normality of ${\Delta_{h_n,p}f(p)-{1\over |\mu|}\Delta_Mf(p)}$ (functional CLT) for a fixed ${p\in M}$ and uniformly in ${f}$.

We establish a new concentration result for regularized risk minimizers which is similar to an oracle inequality. Applying this inequality to regularized least squares minimizers like least squares support vector machines, we show that these algorithms learn with (almost) the optimal rate in some specific situations. In addition, for regression our results suggest that using the loss function $L_{\a}(y,t)=|y -t|^{\a}$ with $\a$ near $1$ may often be preferable to the usual choice of $\a=2$.