New limit theorems for regular diffusion processes with finite speed measure

We derive limit theorems for diffusion processes that have a finite speed measure. First we prove a number of asymptotic properties of the density $rho_t = dmu_t /dmu$ of the empirical measure $mu_t$ with respect to the normalized speed measure $mu$. These results are then used to derive finite dimensional and uniform central limit theorems for integrals of the form sqrt{tint (rho_t-1),dnu, where $nu$ is an arbitrary finite, signed measure on the state space of the diffusion. We also consider a number of interesting special cases, such as uniform central limit theorems for Lebesgue integrals and functional weak convergence of the empirical distribution function

New limit theorems for regular diffusion processes with finite speed measure

We derive limit theorems for diffusion processes that have a finite speed measure. First we prove a number of asymptotic properties of the density $rho_t = dmu_t /dmu$ of the empirical measure $mu_t$ with respect to the normalized speed measure $mu$. These results are then used to derive finite dimensional and uniform central limit theorems for integrals of the form sqrt{tint (rho_t-1),dnu, where $nu$ is an arbitrary finite, signed measure on the state space of the diffusion. We also consider a number of interesting special cases, such as uniform central limit theorems for Lebesgue integrals and functional weak convergence of the empirical distribution function

The stable central limit theorem for local martingales with bounded jumps via Skorohod embedding

The stable central limit theorem for properly normalized local martingales with bounded jumps is proved. Instead of the usual characteristic function-type methods we use an embedding technique in combination with a result on nested Brownian motions. In this approach, the stability of the CLT is explained by the fact that nested Brownian motions are asymptotically independent of any other random element. As was previously shown in the special case of continuous local martingales, the embedding technique leads to short and transparent arguments. In the conclusion we discuss the direction in which further research is needed to make the embedding method applicable in an even larger number of situations

On the uniform convergence of local time and the uniform consistency of density estimators for ergodic diffusions

We prove a theorem on the uniform convergence of the local time of an ergodic diffusion. This result is then used to investigate certain estimators of the invariant density of an ergodic diffusion, including kernel estimators. We show that the pointwise consistency of these estimators can be strengthened to uniform consistency

Uniform convergence of curve estimators for ergodic diffusion processes

For ergodic diffusions, we consider kernel-type estimators for the invariant density, its derivatives and the drift function. Using empirical process theory for martingales, we first prove a theorem regarding the uniform weak convergence of the empirical density. This result is then used to derive uniform weak convergence for the kernel estimator of the invariant density. For kernel estimators of the derivatives of the invariant density and for a nonparametric drift estimator that was proposed by Banon, we give bounds for the rate at which the uniform distance between the estimator and the true curve vanishes. We also consider the problem of estimation from discrete-time observations. In that case, obvious estimators can be constructed by replacing Lebesgue integrals by Riemann sums. We show that these approximations are also uniformly consistent, provided that the bandwidths and the time between the observations are correctly balanced

A series expansion of fractional Brownian motion

Let $B$ be a fractional Brownian motion with Hurst index $H in (0,1)$. Denote by $x_1 < x_2 < cdots$ the positive, real zeros of the Bessel function J_{-H of the first kind of order $-H$, and let $y_1 < y_2 < cdots$ be the positive zeros of J_{1-H. We prove the series representation begin{equation* B_t = sum_{n=1^infty frac{sin x_n t{x_n, X_n + sum_{n=1^infty frac{1-cos y_n t{y_n, Y_n, end{equation* where $X_1, X_2, ldots$ and $Y_1, Y_2, ldots$ are independent, Gaussian random variables with mean zero and Var X_n = 2c_H^2x_n^{-2HJ^{-2_{1-H(x_n), Var Y_n = 2c_H^2y_n^{-2HJ^{-2_{-H(y_n), where the constant $c_H^2$ is defined by c_H^2 = pi^{-1Gamma(1+2H)sin pi H. With probability $1$, both random series converge absolutely and uniformly in $t in [0,1]$

A note on Bernstein-type inequalities for martingales

Several inequalities of Bernstein's type are derived in a unified manner. Some extra light is shed on the classical inequalities and implications are sought for instance for conditionally symmetric martingales and sequences of asymptotically continuous martingales

On Bernstein-type inequalities for martingales

Bernstein-type inequalities for local martingales are derived. The results extend a number of well-known exponential inequalities and yield an asymptotic inequality for a sequence of asymptotically continuous martingales