Estimating Drift Parameters in a Fractional Ornstein Uhlenbeck Process with Periodic Mean

We construct a least squares estimator for the drift parameters of a fractional Ornstein Uhlenbeck process with periodic mean function and long range dependence. For this estimator we prove consistency and asymptotic normality. In contrast to the classical fractional Ornstein Uhlenbeck process without periodic mean function the rate of convergence is slower depending on the Hurst parameter $H$, namely $n^{1-H}$

An Empirical Process Central Limit Theorem for Multidimensional Dependent Data

Let $(U_n(t))_{t\in\R^d}$ be the empirical process associated to an $\R^d$-valued stationary process $(X_i)_{i\ge 0}$. We give general conditions, which only involve processes $(f(X_i))_{i\ge 0}$ for a restricted class of functions $f$, under which weak convergence of $(U_n(t))_{t\in\R^d}$ can be proved. This is particularly useful when dealing with data arising from dynamical systems or functional of Markov chains. This result improves those of [DDV09] and [DD11], where the technique was first introduced, and provides new applications.Comment: to appear in Journal of Theoretical Probabilit

Parking on a Random Tree

Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Renyi's parking problem, alternatively called blocking RSA: at every vertex of the tree a particle (or car) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function.Comment: 7 page

Limit theorems for von Mises statistics of a measure preserving transformation

For a measure preserving transformation $T$ of a probability space $(X,\mathcal F,\mu)$ we investigate almost sure and distributional convergence of random variables of the form $$x \to \frac{1}{C_n} \sum_{i_1<n,...,i_d<n} f(T^{i_1}x,...,T^{i_d}x),\, n=1,2,...,$$ where $f$ (called the \emph{kernel}) is a function from $X^d$ to $\R$ and $C_1, C_2,...$ are appropriate normalizing constants. We observe that the above random variables are well defined and belong to $L_r(\mu)$ provided that the kernel is chosen from the projective tensor product $$L_p(X_1,\mathcal F_1, \mu_1) \otimes_{\pi}...\otimes_{\pi} L_p(X_d,\mathcal F_d, \mu_d)\subset L_p(\mu^d)$$ with $p=d\,r,\, r\ \in [1, \infty).$ We establish a form of the individual ergodic theorem for such sequences. Next, we give a martingale approximation argument to derive a central limit theorem in the non-degenerate case (in the sense of the classical Hoeffding's decomposition). Furthermore, for $d=2$ and a wide class of canonical kernels $f$ we also show that the convergence holds in distribution towards a quadratic form $\sum_{m=1}^{\infty} \lambda_m\eta^2_m$ in independent standard Gaussian variables $\eta_1, \eta_2,...$. Our results on the distributional convergence use a $T$--\,invariant filtration as a prerequisite and are derived from uni- and multivariate martingale approximations

Weak convergence of Vervaat and Vervaat Error processes of long-range dependent sequences

Following Cs\"{o}rg\H{o}, Szyszkowicz and Wang (Ann. Statist. {\bf 34}, (2006), 1013--1044) we consider a long range dependent linear sequence. We prove weak convergence of the uniform Vervaat and the uniform Vervaat error processes, extending their results to distributions with unbounded support and removing normality assumption

Parking on a Random Tree

Abstract Consider an infinite tree with random degrees, i.i.d. over the sites, with a prescribed probability distribution with generating function G(s). We consider the following variation of Rényi&apos;s parking problem, alternatively called blocking RSA (random sequential adsorption): at every vertex of the tree a particle (or &quot;car&quot;) arrives with rate one. The particle sticks to the vertex whenever the vertex and all of its nearest neighbors are not occupied yet. We provide an explicit expression for the so-called parking constant in terms of the generating function. That is, the occupation probability, averaged over dynamics and the probability distribution of the random trees converges in the large-time limit to (1 − α 2 )/2 with 1 α xdx G(x) = 1

The empirical process of a short-range dependent stationary sequence under Gaussian subordination

Consider the stationary sequence X 1 = G ( Z 1 ), X 2 = G ( Z 2 ),..., where G (·) is an arbitrary Borel function and Z 1 , Z 2 ,... is a mean-zero stationary Gaussian sequence with covariance function r(k)=E ( Z 1 Z k +1 ) satisfying r (0)=1 and ∑ k =1 ∞ | r ( k )| m < ∞, where, with I {·} denoting the indicator function and F (·) the continuous marginal distribution function of the sequence { X n }, the integer m is the Hermite rank of the family { I { G (·) ≦ x } − F ( x ): x ∈ R }. Let F n (·) be the empirical distribution function of X 1 ,..., X n . We prove that, as n →∞, the empirical process n 1/2 { F n (·)- F (·)} converges in distribution to a Gaussian process in the space D [−∞,∞].Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47644/1/440_2005_Article_BF01303800.pd