Correction to: The scaling limit behaviour of periodic stable-like processes

Correction to Bernoulli (2006), 12, 551--570 http://projecteuclid.org/euclid.bj/1151525136Comment: Published at http://dx.doi.org/10.3150/07-BEJ5127 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A self-similar process arising from a random walk with random environment in random scenery

In this article, we merge celebrated results of Kesten and Spitzer [Z. Wahrsch. Verw. Gebiete 50 (1979) 5-25] and Kawazu and Kesten [J. Stat. Phys. 37 (1984) 561-575]. A random walk performs a motion in an i.i.d. environment and observes an i.i.d. scenery along its path. We assume that the scenery is in the domain of attraction of a stable distribution and prove that the resulting observations satisfy a limit theorem. The resulting limit process is a self-similar stochastic process with non-trivial dependencies.Comment: Published in at http://dx.doi.org/10.3150/09-BEJ234 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Convergence of U-statistics indexed by a random walk to stochastic integrals of a Levy sheet

We establish limit theorems for U-statistics indexed by a random walk on Z^d and we express the limit in terms of some Levy sheet Z(s,t). Under some hypotheses, we prove that the limit process is Z(t,t) if the random walk is transient or null-recurrent ant that it is some stochastic integral with respect to Z when the walk is positive recurrent. We compare our results with results for random walks in random scenery.Comment: 38 page

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 $n^{1-H}$

Change Point Testing for the Drift Parameters of a Periodic Mean Reversion Process

In this paper we investigate the problem of detecting a change in the drift parameters of a generalized Ornstein-Uhlenbeck process which is defined as the solution of $dX_t=(L(t)-\alpha X_t) dt + \sigma dB_t$, and which is observed in continuous time. We derive an explicit representation of the generalized likelihood ratio test statistic assuming that the mean reversion function $L(t)$ is a finite linear combination of known basis functions. In the case of a periodic mean reversion function, we determine the asymptotic distribution of the test statistic under the null hypothesis

Parameter estimation for the drift of a time-inhomogeneous jump diffusion process

This work deals with parameter estimation for the drift of jump diffusion processes which are driven by a Lévy process and whose drift term is linear in the parameter. In contrast to the commonly used maximum likelihood estimator, our proposed estimator has the practical advantage that its calculation does not require the evaluation of the continuous part of the sample path. In the important case of an Ornstein-Uhlenbeck-type jump diffusion, which is a widely used model, we prove consistency of our estimator

THE RELAXATION SPEED IN THE CASE THE FLOW SATISFIES EXPONENTIAL DECAY OF CORRELATIONS

We study the convergence speed in L 2-norm of the diffusion semi-group toward its equilibrium when the underlying flow satisfies decay of correlation. Our result is some extension of the main theorem given by Constantin, Kiselev, Ryzhik and Zlatoš in [3]. Our proof is based on Weyl asymptotic law for the eigenvalues of the Laplace operator, Sobolev imbedding and some assumption on decay of correlation for the underlying flow

Stable limit theorem for U-statistic processes indexed by a random walk

Let (Sn)n2N be a random walk in the domain of attraction of an a -stable Lévy process and ( (n))n2N a sequence of iid random variables (called scenery). We want to investigate U-statistics indexed by the random walk Sn, that is Un := P 1 i<j n h( (Si); (Sj )) for some symmetric bivariate function h. We will prove the weak convergence without the assumption of finite variance. Additionally, under the assumption of finite moments of order greater than two, we will establish a law of the iterated logarithm for the U-statistic Un

Drift estimation for a periodic mean reversion process

In this paper we propose a periodic, mean-reverting Ornstein-Uhlenbeck process of the form dXt = (L(t) − alpha Xt) dt + sigma dBt, where L(t) is a periodic, parametric function. We apply maximum likelihood estimation for the drift parameters based on time-continuous observations. The estimator is given explicitly and we prove strong consistency and asymptotic normality as the observed number of periods tends to infinity. The essential idea of the asymptotic study is the interpretation of the stochastic process as a sequence of random variables that take values in some function space