Showing papers in "Bernoulli in 2001"

An adaptive Metropolis algorithm

Abstract: A proper choice of a proposal distribution for Markov chain Monte Carlo methods, for example for the Metropolis-Hastings algorithm, is well known to be a crucial factor for the convergence of the algorithm. In this paper we introduce an adaptive Metropolis (AM) algorithm, where the Gaussian proposal distribution is updated along the process using the full information cumulated so far. Due to the adaptive nature of the process, the AM algorithm is non-Markovian, but we establish here that it has the correct ergodic properties. We also include the results of our numerical tests, which indicate that the AM algorithm competes well with traditional Metropolis-Hastings algorithms, and demonstrate that the AM algorithm is easy to use in practical computation.

Mixed fractional Brownian motion

Abstract: We show that the sum of a Brownian motion and a non-trivial multiple of an independent fractional Brownian motion with Hurst parameter H ∈ (0,1] is not a semimartingale if H ∈ (0, ½) ∪ (½, ¾], that it is equivalent to a multiple of Brownian motion if H = ½ and equivalent to Brownian motion if H ∈ ( ¾ , 1]. As an application we discuss the price of a European call option on an asset driven by a linear combination of a Brownian motion and an independent fractional Brownian motion.

Backward stochastic differential equations and Feynman-Kac formula for Levy processes, with applications in finance

Abstract: In this paper we show the existence and uniqueness of a solution for backward stochastic differential equations driven by a Levy process with moments of all orders. The results are important from a pure mathematical point of view as well as in the world of finance: an application to Clark-Ocone and Feynman-Kac formulas for Levy processes is presented. Moreover, the Feynman-Kac formula and the related partial differential integral equation provide an analogue of the famous Black-Scholes partial differential equation and thus can be used for the purpose of option pricing in a Levy market.

Are classes of deterministic integrands for fractional Brownian motion on an interval complete

Abstract: Let BH be a fractional Brownian motion with self-similarity parameter H∈ (0,1) and a>0 be a fixed real number. Consider the integral ∈t0a f(u)\rm dBH(u), where f belongs to a class of non-random integrands ΛH,a. The integral will then be defined in the L2(Ω) sense. One would like ΛH,a to be a complete inner-product space. This corresponds to a desirable situation because then there is an isometry between ΛH,a and the closure of the span generated by BH(u), 0≤ u≤ a. We show in this work that, when H∈(½,1), the classes of integrands ΛH,a one usually considers are not complete inner-product spaces even though they are often assumed in the literature to be complete. Thus, they are isometric not to overline{\mbox{sp}}\{BH(u), 0≤ u≤ a\} but only to a proper subspace. Consequently, there are (random) elements in that closure which cannot be represented by functions f in ΛH,a. We also show, in contrast to the case H∈ (½,1), that there is a class of integrands for fractional Brownian motion BH with H∈ (0,½) on an interval [0,a] which is a complete inner-product space.

Asymptotics of the maximum likelihood estimator for general hidden Markov models

Abstract: In this paper, we consider the consistency and asymptotic normality of the maximum likelihood estimator for a possibly non-stationary hidden Markov model where the hidden state space is a separable and compact space not necessarily finite, and both the transition kernel of the hidden chain and the conditional distribution of the observations depend on a parameter θ. For identifiable models, consistency and asymptotic normality of the maximum likelihood estimator are shown to follow from exponential memorylessness properties of the state prediction filter and geometric ergodicity of suitably extended Markov chains.

Cahn-Hilliard stochastic equation: existence of the solution and of its density

Abstract: We show the existence and uniqueness of a function-valued process solution to the stochastic Cahn-Hilliard equation driven by space-time white noise with a nonlinear diffusion coefficient. Then we show that the solution is locally differentiable in the sense of the Malliavin calculus, and, under some non-degeneracy condition on the diffusion coefficient, that the law of the solution is absolutely continuous with respect to Lebesgue measure.

Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach

Abstract: We address the problem of the validity of the local asymptotic mixed normality (LAMN) property when the model is a multidimensional diffusion process X whose coefficients depend on a scalar parameter θ: the sample (Xk/n)0≤ k≤ n corresponds to an observation of X at equidistant times in the interval [0,1]. We prove that the LAMN property holds true for the likelihood under an ellipticity condition and some suitable smoothness assumptions on the coefficients of the stochastic differential equation. Our method is based on Malliavin calculus techniques: in particular, we derive for the log-likelihood ratio a tractable representation involving conditional expectations.

Mixed shock models

Abstract: Traditionally, shock models are of two kinds. The failure (of a system) is related either to the cumulative effect of a (large) number of shocks or it is caused by a shock which is larger than some critical level. The present paper is devoted to a mixed model, in which the system is supposed to break down either because of one (very) large shock, or as a result of many smaller ones.

Density estimation for spatial linear processes

Abstract: The problem of estimating the marginal densities of a spatial linear process, observed over a grid of $\mathbb {Z}^N$, is considered. Under general conditions, kernel density estimators computed at any $k$-tuple of sites are shown to be asymptotically multivariate normal. Their limiting covariance matrix is also computed. Despite the huge development of nonparametric estimation methods in the analysis of time series data, little has so far been done to introduce them into the context of random fields. The generalization indeed is far from trivial since the points of $\mathbb {Z}^N$ do not have a natural ordering when $N>1$. No mixing conditions are required, but linearity is assumed.

Generalized neyman-pearson lemma via convex duality ⁄

Abstract: We extend the classical Neyman-Pearson theory for testing composite hypotheses versus composite alternatives, using a convex duality approach, first employed by Witting. Results of Aubin and Ekeland from non-smooth convex analysis are used, along with a theorem of Koml6s, in order to establish the existence of a max-min optimal test in considerable generality, and to investigate its properties. The theory is illustrated on representative examples involving Gaussian measures on Euclidean and Wiener space.

On the second-order characteristics of marked point processes

Abstract: This paper aims to combine the central limit theorem with the limit theorems in extreme value theory through a parametrized class of limit theorems where the former ones appear as special cases. To this end the limit distributions of suitably centered and normalized $l_{cp(n)}$-norms of $n$-vectors of positive i.i.d. random variables are investigated. Here, $c$ is a positive constant and $p(n)$ is a sequence of positive numbers that is given intrinsically by the form of the upper tail behavior of the random variables. A family of limit distributions is obtained if $c$ runs over the positive real axis. The normal distribution and the extreme value distributions appear as the endpoints of these families, namely, for $c =0 +$ and $c = \infty$, respectively.

On Gaussian and Bernoulli covariance representations

Abstract: We discuss several applications, to large deviations for smooth functions of Gaussian random vectors, of a covariance representation in Gauss space. The existence of this type of representation characterizes Gaussian measures. New representations for Bernoulli measures are also derived, recovering some known inequalities.

Local polynomial estimation with a FARIMA-GARCH error process

Abstract: This paper considers estimation of the trend function $g$ as well as its $ u$th derivative $g^{( u)}$ in a so-called semi-parametric FARIMA-GARCH model by local polynomial fits. The focus is on the derivation of the asymptotic normality of $\hat g^{( u)}$. A central limit theorem based on martingale theory is developed. Asymptotic normality of the sample mean of a FARIMA-GARCH process is proved. These results are then used to show the asymptotic normality of $\hat g^{( u)}$. As an auxiliary result, the weak consistency of a weighted sum is obtained for second-order stationary time series with short or long memory under very weak conditions. Formulae for the mean integrated square error and the asymptotically optimal bandwidth of $\hat g^{( u)}$ are also given..

Confidence intervals for the tail index

Abstract: One of the best-known estimators for the tail index of a heavy-tailed distribution is the Hill estimator. In this paper, confidence intervals based on the asymptotic normal approximation of the Hill estimator are studied. The coverage accuracy is evaluated and the theoretical optimal choice of the sample fraction for the one-sided confidence interval is given. One surprising finding is that the order of optimal coverage accuracy for the one-sided confidence interval depends on the sign of the second-order regular variation.

Adaptive estimation of the spectrum of a stationary Gaussian sequence

Abstract: In this paper, we study the problem of nonparametric adaptive estimation of the spectral density f of a stationary Gaussian sequence. For this purpose, we consider a collection of finite-dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on possibly irregular grids or spaces of trigonometric polynomials). We estimate the spectral density by a projection estimator based on the periodogram and constructed on a data-driven choice of linear space from the collection. This data-driven choice is made via the minimization of a penalized projection contrast. The penalty function depends on If In,, but we give results including the estimation of this bound. Moreover, we give extensions to the case of unbounded spectral densities (long-memory processes). In all cases, we state non-asymptotic risk bounds in L2-norm for our estimator, and we show that it is adaptive in the minimax sense over a large class of Besov balls.

The Riemann zeta distribution

Abstract: Let $\zeta$ be the Riemann zeta function. Khinchine (1938) proved that the function $f_\sigma(t)=\zeta(\sigma + $i $t)/\zeta(\sigma)$, where $\sigma > 1$ and $t$ is real, is an infinitely divisible characteristic function. We investigate further the fundamental properties of the corresponding distribution of $f_\sigma$, the Riemann zeta distribution, including its support and unimodality. In particular, the Riemann zeta random variable is represented as a linear function of infinitely many independent geometric random variables. To extend Khinchine's result, we construct the Dirichlet-type characteristic functions of discrete distributions and provide a sufficient condition for the infinite divisibility of these characteristic functions. By way of applications, we give probabilistic proofs for some identities in number theory, including a new identity for the reciprocal of the Riemann zeta function.

Remarks on the maximum correlation coefficient

Abstract: The maximum correlation coefficient between partial sums of independent and identically distributed random variables with finite second moment equals the classical (Pearson) correlation coefficient between the sums, and thus does not depend on the distribution of the random variables. This result is proved, and relations between the linearity of regression of each of two random variables on the other and the maximum correlation coefficient are discussed.

Optimal harvesting from interacting populations in a stochastic environment

Abstract: Consider n populations whose sizes are given by stochastic differential equations driven by m-dimensional Brownian motion. We study the following problem: what harvesting strategy from the n populations maximizes the expected total income from the harvest? We formulate this as a (singular) stochastic control problem and give sufficient conditions for the existence of an optimal strategy. Our results lead to the one-at-a-time principle that it is almost surely never optimal to harvest from more than one population at a time.

A generalized class of Lyons-Zheng processes

Abstract: Generalizing work by Lyons and Zheng, we study Dirichlet processes admitting a decomposition into the sum of a forward and a backward local martingale plus a bounded variation process. We develop a framework of stochastic calculus for these processes and deal with existence and uniqueness for stochastic differential equations driven by such processes. In particular, Bessel processes turn out to be an interesting example of Lyons-Zheng processes.

Steady State Distribution of the Buffer Content for M/G/infinity Input Fluid Queues

Abstract: We consider a fluid queue with on periods initiated by a Poisson process and having a long-tailed distribution This queue has long-range dependence, and we compute the asymptotic behaviour of the steady-state distribution of the buffer content The tail of this distribution is much heavier than the tail of the buffer content distribution of a queue which does not possess long-range dependence and which has light-tailed on periods and the same traffic intensity

On the effect of covariance function estimation on the accuracy of kriging predictors

Abstract: The kriging procedure gives an optimal linear predictor of a spatial process at a point x0, given observations of the process at other locations x1,...,xn, taking into account the spatial dependence of the observations. The kriging predictor is optimal if the weights are calculated from the correct underlying covariance structure. In practice, this covariance structure is unknown and is estimated from the data. An important, but not very well understood, problem in kriging theory is the effect on the accuracy of the kriging predictor of substituting the optimal weights by weights derived from the estimated covariance structure. We show that the effect of estimation is negligible asymptotically if the joint Gaussian distributions of the process at x0,...,xn under the true and the estimated covariance are contiguous almost surely. We consider a number of commonly used parametric covariance models where this can indeed be achieved.

Markov chain Monte Carlo estimation of the law of the mean of a Dirichlet process

Abstract: a support which is not doubly infinite. We use this to study an approximation procedure for /&a, and evaluate the approximation error in simulating /a using this chain. We include examples for a comparison with some of the existing procedures for approximating &a, and show that the Markov chain approximation compares well with other methods.

Testing additivity by kernel-based methods - what is a reasonable test?

Abstract: In the common nonparametric regression model with high dimensional predictor several tests for the hypothesis of an additive regression are investigated. The corresponding test statistics are either based on the diiferences between a fit under the assumption of additivity and a fit in the general model or based on residuals under the assumption of additivity. For all tests asymptotic normality is established under the null hypothesis of additivity and under fixed alternatives with different rates of convergence corresponding to both cases. These results are used for a comparison of the different methods. It is demonstrated that a statistic based on an empirical L1 - distance of the Nadaraya Watson and the marginal integration estimator yields the asymptotically most efficient procedure if these are compared with respect to the asymptotic behaviour under fixed alternatives.

Nonparametric regression with dependent errors

Abstract: We study minimax rates of convergence for nonparametric regression under a random design with dependent errors. It is shown that when the errors are independent of the explanatory variables, long-range dependence among the errors does not necessarily hurt regression estimation, which at first glance contradicts with earlier results by Hall and Hart, Wang, and Johnstone and Silverman under a fixed design. In fact we show that, in general, the minimax rate of convergence under the square $L_2$ loss is simply at the worse of two quantities: one determined by the massiveness of the class alone and the other by the severity of the dependence among the errors alone. The clear separation of the effects of the function class and dependence among the errors in determining the minimax rate of convergence is somewhat surprising. Examples of function classes under different covariance structures including both short- and long-range dependences are given.

Adaptive estimation of the fractional differencing coefficient

Abstract: Semi-parametric estimation of the fractional differencing coefficient d of a long-range dependent stationary time series has received substantial attention in recent years. Some of the so-called local estimators introduced early on were proved rate-optimal over relevant classes of spectral densities. The rates of convergence of these estimators are limited to n2/5, where n is the sample size. This paper focuses on the fractional exponential (FEXP) or broadband estimator of d. Minimax rates of convergence over classes of spectral densities which are smooth outside the zero frequency are obtained, and the FEXP estimator is proved rate-optimal over these classes. On a certain functional class which contains the spectral densities of FARIMA processes, the rate of convergence of the FEXP estimator is (n/log(n))1/2, thus making it a reasonable alternative to parametric estimators. As usual in semiparametric estimation problems, these rate-optimal estimators are infeasible, since they depend on an unknown smoothness parameter defining the functional class. A feasible adaptive version of the broadband estimator is constructed. It is shown that this estimator is minimax rate-optimal up to a factor proportional to the logarithm of the sample size.

On large deviations in the Gaussian autoregressive process: stable, unstable and explosive cases

Abstract: For the Gaussian autoregressive process, the asymptotic behaviour of the Yule‐Walker estimator is totally different in the stable, unstable and explosive cases. We show that, irrespective of this trichotomy, this estimator shares quite similar large deviation properties in the three situations. However, in the explosive case, we obtain an unusual rate function with a discontinuity point at its minimum.

A mixture of the exclusion process and the voter model

Abstract: We consider a one-dimensional nearest-neighbour interacting particle system, which is a mixture of the simple exclusion process and the voter model. The state space is taken to be the countable set of the configurations that have a finite number of particles to the right of the origin and a finite number of empty sites to the left of it. We obtain criteria for the ergodicity and some other properties of this system using the method of Lyapunov functions.

A new fluctuation identity for Levy processes and some applications

Abstract: Let τ and H be respectively the ladder time and ladder height processes associated with a given Levy process X. We give an identity in law between (τ,H) and (X,H*), H* being the right-continuous inverse of the process H. This allows us to obtain a relationship between the entrance law of X and the entrance law of the excursion measure away from 0 of the reflected process (Xt- infs≤tXs, t ≥0). In the stable case, some explicit calculations are provided. These results also lead to an explicit form of the entrance law of the Levy process conditioned to stay positive.

Martingale convergence and the functional equation in the multi-type branching random walk

Abstract: A generalization of Biggins Martingale Convergence Theorem is proved for the multitype branching random walk The proof appeals to modern techniques involving the construction of sizebiased measures on the space of marked trees generated by the branching process As a simple conse quence we obtain existence and uniqueness of solutions within a specied class to a system of functional equations

Analysis of change-point estimators under the null hypothesis

Abstract: We consider estimators for the change-point in a sequence of independent observations. These are defined as the maximizing points of weighted U-statistic type processes. Our investigations focus on the behaviour of the estimators in the case of independent and identically distributed random variables (null hypothesis of no change), but contiguous alternatives in the sense of Oosterhoff and van Zwet are also taken into account. If the weight functions belong to the Chibisov-O'Reilly class we derive convergence in distribution, including a special Berry-Esseen result. The limit variable is the almost sure unique maximizing point of a weighted (standard or reflected) Brownian bridge with drift. For general weight functions the limiting null distribution is analytically not known. However, in the special case where no weight functions are involved it is known that the maximizer of a standard Brownian bridge is uniformly distributed on the unit interval. A corresponding result for the reflected Brownian bridge seems to be unknown in the literature. In this paper we fill this gap and actually compute the common density of the maximum and its location for a reflected Brownian bridge. From this one can find the density of the maximizer, which analytically can be expressed in terms of a series. In a special case even the finite sample size distribution of our estimator is established. Besides distributional results, we also determine the almost sure set of cluster points.