# Stochastic Processes and their Applications

Published by Elsevier

Print ISSN: 0304-4149

Published by Elsevier

Print ISSN: 0304-4149

Publications

We investigate the high resolution quantization and entropy coding problem for solutions of stochastic differential equations under Lp[0,1]-norm distortion. We find explicit high resolution formulas in terms of the average diffusion coefficient seen by the process. The proof is based on a decoupling method introduced in a former article by the author. Given that link it remains to analyze the coding problem for a concatenation of Wiener processes and to solve the corresponding rate allocation problem.

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Recent optimal scaling theory has produced a condition for the asymptotically optimal acceptance rate of Metropolis algorithms to be the well-known 0.234 when applied to certain multi-dimensional target distributions. These d-dimensional target distributions are formed of independent components, each of which is scaled according to its own function of d. We show that when the condition is not met the limiting process of the algorithm is altered, yielding an asymptotically optimal acceptance rate which might drastically differ from the usual 0.234. Specifically, we prove that as d→∞ the sequence of stochastic processes formed by say the i∗th component of each Markov chain usually converges to a Langevin diffusion process with a new speed measure υ, except in particular cases where it converges to a one-dimensional Metropolis algorithm with acceptance rule α∗. We also discuss the use of inhomogeneous proposals, which might prove to be essential in specific cases.

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We show that under mild conditions the joint densities Px1,...,xn) of the general discrete time stochastic process Xn on can be computed via Px1,...,xn(x1,...,xn) = ||[phi]T(x1)...T(xn)||2 where [phi] is in a Hilbert space , and T (x), x [epsilon] are linear operators on . We then show how the Central Limit Theorem can easily be derived from such representations.

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We consider the free boundary condition Gibbs measure of the Potts model on a random tree. We provide an explicit temperature interval below the ferromagnetic transition temperature for which this measure is extremal, improving older bounds of Mossel and Peres. In information theoretic language extremality of the Gibbs measure corresponds to non-reconstructability for symmetric q-ary channels. The bounds for the corresponding threshold value of the inverse temperature are optimal for the Ising model and differ from the Kesten Stigum bound by only 1.50% in the case q=3 and 3.65% for q=4, independently of d. Our proof uses an iteration of random boundary entropies from the outside of the tree to the inside, along with a symmetrization argument.

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COGARCH is an extension of the GARCH time series concept to continuous time, which has been suggested by Klüppelberg, Lindner and Maller [C. Klüppelberg, A. Lindner, R. Maller, A continuous-time GARCH process driven by a Lévy process: Stationarity and second order behaviour, Journal of Applied Probability 41 (2004) 601-622]. We show that any COGARCH process can be represented as the limit in law of a sequence of GARCH(1,1) processes. As a by-product we derive the infinitesimal generator of the bivariate Markov process representation of COGARCH. Moreover, we argue heuristically that COGARCH and the classical bivariate diffusion limit of Nelson [D. Nelson, ARCH models as diffusion approximations, Journal of Econometrics 45 (1990) 7-38] are probably the only continuous-time limits of GARCH.

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The squares of a GARCH(p,q) process satisfy an ARMA equation with white noise innovations and parameters which are derived from the GARCH model. Moreover, the noise sequence of this ARMA process constitutes a strongly mixing stationary process with geometric rate. These properties suggest to apply classical estimation theory for stationary ARMA processes. We focus on the Whittle estimator for the parameters of the resulting ARMA model. Giraitis and Robinson (2000) show in this context that the Whittle estimator is strongly consistent and asymptotically normal provided the process has finite 8th moment marginal distribution. We focus on the GARCH(1,1) case when the 8th moment is infinite. This case corresponds to various real-life log-return series of financial data. We show that the Whittle estimator is consistent as long as the 4th moment is finite and inconsistent when the 4th moment is infinite. Moreover, in the finite 4th moment case rates of convergence of the Whittle estimator to the true parameter are the slower, the fatter the tail of the distribution. These findings are in contrast to ARMA processes with iid innovations. Indeed, in the latter case it was shown by Mikosch et al. (1995) that the rate of convergence of the Whittle estimator to the true parameter is the faster, the fatter the tails of the innovations distribution. Thus the analogy between a squared GARCH process and an ARMA process is misleading insofar that one of the classical estimation techniques, Whittle estimation, does not yield the expected analogy of the asymptotic behavior of the estimators.

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In this paper we present a central limit theorem for general functions of the increments of Brownian semimartingales. This provides a natural extension of the results derived in [O.E. Barndorff-Nielsen, S.E. Graversen, J. Jacod, M. Podolskij, N. Shephard, A central limit theorem for realised power and bipower variations of continuous semimartingales, in: From Stochastic Analysis to Mathematical Finance, Festschrift for Albert Shiryaev, Springer, 2006], where the central limit theorem was shown for even functions. We prove an infeasible central limit theorem for general functions and state some assumptions under which a feasible version of our results can be obtained. Finally, we present some examples from the literature to which our theory can be applied.

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This paper describes the limiting behaviour of tail empirical processes associated with long memory stochastic volatility models. We show that such a process has dichotomous behaviour, according to an interplay between the Hurst parameter and the tail index. On the other hand, the tail empirical process with random levels never suffers from long memory. This is very desirable from a practical point of view, since such a process may be used to construct the Hill estimator of the tail index. To prove our results we need to establish new results for regularly varying distributions, which may be of independent interest.

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We derive a functional central limit theorem for the empirical spectral measure or discretely averaged (integrated) periodogram of a multivariate long range dependent stochastic process in a degenerating neighborhood of the origin. We show that, under certain restrictions on the memory parameters, this local empirical spectral measure converges weakly to a Gaussian process with independent increments. Applications to narrow-band frequency domain estimation in time series regression with long range dependence, and to local (to the origin) goodness-of-fit testing are offered.

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The purpose of this note is to correct an error in Baltrunas etÂ al. (2004)Â [1], and to give a more detailed argument to a formula whose validity has been questioned over the years. These details close a gap in the proof of TheoremÂ 4.1 as originally stated, the validity of which is hereby strengthened.

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The generalised Ornstein-Uhlenbeck process constructed from a bivariate Lévy process ([xi]t,[eta]t)t[greater-or-equal, slanted]0 is defined aswhere V0 is an independent starting random variable. The stationarity of the process is closely related to the convergence or divergence of the Lévy integral . We make precise this relation in the general case, showing that the conditions are not in general equivalent, though they are for example if [xi] and [eta] are independent. Characterisations are expressed in terms of the Lévy measure of ([xi],[eta]). Conditions for the moments of the strictly stationary distribution to be finite are given, and the autocovariance function and the heavy-tailed behaviour of the stationary solution are also studied.

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Motivated by recent studies in financial mathematics and other areas, we investigate the exponential functional of a Lévy process X(t),t[greater-or-equal, slanted]0. In particular, we investigate its tail asymptotics. We show that, depending on the right tail of X(1), the tail behavior of Z is exponential, Pareto, or extremely heavy-tailed.

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Stochastic variables associated to a single-server queueing system with finite population are shown to weakly converge, on some time regions, to Gaussian processes, Brownian motions or stochastic integrals on such when the population size increases. Queue length, unfinished work, storage occupied (in a computer system model) and idle time show different limiting behaviour, depending on the arrival and service distribution.

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We give a new characterization for the convergence in distribution to a standard normal law of a sequence of multiple stochastic integrals of a fixed order with variance one, in terms of the Malliavin derivatives of the sequence. We also give a new proof of the main theorem in [D. Nualart, G. Peccati, Central limit theorems for sequences of multiple stochastic integrals, Ann. Probab. 33 (2005) 177–193] using techniques of Malliavin calculus. Finally, we extend our result to the multidimensional case and prove a weak convergence result for a sequence of square integrable random vectors, giving an application.

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We obtain the rate of growth of long strange segments and the rate of decay of infinite horizon ruin probabilities for a class of infinite moving average processes with exponentially light tails. The rates are computed explicitly. We show that the rates are very similar to those of an i.i.d. process as long as the moving average coefficients decay fast enough. If they do not, then the rates are significantly different. This demonstrates the change in the length of memory in a moving average process associated with certain changes in the rate of decay of the coefficients.

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We prove that for any $\alpha$-mixing stationnary process the hitting time of any $n$-string $A_n$ converges, when suitably normalized, to an exponential law. We identify the normalization constant $\lambda(A_n)$. A similar statement holds also for the return time. To establish this result we prove two other results of independent interest. First, we show a relation between the rescaled hitting time and the rescaled return time, generalizing a theorem by Haydn, Lacroix and Vaienti. Second, we show that for positive entropy systems, the probability of observing any $n$-string in $n$ consecutive observations, goes to zero as $n$ goes to infinity.

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We introduce a broad class of self-similar processes $\{Z(t),t\ge 0\}$ called
generalized Hermite process. They have stationary increments, are defined on a
Wiener chaos with Hurst index $H\in (1/2,1)$, and include Hermite processes as
a special case. They are defined through a homogeneous kernel $g$, called
"generalized Hermite kernel", which replaces the product of power functions in
the definition of Hermite processes. The generalized Hermite kernels $g$ can
also be used to generate long-range dependent stationary sequences forming a
discrete chaos process $\{X(n)\}$. In addition, we consider a
fractionally-filtered version $Z^\beta(t)$ of $Z(t)$, which allows $H\in
(0,1/2)$. Corresponding non-central limit theorems are established. We also
give a multivariate limit theorem which mixes central and non-central limit
theorems.

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We study the structure of point processes N with the property that the vary in a finite-dimensional space where [theta]t is the shift and the [sigma]-field generated by the counting process up to time t. This class of point processes is strictly larger than Neuts' class of Markovian arrival processes. On the one hand, it allows for more general features like interarrival distributions which are matrix-exponential rather than phase type, on the other the probabilistic interpretation is a priori less clear. Nevertheless, the properties are very similar. In particular, finite-dimensional distributions of interarrival times, moments, Laplace transforms, Palm distributions, etc., are shown to be given by two fundamental matrices C, D just as for the Markovian arrival process. We also give a probabilistic interpretation in terms of a piecewise deterministic Markov process on a compact convex subset of , whose jump times are identical to the epochs of N.

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The asymptotic distribution of the maximum Mn=max1⩽t⩽nξt in a stationary normal sequence ξ1,ξ,… depends on the correlation rt between ξ0 and ξt. It is well known that if rt log t → 0 as t → ∞ or if Σr2t<∞, then the limiting distribution is the same as for a sequence of independent normal variables. Here it is shown that this also follows from a weaker condition, which only puts a restriction on the number of t-values for which rt log t islarge. The condition gives some insight into what is essential for this asymptotic behaviour of maxima. Similar results are obtained for a stationary normal process in continuous time.

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The Silences of the Archives, the Reknown of the Story.
The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, before captivated judges and a guilty feeling Bertrande de Rols. The historian wanted to go beyond the known story in order to discover the world of the heroes. This research led to disappointments and surprizes as documents were discovered concerning the environment of Artigat’s inhabitants and bearing directly on the main characters thanks to notarial contracts. Along the way, study of the works of Coras and Lesueur took a new direction. Coming back to the affair a quarter century later did not result in finding new documents (some are perhaps still buried in Spanish archives), but by going back over her tracks, the historian could only be struck by the silences of the archives that refuse to reveal their secrets and, at the same time, by the possible openings they suggest, by the intuition that almost invisible threads link here and there characters and events.

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We study the problem of estimating autoregressive parameters when the observations are from an AR process with innovations in the domain of attraction of a stable law. We show that non-degenerate limit laws exist for M-estimates if the loss function is sufficiently smooth; these results remain valid if location and scale are also estimated. For least absolute deviation (LAD) estimates, similar results hold under conditions on the innovations distribution near 0. We also discuss, under moment conditions on the innovations, consistency properties for M-estimators corresponding to the class of loss functions, ϱ(x) = |x |γ for some γ > 0.

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Consider the nonparametric estimation of a multivariate regression function and its derivatives for a regression model with long-range dependent errors. We adopt local linear fitting approach and establish the joint asymptotic distributions for the estimators of the regression function and its derivatives. The nature of asymptotic distributions depends on the amount of smoothing resulting in possibly non-Gaussian distributions for large bandwidth and Gaussian distributions for small bandwidth. It turns out that the condition determining this dichotomy is different for the estimates of the regression function than for its derivatives; this leads to a double bandwidth dichotomy whereas the asymptotic distribution for the regression function estimate can be non-Gaussian whereas those of the derivatives estimates are Gaussian. Asymptotic distributions of estimates of derivatives in the case of large bandwidth are the scaled version of that for estimates of the regression function, resembling the situation of estimation of cumulative distribution function and densities under long-range dependence. The borderline case between small and large bandwidths is also examined.

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Sufficient conditions are given for linear processes and ARMA processes to have the Gaswirth and Rubin mixing condition. The mixing rates are also determined.

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We show that sufficient conditions in terms of moments for cumulative processes (additive functionals of regenerative processes) to satisfy the central limit theorem and the weak law of large numbers established in Glynn and Whitt (Stochastic Process. Appl. 47 (1993) 299-314) are also necessary, as previously conjectured.

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The paper obtains a functional limit theorem for the empirical process of a stationary moving average process Xt with i.i.d. innovations belonging to the domain of attraction of a symmetric [alpha]-stable law, 1<[alpha]<2, with weights bj decaying as j-[beta], 1<[beta]<2/[alpha]. We show that the empirical process (normalized by N1/[alpha][beta]) weakly converges, as the sample size N increases, to the process cx+L++cx-L-, where L+,L- are independent totally skewed [alpha][beta]-stable random variables, and cx+,cx- are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(Xs) is an [alpha][beta]-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/[alpha]<[beta]<1 and 2/[alpha]<[beta] studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.

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Letting the initial condition of a PDE be random is interesting when considering complex phenomena. For 2D-Navier-Stokes equations, it is for instance an attempt to take into account the turbulence arising with high velocities and low viscosities. The solutions of these PDEs are random and their laws are called statistical solutions. We start by studying McKean-Vlasov equations with initial conditions parameterized by a real random variable [theta], and link their weak measure solutions to the laws of nonlinear SDEs, for which the drift coefficients are expressed as conditional expectations in the diffusions' laws given [theta]. We propose an original stochastic particle method to compute the first-order moments of the statistical solutions, obtained by approximating the conditional expectations by wavelet regression estimators. We establish a convergence rate that improves the ones obtained for existing methods with Nadaraya-Watson kernel estimators. We then carry over these results to 2D-Navier-Stokes equations and compute some physical quantities of interest, like the mean velocity vector field. Numerical simulations illustrate the method and allow us to test its robustness.

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We consider stochastic Navier-Stokes equations in a 2D-bounded domain with
the Navier with friction boundary condition. We establish the existence and the
uniqueness of the solutions and study the vanishing viscosity limit. More
precisely, we prove that solutions of stochastic Navier-Stokes equations
converge, as the viscosity goes to zero, to solutions of the corresponding
stochastic Euler equations.

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In this paper one specifies the ergodic behavior of the 2D-stochastic Navier-Stokes equation by giving a Large Deviation Principle for the occupation measure for large time. It describes the exact rate of exponential convergence. The considered random force is non-degenerate and compatible with the strong Feller property.

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We reinvestigate the 2D problem of the inhomogeneous incipient infinite cluster where, in an independent percolation model, the density decays to pc with an inverse power, [lambda], of the distance to the origin. Assuming the existence of critical exponents (as is known in the case of the triangular site lattice) if the power is less than 1/[nu], with [nu] the correlation length exponent, we demonstrate an infinite cluster with scale dimension given by DH=2-[beta][lambda]. Further, we investigate the critical case [lambda]c=1/[nu] and show that iterated logarithmic corrections will tip the balance between the possibility and impossibility of an infinite cluster.

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By adapting the renormalization techniques of Pisztora (Probab. Theory Relat. Fields 104 (1996) 427), we establish surface order large deviations estimates for FK-percolation on with parameter q[greater-or-equal, slanted]1 and for the corresponding Potts models. Our results are valid up to the exponential decay threshold of dual connectivities which is widely believed to agree with the critical point.

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We study a natural dependent percolation model introduced by Häggström. Consider subcritical Bernoulli bond percolation with a fixed parameter p<pc. We define a dependent site percolation model by the following procedure: for each bond cluster, we colour all vertices in the cluster black with probability r and white with probability 1−r, independently of each other. On the square lattice, defining the critical probabilities for the site model and its dual, rc(p) and respectively, as usual, we prove that for all subcritical p. On the triangular lattice, where our method also works, this leads to rc(p)=1/2, for all subcritical p. On both lattices, we obtain exponential decay of cluster sizes below rc(p), divergence of the mean cluster size at rc(p), and continuity of the percolation function in r on [0,1]. We also discuss possible extensions of our results, and formulate some natural conjectures. Our methods rely on duality considerations and on recent extensions of the classical RSW theorem.

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The paper is concerned with the existence and uniqueness of a strong solution
to a two-dimensional backward stochastic Navier-Stokes equation with nonlinear
forcing, driven by a Brownian motion. We use the spectral approximation and the
truncation and variational techniques. The methodology features an interactive
analysis on basis of the regularity of the deterministic Navier-Stokes dynamics
and the stochastic properties of the It\^o-type diffusion processes.

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The backward two-dimensional stochastic Navier–Stokes equations (BSNSEs, for short) with suitable perturbations are studied in this paper, over bounded domains for incompressible fluid flow. A priori estimates for adapted solutions of the BSNSEs are obtained which reveal a pathwise L∞(H) bound on the solutions. The existence and uniqueness of solutions are proved by using a monotonicity argument for bounded terminal data. The continuity of the adapted solutions with respect to the terminal data is also established.

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In this paper we consider critical site percolation on the triangular lattice
in the upper half-plane. Let $u_1, u_2$ be two sites on the boundary and $w$ a
site in the interior of the half-plane. It was predicted by Simmons, Kleban and
Ziff in a paper from 2007 that the ratio $\mathbb{P}(nu_1 \leftrightarrow nu_2
\leftrightarrow nw)^{2}\,/\,\mathbb{P}(nu_1 \leftrightarrow
nu_2)\cdot\mathbb{P}(nu_1 \leftrightarrow nw)\cdot\mathbb{P}(nu_2
\leftrightarrow nw)$ converges to $K_F$ as $n \to \infty$, where
$x\leftrightarrow y$ means that $x$ and $y$ are in the same open cluster, and
$K_F$ is an explicitly known constant. Beliaev and Izyurov proved in a paper in
2012 an analog of this factorization in the scaling limit. We prove, using
their result and a quite general coupling argument, the earlier mentioned
prediction. Furthermore we prove a factorization formula for the probability
$\mathbb{P}(nu_2 \leftrightarrow [nu_1,nu_1+s];\, nw \leftrightarrow
[nu_1,nu_1+s])$, where $s>0$.

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The motion of a finite number of point vortices on a two-dimensional periodic domain is considered. In the deterministic case it is known to be well posed only for almost every initial configuration. Coalescence of vortices may occur for certain initial conditions. We prove that when ageneric stochastic perturbation compatible with the Eulerian description is introduced, the point vortex motion becomes well posed for every initial configuration, in particular coalescence disappears.

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We study versions of the contact process with two or three states, and with
infections occurring at a rate depending on the infection density. Motivated by
a model for vegetation patterns in arid landscapes, we focus on percolation
under invariant measures of such processes. We prove that the percolation
transition is \emph{sharp} (for one of the three-state processes we consider
this requires a reasonable assumption). This is shown to contradict a form of
`robust critical behaviour' (with power law cluster size distribution for a
range of parameter values) suggested in [13].

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We derive "quenched" subdiffusive lower bounds for the exit time tau(n) from
a box of size n for the simple random walk on the planar invasion percolation
cluster. The first part of the paper is devoted to proving an almost sure
analog of H. Kesten's subdiffusivity theorem for the random walk on the
incipient infinite cluster and the invasion percolation cluster using ideas of
M. Aizenman, A. Burchard and A. Pisztora. The proof combines lower bounds on
the instrinsic distance in these graphs and general inequalities for reversible
Markov chains. In the second part of the paper, we present a sharpening of
Kesten's original argument, leading to an explicit almost sure lower bound for
tau(n) in terms of percolation arm exponents. The methods give tau(n) \geq
n^{2+epsilon_0 + kappa}, where epsilon_0>0 depends on the instrinsic distance
and (assuming the exact value of the backbone exponent) kappa can be taken to
be 17/384 on the hexagonal lattice.

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A uniqueness result is proven for the infinitesimal generator associated with the 2D Euler flow with periodic boundary conditions in the space L2([mu]) with respect to the natural Gibbs measure [mu] given by the enstrophy. This result remains true for the generator of the stochastic process associated with a 2D Navier-Stokes equation perturbed by a space-time Gaussian white noise force. The corresponding Liouville operator N defined on the space of smooth cylinder bounded functions has a unique skew-adjoint m-dissipative extension in the class of closed operators in L2([mu])xV' where .

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Let $N$ be a positive integer. We consider pseudo-Brownian motion $X=(X(t))_{t\ge 0}$ driven by the high-order heat-type equation $∂/∂t=(-1)^{N-1}∂^{2N}/∂x^{2N}$. Let us introduce the first exit time {\tau}ab from a bounded interval $(a,b)$ by $X$ ($a,b\in\mathbb{R}$). In this paper, we provide a representation of the joint pseudo-distribution of the vector $(\tau_{ab},X(\tau_{ab}))$ by means of Vandermonde-like determinants. The method we use is based on the Feynman-Kac functional related to pseudo-Brownian motion which leads to a boundary value problem. In particular, the pseudo-distribution of the location of $X$ at time $\tau_{ab}$, namely X(\tau_{ab})$, admits a fine expression involving famous Hermite interpolating polynomials.

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We study the distribution of the stochastic integral [integral operator]0t8e-Rt dPt where R is a Brownian motion with positive drift and P is an independent compound Poisson process. We show that in the special case when the jumps of P are exponentially distributed, the integral has the same distribution as that of a gamma variable divided by an independent beta variable.

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We consider partial sums S-n = X-1 + X-2 + ... + X-n, n is an element of N, of i.i.d. random variables with moments E(X-1) = 0, E(X-1(2)) = sigma(2) and sup{t is an element of R:E(exp((t\X-1\)(1/p)sgn(X-1)) < infinity} is an element of (0, infinity) and show that lim(n-->infinity){ma(x0 less than or equal to j<n) max(1 less than or equal to k less than or equal to n-j) Sj+k-S-j/phi(k/(log n)(2p-1))(log n)(p)} = 1 a.s. with some explicit function rp(). A related result for random variables with exponentially thin tails has recently been shown by Steinebach, extending a result given by Shao. (C) 2000 Elsevier Science B.V. All rights reserved. MSC. Primary 60 F 15; Secondary 60 G 70.

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This paper obtains a uniform reduction principle for the empirical process of a stationary moving average time series {Xt} with long memory and independent and identically distributed innovations belonging to the domain of attraction of symmetric [alpha]-stable laws, 1<[alpha]<2. As a consequence, an appropriately standardized empirical process is shown to converge weakly in the uniform-topology to a degenerate process of the form f Z, where Z is a standard symmetric [alpha]-stable random variable and f is the marginal density of the underlying process. A similar result is obtained for a class of weighted empirical processes. We also show, for a large class of bounded functions h, that the limit law of (normalized) sums [summation operator]s=1nh(Xs) is symmetric [alpha]-stable. An application of these results to linear regression models with moving average errors of the above type yields that a large class of M-estimators of regression parameters are asymptotically equivalent to the least-squares estimator and [alpha]-stable. This paper thus extends various well-known results of Dehling-Taqqu and Koul-Mukherjee from finite variance long memory models to infinite variance models of the above type.

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We study the three-dimensional stochastic Navier-Stokes equations with additive white noise, in the context of spatially homogeneous solutions in , i.e. solutions with a law invariant under space translations. We prove the existence of such a solution, with the additional property of being suitable in the sense of Caffarelli, Kohn and Nirenberg: it satisfies a localized version of the energy inequality.

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We prove that the any Markov solution to the 3D stochastic Navier-Stokes equations driven by a mildly degenerate noise (i.e.all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility. Comment: 29 pages

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The existence of martingale solutions of the hydrodynamic-type equations in
3D possibly unbounded domains is proved. The construction of the solution is
based on the Faedo-Galerkin approximation. To overcome the difficulty related
to the lack of the compactness of Sobolev embeddings in the case of unbounded
domain we use certain Fr\'{e}chet space.
We use also compactness and tightness criteria in some nonmetrizable spaces
and a version of the Skorokhod Theorem in non-metric spaces. The general
framework is applied to the stochastic Navier-Stokes, magneto-hydrodynamic
(MHD) and the Boussinesq equations.

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This paper considers the supremum m of the service times of the customers served in a busy period of the M[+45 degree rule]G[+45 degree rule]1 queueing system. An implicit expression for the distribution m(w) of m is derived. This expression leads to some bounds for m(w), while it can also be used to obtain numerical results. The tail behaviour of m(w) is investigated, too. The results are particularly useful in the analysis of a class of tandem queueing systems.

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We derive functional limit theorems for the integrated periodogram of linear processes whose innovations may have finite or infinite variance, and which may exhibit long memory. The results are applied to obtain corresponding Kolmogorov-Smirnov and Cramér-von Mises goodness-of-fit tests.

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