An example of stability of singular spectrum under smooth perturbations

Random Fourier series with Gaussian coefficients and very large gaps are used to construct a class of continuous stationary Gaussian processes whose sample functions X(t) have remarkable properties. Almost surely the function X(t) has an essential range of Lebesgue measure zero and for each smooth function Y(t) the multiplication operator (M_x+yf) (t) [X(t)+Y(t)]f(t) on L²[0,2] has only singular continuous spectrum.     

Stochastic realization of a Gaussian stochastic control system

The stochastic realization problem is considered of representing a stationary Gaussian process as the observation process of a Gaussian stochastic control system. The problem formulation includes that the lastm components of the observation process form the Gaussian white noise input process to the system. Identifiability of this class of systems motivates the problem. The results include a necessary and sufficient condition for the existence of a stochastic realization. A subclass of Gaussian stochastic control systems is defined that is almost a canonical form for this stochastic realization problem. For a structured Gaussian stochastic control system an equivalent condition for identifiability of the parametrization is stated.The research of this paper is supported in part by the Commission of the European Communities through the SCIENCE Program by the projectSystem Identification with contract number SC1-CT92-0779.     

Extension of stationary stochastic processes

Summary LetX be an arbitrary Hausdorff space, and consider a stationary stochastic process inX with time interval [0, 1], i.e. a tight probability onX ^{[0, 1]}, equipped with the Borel -field of the product space. We prove the existence of a stationary extension of this process to ₀ ⁺ . Furthermore, we show that the extended process may be chosen to have continuous paths if the original process has this property. Under stronger topological assumptions, we derive the corresponding results whenX ^{[0, 1]} is equipped with the product of the Borel -fields.Corporate Research and Development, SIEMENS AG, D-81730 Munich, Germany     

Large Deviations for Quadratic Functionals of Gaussian Processes

The Large Deviation Principle (LDP) is derived for several quadratic additive functionals of centered stationary Gaussian processes. For example, the rate function corresponding to is the Fenchel-Legendre transform of where X _t is a continuous time process with the bounded spectral density f(s). This spectral density condition is strictly weaker than the one necessary for the LDP to hold for all bounded continuous functionals. Similar results are obtained for the energy of multivariate discrete-time Gaussian processes and in the regime of moderate deviations, the latter yielding the corresponding Central Limit Theorems.     

Compactness of probability measures and the uniform convergence of certain stochastic series

We give the conditions which ensure the compactness of the probability measures _n, n1, generated by Gaussian processes the realizations of which are continuous with unit probability in [0, 1]. We also give the conditions for the uniform convergence of stochastic series of the form _{k=1 2k}(t), where the _k(t) are independent Gaussian processes the realizations of which are continuous with unit probability in [0, 1].Translated from Matematicheskie Zametki, Vol. 12, No. 4, pp. 443–451, October, 1972.In conclusion the author wishes to express his deep gratitude to Yu. V. Kozachenko for formulating the problem and for his attention to the paper.     

Approximation of stationary solutions to SDEs driven by multiplicative fractional noise

In a previous paper, we studied the ergodic properties of an Euler scheme of a stochastic differential equation with a Gaussian additive noise in order to approximate the stationary regime of such an equation. We now consider the case of multiplicative noise when the Gaussian process is a fractional Brownian motion with Hurst parameter H>1/2$H>1/2$ and obtain some (functional) convergence properties of some empirical measures of the Euler scheme to the stationary solutions of such SDEs.     

Entropy meaning of summability of the logarithm

We consider the mutual relations between the concepts of sets of uniqueness for analytic functions, the loss of entropy in nondetermined stationary linear filters, Szegö's theorem and the familiar condition of summability of the logarithm. The goal of the paper is to give the physical meaning of these mutual relations. Here we take the concept of linear stationary filter and loss of entropy in it as basic. In the first part of the paper the account is given for the case of discrete time, and in the second part we give the method of passing to continuous time. To this end we introduce the concept of stationary sampling system. This is a sequence of functions from L²(), which transforms any stationary Gaussian process with continuous correlation function into a stationary Gaussian process with discrete time. Such systems can be described in terms of the Fourier transform. Laguerre systems where z is a fixed point in the upper half-plane, play a special role among all sampling systems. If z=i, then is a classical Laguerre function on the line up to a multiplicative constant. Laguerre sampling systems allow one to give the entropy meaning to the values of the harmonic extension to the upper half-plane of the logarithm of the spectral density of the processTranslated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 73, pp. 152–187, 1977.     

Discrete Spectrum of Nonstationary Stochastic Processes on Groups

Vector-valued, asymptotically stationary stochastic processes on -compact locally compact abelian groups are studied. For such processes, we introduce a stationary spectral measure and show that it is discrete if and only if the asymptotically stationary covariance function is almost periodic. Using an almost periodic Fourier transform we recover the discrete part of the spectral measure and construct a natural, consistent estimator for the latter from samples of the process.     

Quantum stochastic calculus and quantum Gaussian processes

In this lecture we present a brief outline of boson Fock space stochastic calculus based on the creation, conservation and annihilation operators of free field theory, as given in the 1984 paper of Hudson and Parthasarathy [9]. We show how a part of this architecture yields Gaussian fields stationary under a group action. Then we introduce the notion of semigroups of quasifree completely positive maps on the algebra of all bounded operators in the boson Fock space Γ(? ⁿ ) over ? ⁿ . These semigroups are not strongly continuous but their preduals map Gaussian states to Gaussian states. They were first introduced and their generators were shown to be of the Lindblad type by Vanheuverzwijn [19]. They were recently investigated in the context of quantum information theory by Heinosaari et al. [7]. Here we present the exact noisy Schrödinger equation which dilates such a semigroup to a quantum Gaussian Markov process.     

A functional central limit theorem for empirical processes under a strong mixing condition

This paper introduces a functional central limit theorem for empirical processes endowed with real values from a strictly stationary random field that satisfies an interlaced mixing condition. We proceed by using a common technique from Billingsley (Convergence of probability measures, Wiley, New York, 1999), by first obtaining the limit theorem for the case where the random variables of the strictly stationary ???-mixing random field are uniformly distributed on the interval [0, 1]. We then generalize the result to the case where the absolutely continuous marginal distribution function is not longer uniform. In this case we show that the empirical process endowed with values from the ???-mixing stationary random field, due to the strong mixing condition, doesn??t converge in distribution to a Brownian bridge, but to a continuous Gaussian process with mean zero and the covariance given by the limit of the covariance of the empirical process. The argument for the general case holds similarly by the application of a standard variant of a result of Billingsley (1999) for the space D(???, ??).     

Distribution function of the absolute maximum of random harmonic oscillations

An analytical expression is derived for the distribution function of the absolute maximum of a Gaussian stationary process with correlation function p(t)=₁ ²+₂ ²cost t.Translated from Statisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 144–147, 1986.     

Extreme value theory for continuous parameter stationary processes

Summary In this paper the central distributional results of classical extreme value theory are obtained, under appropriate dependence restrictions, for maxima of continuous parameter stochastic processes. In particular we prove the basic result (here called Gnedenko's Theorem) concerning the existence of just three types of non-degenerate limiting distributions in such cases, and give necessary and sufficient conditions for each to apply. The development relies, in part, on the corresponding known theory for stationary sequences.The general theory given does not require finiteness of the number of upcrossings of any levelx. However when the number per unit time is a.s. finite and has a finite mean(x), it is found that the classical criteria for domains of attraction apply when(x) is used in lieu of the tail of the marginal distribution function. The theory is specialized to this case and applied to give the general known results for stationary normal processes for which(x) may or may not be finite).A general Poisson convergence theorem is given for high level upcrossings, together with its implications for the asymptotic distributions ofr ^th largest local maxima.This work was supported by the Office of Naval Research under Contract N00014-75-C-0809, and in part by the Danish natural Science research Council     

Local nondeterminism and local times for stable processes

Summary Our main theorem gives sufficient conditions for symmetric stable processes and fields to have a jointly continuous local time. The approach is through the L ^p representation for such processes. We develop a measure of dependence for vectors in a normed linear space and use that to analyze the probabilistic independence of the increments of a stable process. Local nondeterminism is defined for stable processes and shown to be equivalent to locally approximately independent increments. Sufficient conditions for several classes of stable processes to be local nondeterministic are given. These ideas are extended to multidimensional stable random fields and we prove existence of jointly continuous local times. The results extend most Gaussian results to their stable analogs.     

Position and Height of the Global Maximum of a Twice Differentiable Stochastic Process

For a stochastic process with absolutely continuous sample path derivative, a formula for the joint density of (T, Z), the position and height of the global maximum of in a closed interval, is given. The formula is derived using the generalized Rices formula. The presented result can be applied both to stationary and non-stationary processes under mild assumptions on the process. The formula for the density is explicit but involves integrals that have to be computed using numerical integration. The computation of the density is discussed and some numerical examples are given.     

Local field brownian motion

A local field is any locally compact, non-discrete field other than the field of real numbers or the field of complex numbers. There is a natural notion of Gaussian measures on a local field vector space. We construct and study a specific local field Gaussian stochastic process taking values in a finite dimensional local field vector space and indexed by another finite dimensional local field vector space. This process has a structure that strongly reflects the algebraic and geometric structure of the underlying index space and, as such, plays the same role in the local field setting that standard Brownian motion and the related multiparameter processes such as Lévy's multiparameter Brownian motion play in a Euclidean context. We investigate the theory of additive functionals and the related potential theory for this process and show that it strongly resembles the Euclidean prototype. As a particular consequence of this investigation, we find that a local time process exists when the process hits points. We give two intrinsic constructions of the local time at a given level. These constructions are analogous to the dilation construction of Kingman and the Hausdorff measure construction of Taylor and Wendel in the Euclidean case. Finally, the local time is shown to be continuous as a measure valued stochastic process indexed by the levèl at which it is evaluated.Research supported in part by an NSF Grant and Presidential Young Investigator Award.     

On infinitely divisible semimartingales

Stricker’s theorem states that a Gaussian process is a semimartingale in its natural filtration if and only if it is the sum of an independent increment Gaussian process and a Gaussian process of finite variation, see Stricker (Z Wahrsch Verw Geb 64(3):303–312, 1983). We consider extensions of this result to non Gaussian infinitely divisible processes. First we show that the class of infinitely divisible semimartingales is so large that the natural analog of Stricker’s theorem fails to hold. Then, as the main result, we prove that an infinitely divisible semimartingale relative to the filtration generated by a random measure admits a unique decomposition into an independent increment process and an infinitely divisible process of finite variation. Consequently, the natural analog of Stricker’s theorem holds for all strictly representable processes (as defined in this paper). Since Gaussian processes are strictly representable due to Hida’s multiplicity theorem, the classical Stricker’s theorem follows from our result. Another consequence is that the question when an infinitely divisible process is a semimartingale can often be reduced to a path property, when a certain associated infinitely divisible process is of finite variation. This gives the key to characterize the semimartingale property for many processes of interest. Along these lines, using Basse-O’Connor and Rosiński (Stoch Process Appl 123(6):1871–1890, 2013a), we characterize semimartingales within a large class of stationary increment infinitely divisible processes; this class includes many infinitely divisible processes of interest, including linear fractional processes, mixed moving averages, and supOU processes, as particular cases. The proof of the main theorem relies on series representations of jumps of càdlàg infinitely divisible processes given in Basse-O’Connor and Rosiński (Ann Probab 41(6):4317–4341, 2013b) combined with techniques of stochastic analysis.     

The functional iterated logarithm law for stochastic processes represented by multiple Wiener integrals

Summary In a previous paper the authors obtained a functional law of the iterated logarithm for a class of self-similar processes with stationary increments, which are represented by multiple Wiener integrals. This result is extended to a certain class of processes represented by multiple Wiener integrals which converge to with an appropriate normalization. As an application a functional log log law for nonlinear functionals of some stationary Gaussian processes is given.     

Fractional Brownian Density Process and Its Self-Intersection Local Time of Order k

The fractional Brownian density process is a continuous centered Gaussian ( ^d )-valued process which arises as a high-density fluctuation limit of a Poisson system of independent d-dimensional fractional Brownian motions with Hurst parameter H. ( ( ^d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure of the (initial) Poisson random measure on ^d is either the Lebesgue measure or a finite measure, then the density process has self-intersection local time of order k 2 if and only if Hd < k/(k – 1). The latter is also the necessary and sufficient condition for existence of multiple points of order k for d-dimensional fractional Brownian motion, as proved by Talagrand¹². This result extends to a non-Markovian case the relationship known for (Markovian) symmetric -stable Lévy processes and their corresponding density processes. New methods are used in order to overcome the lack of Markov property. Other properties of the fractional Brownian density process are also given, in particular the non-semimartingale property in the case H 1/2, which is obtained by a general criterion for the non-semimartingale property of real Gaussian processes that we also prove.     

On the martingale problem for Banach space valued stochastic differential equations

LetE denote a real separable Banach space and letZ=(Z(t, f) be a family of centered, homogeneous, Gaussian independent increment processes with values inE, indexed by timet0 and the continuous functionsf:[0,t] E. If the dependence ont andf fulfills some additional properties,Z is called a gaussian random field. For continuous, adaptedE-valued processesX a stochastic integral processY = ₀ ^. Z(t, X)(dt) is defined, which is a continuous local martingale with tensor quadratic variation[Y] = ₀ ^. Q(t, X)dt, whereQ(t, f) denotes the covariance operator ofZ(t, f).Y is called a solution of the homogeneous Gaussian martingale problem, ifY = ₀ ^. Z(t, Y)(dt). Such solutions occur naturally in connection with stochastic differential equations of the type (D):dX(t)=G(t, X) dt+Z(t, X)(dt), whereG is anE-valued vector field. It is shown that a solution of (D) can be obtained by a kind of variation of parameter method, first solving a deterministic integral equation only involvingG and then solving an associated homogeneous martingale problem.