Asymptotic normality of the additive regression components for continuous time processes

In multivariate regression estimation, the rate of convergence depends on the dimension of the regressor. This fact, known as the curse of the dimensionality, motivated several works. The additive model, introduced by Stone [C.J. Stone, Additive regression and other nonparametric models, Ann. Statist. 13 (2) (1985) 689–705. [9]], offers an efficient response to this problem. In the setting of continuous time processes, using the marginal integration method, we obtain the quadratic convergence rate and the asymptotic normality of the components of the additive model. To cite this article: M. Debbarh, B. Maillot, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

The stable manifold theorem for non-linear stochastic systems with memory: II. The local stable manifold theorem

We state and prove a Local Stable Manifold Theorem (Theorem 4.1) for non-linear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde's)). We introduce the notion of hyperbolicity for stationary trajectories of sfde's. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary trajectory. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle, together with interpolation arguments.     

Random-Time Isotropic Fractional Stable Fields

Generalizing both Substable Fractional Stable Motions (FSMs) and Indicator FSMs, we introduce α-stabilized subordination, a procedure which produces new FSMs (H-self-similar, stationary increment symmetric α-stable processes) from old ones. We extend these processes to isotropic stable fields which have stationary increments in the strong sense, i.e., processes which are invariant under Euclidean rigid motions of the multi-dimensional time parameter. We also prove a Stable Central Limit Theorem which provides an intuitive picture of α-stabilized subordination. Finally we show that α-stabilized subordination of Linear FSMs produces null-conservative FSMs, a class of FSMs introduced by Samorodnitsky (Ann. Probab. 33(5):1782–1803, 2005).     

Wiener integrals, Malliavin calculus and covariance measure structure

We introduce the notion of covariance measure structure for square integrable stochastic processes. We define Wiener integral, we develop a suitable formalism for stochastic calculus of variations and we make Gaussian assumptions only when necessary. Our main examples are finite quadratic variation processes with stationary increments and the bifractional Brownian motion.     

Space–time fractal properties of the forest-fire series in central Italy

The space–time fractality of the forest-fire sequence (1997–2003) occurred in the Tuscany Region (central Italy), one of the most vulnerable to wildfires in Italy, has been approached by using spatial and temporal fractal tools. The fractal exponent α, estimated by the Fano factor method, characterises the time-clustering behaviour of the set of fires, while the correlation dimension D_c, calculated by means of the correlation integral method, gives information on the space-clustering behaviour of the sequence of fires. We found that (i) the investigated fire set is globally characterized by space–time clustering behaviour; (ii) α and D_c decreases and increases, respectively with the increase of the threshold size of burned area; (iii) the time variation of α shows a tendency towards Poissonian processes in correspondence of the largest events.     

Fokker–Planck equation on fractal curves

A Fokker–Planck equation on fractal curves is obtained, starting from Chapmann–Kolmogorov equation on fractal curves. This is done using the recently developed calculus on fractals, which allows one to write differential equations on fractal curves. As an important special case, the diffusion and drift coefficients are obtained, for a suitable transition probability to get the diffusion equation on fractal curves. This equation is of first order in time, and, in space variable it involves derivatives of order α, α being the dimension of the curve. An exact solution of this equation with localized initial condition shows departure from ordinary diffusive behavior due to underlying fractal space in which diffusion is taking place and manifests a subdiffusive behavior. We further point out that the dimension of the fractal path can be estimated from the distribution function.     

Backbone fractal dimension and fractal hybrid orbital of protein structure

Fractal geometry analysis provides a useful and desirable tool to characterize the configuration and structure of proteins. In this paper we examined the fractal properties of 750 folded proteins from four different structural classes, namely (1) the α-class (dominated by α-helices), (2) the β-class (dominated by β-pleated sheets), (3) the (α/β)-class (α-helices and β-sheets alternately mixed) and (4) the (α + β)-class (α-helices and β-sheets largely segregated) by using two fractal dimension methods, i.e. “the local fractal dimension” and “the backbone fractal dimension” (a new and useful quantitative parameter). The results showed that the protein molecules exhibit a fractal behavior in the range of 1 ? N ? 15 (N is the number of the interval between two adjacent amino acid residues), and the value of backbone fractal dimension is distinctly greater than that of local fractal dimension for the same protein. The average value of two fractal dimensions decreased in order of α > α/β > α + β > β. Moreover, the mathematical formula for the hybrid orbital model of protein based on the concept of backbone fractal dimension is in good coincidence with that of the similarity dimension. So it is a very accurate and simple method to analyze the hybrid orbital model of protein by using the backbone fractal dimension.     

Ergodicity of scalar stochastic differential equations with Hölder continuous coefficients

It is well-known that for a one dimensional stochastic differential equation driven by Brownian noise, with coefficient functions satisfying the assumptions of the Yamada–Watanabe theorem (Yamada and Watanabe, 1971, [31,32]) and the Feller test for explosions (Feller, 1951, 1954), there exists a unique stationary distribution with respect to the Markov semigroup of transition probabilities. We consider systems on a restricted domain $D$ of the phase space $\mathbb{R}$ and study the rate of convergence to the stationary distribution. Using a geometrical approach that uses the so called free energy function on the density function space, we prove that the density functions, which are solutions of the Fokker–Planck equation, converge to the stationary density function exponentially under the Kullback–Leibler divergence, thus also in the total variation norm. The results show that there is a relation between the Bakry–Émery curvature dimension condition and the dissipativity condition of the transformed system under the Fisher–Lamperti transformation. Several applications are discussed, including the Cox–Ingersoll–Ross model and the Ait-Sahalia model in finance and the Wright–Fisher model in population genetics.     

An explicit solution to the Skorokhod embedding problem for functionals of excursions of Markov processes

We develop an explicit non-randomized solution to the Skorokhod embedding problem in an abstract setup of signed functionals of excursions of Markov processes. Our setting allows us to solve the Skorokhod embedding problem, in particular, for the age process of excursions of a Markov process, for diffusions and their signed age processes, for Azéma’s martingale and for Bessel processes of dimension smaller than 2.This work is a continuation and an important generalization of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110]. Our methodology is based on excursion theory and the solution to the Skorokhod embedding problem is described in terms of the Itô measure of the functional. We also derive an embedding for positive functionals and we correct a mistake in the formula of Obłój and Yor [J. Obłój, M. Yor, An explicit Skorokhod embedding for the age of Brownian excursions and Azéma martingale, Stochastic Process. Appl. 110 (1) (2004) 83–110] for measures with atoms.     

Fractional Fokker–Planck equation

By using the definition of the characteristic function and Kramers–Moyal Forward expansion, one can obtain the Fractional Fokker–Planck Equation (FFPE) in the domain of fractal time evolution with a critical exponent α (0<α⩽1). Two different classes of fractional differential operators, Liouville–Riemann (L–R) and Nishimoto (N) are used to represent the fractal differential operators in time. By applying the technique of eigenfunction expansion to get the solution of FFPE, one finds that the time part of eigenfunction expansion in terms of L–R represents the waiting time density Ψ(t), which gives the relation between fractal time evolution and the theory of continuous time random walk (CTRW). From the principle of maximum entropy, the structure of the distribution function can be known.     

Convergence analysis of stationary points in sample average approximation of stochastic programs with second order stochastic dominance constraints

Sample average approximation (SAA) method has recently been applied to solve stochastic programs with second order stochastic dominance (SSD) constraints. In particular, Hu et al. (Math Program 133:171–201, 2012) presented a detailed convergence analysis of $\epsilon $ -optimal values and $\epsilon $ -optimal solutions of sample average approximated stochastic programs with polyhedral SSD constraints. In this paper, we complement the existing research by presenting convergence analysis of stationary points when SAA is applied to a class of stochastic minimization problems with SSD constraints. Specifically, under some moderate conditions we prove that optimal solutions and stationary points obtained from solving sample average approximated problems converge with probability one to their true counterparts. Moreover, by exploiting some recent results on large deviation of random functions and sensitivity analysis of generalized equations, we derive exponential rate of convergence of stationary points.     

A generalized finite type condition for iterated function systems

We study iterated function systems (IFSs) of contractive similitudes on Rd with overlaps. We introduce a generalized finite type condition which extends a more restrictive condition in [S.-M. Ngai, Y. Wang, Hausdorff dimension of self-similar sets with overlaps, J. London Math. Soc. (2) 63 (3) (2001) 655-672] and allows us to include some IFSs of contractive similitudes whose contraction ratios are not exponentially commensurable. We show that the generalized finite type condition implies the weak separation property. Under this condition, we can identify the attractor of the IFS with that of a graph-directed IFS, and by modifying a setup of Mauldin and Williams [R.D. Mauldin, S.C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811-829], we can compute the Hausdorff dimension of the attractor in terms of the spectral radius of certain weighted incidence matrix.     

Convergence of nonhomogeneous stochastic chains with countable states

The work started by V. M. Maksimov [1970, Theory Probab. Appl.15, 604–618], and continued by A. Mukherjea [1980, Trans. Amer. Math. Soc.263, 505–520], is extended, and completed with respect to certain aspects. Infinite-dimensional stochastic chains are considered in the framework of Mukherjea [loc. cit.]; backward products of stochastic matrices and their convergence are also considered. The main theme centers around understanding how the convergence of products (backward and forward, finite and infinite dimensional) takes place and what it means in terms of various types of asymptotic behavior of the individual stochastic matrices in the chain. The study is based on establishing the existence of a basis for convergent chains. The basis then makes it possible to describe properly various aspects of convergence. All results are new; they are also complete at least in the sense they have been presented and suitable examples (or counter-examples) are presented to justify the assumptions involved.     

Stochastic clearing systems

A stochastic clearing system is characterized by a non-decreasing stochastic input process $\text{{Y(t), t ≧ 0}}$, where Y(t) is the cumulative quantity entering the system in [0, t], and an output mechanism that intermittently and instantaneously clears the system, that is, removes all the quantity currently present. Examples may be found in the theory of queues, inventories, and other stochastic service and storage systems. In this paper we derive an explicit expression for the stationary (in some cases, limiting) distribution of the quantity in the system, under the assumption that the clearing instants are regeneration points and, in particular, first entrance times into sets of the form {y: y>q}. The expression is in terms of the sojourn measure W associated with $\text{{Y(t), t ≧ 0}: W{A} =}\text{E}${time spent in A by Y(t), 0 ≤ t < ∞}. The results are applied to compound input processes and processes with stationary independent increments. In particular, we show that, contrary to a wide-spread belief, the uniform stationary distribution characteristic of deterministic models does not usually carry over to genuinely stochastic models.     

A matricial extension of the Helson-Sarason theorem and a characterization of some multivariate linearly completely regular processes

We generalize the theorems of Helson-Szegö and Helson-Sarason for matricial measures. We study two-weighted inequalities for the Hilbert transform in [0, 2π] and in R and give a characterization for the positivity of the angle between past and future of multivariate weakly stationary stochastic processes, in the discrete and the continuous case. We also characterize the multivariate weakly stationary stochastic processes that are linearly completely regular and study the rate of convergence of the maximal correlation coefficient.     

Limit theorems for the canonical von Mises statistics with dependent data

We study the limit behavior of the canonical (i.e., degenerate) von Mises statistics based on samples from a sequence of weakly dependent stationary observations satisfying the ψ-mixing condition. The corresponding limit distributions are defined by the multiple stochastic integrals of nonrandom functions with respect to the nonorthogonal Hilbert noises generated by Gaussian processes with nonorthogonal increments.     

Convergence theorems for implicit iteration process for a finite family of continuous pseudocontractive mappings

In this paper, a necessary and sufficient conditions for the strong convergence to a common fixed point of a finite family of continuous pseudocontractive mappings are proved in an arbitrary real Banach space using an implicit iteration scheme recently introduced by Xu and Ori [H.K. Xu, R.G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Fuct. Anal. Optim. 22 (2001) 767-773] in condition αn∈(0,1], and also strong and weak convergence theorem of a finite family of strictly pseudocontractive mappings of Browder-Petryshyn type is obtained. The results presented extend and improve the corresponding results of M.O. Osilike [M.O. Osilike, Implicit iteration process for common fixed points of a finite family of strictly pseudocontractive maps, J. Math. Anal. Appl. 294 (2004) 73-81].     

Vector-valued fractal functions: Fractal dimension and fractional calculus

There are many research available on the study of a real-valued fractal interpolation function and fractal dimension of its graph. In this paper, our main focus is to study the dimensional results for a vector-valued fractal interpolation function and its Riemann–Liouville fractional integral. Here, we give some results which ensure that dimensional results for vector-valued functions are quite different from real-valued functions. We determine interesting bounds for the Hausdorff dimension of the graph of a vector-valued fractal interpolation function. We also obtain bounds for the Hausdorff dimension of the associated invariant measure supported on the graph of a vector-valued fractal interpolation function. Next, we discuss more efficient upper bound for the Hausdorff dimension of measure in terms of probability vector and contraction ratios. Furthermore, we determine some dimensional results for the graph of the Riemann–Liouville fractional integral of a vector-valued fractal interpolation function.     

Wiener path intersections and local time

We study intersection properties of Wiener processes in the plane. For each positive integer k we show that k independent Wiener processes intersect almost surely in a set of Hausdorff dimension two, and that the set of points a single process visits at least k distinct times also has dimension two. We construct a functional on configurations of k independent Wiener processes that measures the extent to which the trajectories of the k processes intersect. We prove certain L^p estimates for this functional and show that it is a local time for a certain vector-valued multiparameter stochastic process.     

Spatial and Spatiotemporal Karhunen-Loève-Type Representations on Fractal Domains

Abstract

We study the spectral properties of spatial and spatiotemporal Gaussian random fields defined as the solutions to stochastic elliptic, parabolic, and hyperbolic fractional pseudodifferential equations on compact fractal domains. The fractal dimension of the domain modifies the asymptotic properties of the eigenvalues that define the pure point spectra of the covariance functions of the solutions and their Karhunen-Loève-type expansions. The eigenfunction systems involved constitute orthogonal bases of the corresponding trace spaces on fractal sets. The Hölder exponent of the sample paths of the random fields is computed in terms of the fractional order of mean-quadratic variation on their increments. Such an exponent also depends on the Hausdorff dimension of the domain.