Showing papers in "Journal of Theoretical Probability in 2007"

Bracketing Metric Entropy Rates and Empirical Central Limit Theorems for Function Classes of Besov- and Sobolev-Type

Abstract: We derive ℒ r (μ)-bracketing metric and sup-norm metric entropy rates of bounded subsets of general function spaces defined over ℝ d or, more generally, over Borel subsets thereof, by adapting results of Haroske and Triebel (Math. Nachr. 167, 131–156, 1994; 278, 108–132, 2005). The function spaces covered are of (weighted) Besov, Sobolev, Holder, and Triebel type. Applications to the theory of empirical processes are discussed. In particular, we show that (norm-)bounded subsets of the above mentioned spaces are Donsker classes uniformly in various sets of probability measures.

Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion

Abstract: In this article, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H. We consider two cases. If H>1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution. This result extends those contained in J. Complex. 22(4), 459–474, 2006 and C.R. Acad. Sci. Paris, Ser. I 340(8), 611–614, 2005. When 1/6

Mixing Limit Theorems for Ergodic Transformations

Abstract: We show that distributional and weak functional limit theorems for ergodic processes often hold for arbitrary absolutely continuous initial distributions. This principle is illustrated in the setup of ergodic sums, renewal-theoretic variables, and hitting times for ergodic measure preserving transformations.

Random Flights in Higher Spaces

Abstract: We consider in this paper random flights in ℝd performed by a particle changing direction of motion at Poisson times. Directions are uniformly distributed on hyperspheres S1d. We obtain the conditional characteristic function of the position of the particle after n changes of direction. From this characteristic function we extract the conditional distributions in terms of (n+1)−fold integrals of products of Bessel functions. These integrals can be worked out in simple terms for spaces of dimension d=2 and d=4. In these two cases also the unconditional distribution is determined in explicit form. Some distributions connected with random flights in ℝ3 are discussed and in some special cases are analyzed in full detail. We point out that a strict connection between these types of motions with infinite directions and the equation of damped waves holds only for d=2.

A Sharp Form of the Cramér-Wold Theorem

Abstract: The Cramer–Wold theorem states that a Borel probability measure P on ℝd is uniquely determined by its one-dimensional projections. We prove a sharp form of this result, addressing the problem of how large a subset of these projections is really needed to determine P. We also consider extensions of our results to measures on a separable Hilbert space.

On the Weak Invariance Principle for Non-Adapted Sequences under Projective Criteria

Abstract: In this paper we study the central limit theorem and its weak invariance principle for sums of non-adapted stationary sequences, under different normalizations. Our conditions involve the conditional expectation of the variables with respect to a given σ-algebra, as done in Gordin (Dokl. Akad. Nauk SSSR 188, 739–741, 1969) and Heyde (Z. Wahrsch. verw. Gebiete 30, 315–320, 1974). These conditions are well adapted to a large variety of examples, including linear processes with dependent innovations or regular functions of linear processes.

Discrete and continuous time modulated random walks with heavy-tailed increments

Abstract: We consider a modulated process S which, conditional on a background process X, has independent increments. Assuming that S drifts to −∞ and that its increments (jumps) are heavy-tailed (in a sense made precise in the paper), we exhibit natural conditions under which the asymptotics of the tail distribution of the overall maximum of S can be computed. We present results in discrete and in continuous time. In particular, in the absence of modulation, the process S in continuous time reduces to a Levy process with heavy-tailed Levy measure. A central point of the paper is that we make full use of the so-called “principle of a single big jump” in order to obtain both upper and lower bounds. Thus, the proofs are entirely probabilistic. The paper is motivated by queueing and Levy stochastic networks.

Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices

Abstract: Consider the ensemble of real symmetric Toeplitz matrices, each independent entry an i.i.d. random variable chosen from a fixed probability distribution p of mean 0, variance 1, and finite higher moments. Previous investigations showed that the limiting spectral measure (the density of normalized eigenvalues) converges weakly and almost surely, independent of p, to a distribution which is almost the standard Gaussian. The deviations from Gaussian behavior can be interpreted as arising from obstructions to solutions of Diophantine equations. We show that these obstructions vanish if instead one considers real symmetric palindromic Toeplitz matrices, matrices where the first row is a palindrome. A similar result was previously proved for a related circulant ensemble through an analysis of the explicit formulas for eigenvalues. By Cauchy’s interlacing property and the rank inequality, this ensemble has the same limiting spectral distribution as the palindromic Toeplitz matrices; a consequence of combining the two approaches is a version of the almost sure Central Limit Theorem. Thus our analysis of these Diophantine equations provides an alternate technique for proving limiting spectral measures for certain ensembles of circulant matrices.

The Central Limit Problem for Random Vectors with Symmetries

Abstract: Motivated by the central limit problem for convex bodies, we study normal approximation of linear functionals of high-dimensional random vectors with various types of symmetries. In particular, we obtain results for distributions which are coordinatewise symmetric, uniform in a regular simplex, or spherically symmetric. Our proofs are based on Stein’s method of exchangeable pairs; as far as we know, this approach has not previously been used in convex geometry. The spherically symmetric case is treated by a variation of Stein’s method which is adapted for continuous symmetries.

Intersection local time for two independent fractional Brownian motions

Abstract: Let B H and $\widetilde{B}^{H}$ be two independent, d-dimensional fractional Brownian motions with Hurst parameter H∈(0,1). Assume d≥2. We prove that the intersection local time of B H and $\widetilde{B}^{H}$ $$I(B^{H},\widetilde{B}^{H})=\int_{0}^{T}\int_{0}^{T}\delta(B_{t}^{H}-\widetilde{B}_{s}^{H})dsdt$$ exists in L 2 if and only if Hd<2.

The Spectrum of a Random Geometric Graph is Concentrated

Abstract: Consider n points, x 1,... , x n , distributed uniformly in [0, 1] d . Form a graph by connecting two points x i and x j if $$\Vert x_i - x_j\Vert \leq r(n)$$ . This gives a random geometric graph, $$G({\mathcal {X}}_n;r(n))$$ , which is connected for appropriate r(n). We show that the spectral measure of the transition matrix of the simple random walk on $$G({\mathcal {X}}_n; r(n))$$ is concentrated, and in fact converges to that of the graph on the deterministic grid.

On the Lower Tail Probabilities of Some Random Sequences in lp

Abstract: We investigate the behaviour of the logarithmic small deviation probability of a sequence (σnθn) in lp, 0

Critical Behavior in Almost Sure Central Limit Theory

Abstract: Let X1,X2,… be i.i.d. random variables with EX1=0, EX12=1 and let Sk=X1+⋅⋅⋅+Xk. We study the a.s. convergence of the weighted averages $$D_{N}^{-1}\sum_{k=1}^{N}d_{k}I\biggl\{\frac{S_{k}}{\sqrt{k}}\leq x\biggr\},$$ where (dk) is a positive sequence with DN=∑k=1Ndk→∞. By the a.s. central limit theorem, the above averages converge a.s. to Φ(x) if dk=1/k (logarithmic averages) but diverge if dk=1 (ordinary averages). Under regularity conditions, we give a fairly complete solution of the problem for what sequences (dk) the weighted averages above converge, resp. the corresponding LIL and CLT hold. Our results show that logarithmic averaging, despite its prominent role in a.s. central limit theory, is far from optimal and considerably stronger results can be obtained using summation methods near ordinary (Cesaro) summation.

Laws of the Iterated Logarithm for the Local U-Statistic Process

Abstract: Laws of the iterated logarithm are established for the local U-statistic process. This entails the development of probability inequalities and moment bounds for U-processes that should be of separate interest. The local U-statistic process is based upon an estimator of the density of a function of several i.i.d. variables proposed by Frees (J. Am. Stat. Assoc. 89, 517–525, 1994). As a consequence, our results are directly applicable to the derivation of exact rates of uniform in bandwidth consistency in the sup and in the L p norms for these estimators.

Uniform Distributions on the Natural Numbers

Abstract: We compare the following three notions of uniformity for a finitely additive probability measure on the set of natural numbers: that it extend limiting relative frequency, that it be shift-invariant, and that it map every residue class mod m to 1/m. We find that these three types of uniformity can be naturally ordered. In particular, we prove that the set L of extensions of limiting relative frequency is a proper subset of the set S of shift-invariant measures and that S is a proper subset of the set R of measures which map residue classes uniformly. Moreover, we show that there are subsets G of ℕ for which the range of possible values μ(G) for μ∈L is properly contained in the set of values obtained when μ ranges over S, and that there are subsets G which distinguish S and R analogously.

Normal Approximations for Descents and Inversions of Permutations of Multisets

Abstract: Normal approximations for descents and inversions of permutations of the set {1,2,…,n} are well known. We consider the number of inversions of a permutation π(1),π(2),…,π(n) of a multiset with n elements, which is the number of pairs (i,j) with 1≤i π(j). The number of descents is the number of i in the range 1≤i π(i+1). We prove that, appropriately normalized, the distribution of both inversions and descents of a random permutation of the multiset approaches the normal distribution as n→∞, provided that the permutation is equally likely to be any possible permutation of the multiset and no element occurs more than αn times in the multiset for a fixed α with 0<α<1. Both normal approximation theorems are proved using the size bias version of Stein’s method of auxiliary randomization and are accompanied by error bounds.

On Several Two-Boundary Problems for a Particular Class of Lévy Processes

Abstract: Several two-boundary problems are solved for a special Levy process: the Poisson process with an exponential component. The jumps of this process are controlled by a homogeneous Poisson process, the positive jump size distribution is arbitrary, while the distribution of the negative jumps is exponential. Closed form expressions are obtained for the integral transforms of the joint distribution of the first exit time from an interval and the value of the overshoot through boundaries at the first exit time. Also the joint distribution of the first entry time into the interval and the value of the process at this time instant are determined in terms of integral transforms.

Continuity in Law with Respect to the Hurst Parameter of the Local Time of the Fractional Brownian Motion

Abstract: We give a result of stability in law of the local time of the fractional Brownian motion with respect to small perturbations of the Hurst parameter. Concretely, we prove that the law (in the space of continuous functions) of the local time of the fractional Brownian motion with Hurst parameter H converges weakly to that of the local time of $$B^{H_0}$$ , when H tends to H 0.

A Functional Non-Central Limit Theorem for Jump-Diffusions with Periodic Coefficients Driven by Stable Lévy-Noise

Abstract: We prove a functional non-central limit theorem for jump-diffusions with periodic coefficients driven by stable Levy-processes with stability index α>1. The limit process turns out to be an α-stable Levy process with an averaged jump-measure. Unlike in the situation where the diffusion is driven by Brownian motion, there is no drift related enhancement of diffusivity.

A Generalization of Strassen’s Functional LIL

Abstract: Let X1,X2,… be a sequence of i.i.d. mean zero random variables and let Sn denote the sum of the first n random variables. We show that whenever we have with probability one, lim sup n→∞|Sn|/cn=α0<∞ for a regular normalizing sequence {cn}, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.

Berry–Esseen for Free Random Variables

Abstract: An analogue of the Berry–Esseen inequality is proved for the speed of convergence of free additive convolutions of bounded probability measures. The obtained rate of convergence is of the order n−1/2, the same as in the classical case. An example with binomial measures shows that this estimate cannot be improved without imposing further restrictions on convolved measures.

On Uniqueness of Maximal Coupling for Diffusion Processes with a Reflection

Abstract: A maximal coupling of two diffusion processes makes two diffusion particles meet as early as possible We study the uniqueness of maximal couplings under a sort of ‘reflection structure’ which ensures the existence of such couplings In this framework, the uniqueness in the class of Markovian couplings holds for the Brownian motion on a Riemannian manifold whereas it fails in more singular cases We also prove that a Kendall-Cranston coupling is maximal under the reflection structure

Reflected Backward SDEs with Two Barriers Under Monotonicity and General Increasing Conditions

Abstract: In this paper, we prove the existence and uniqueness result of the reflected BSDE with two continuous barriers under monotonicity and general increasing condition on y, with Lipschitz condition on z

Cumulants for Random Matrices as Convolutions on the Symmetric Group, II

Abstract: In a previous paper we defined some “cumulants of matrices” which naturally converge toward the free cumulants of the limiting non commutative random variables when the size of the matrices tends to infinity. Moreover these cumulants satisfied some of the characteristic properties of cumulants whenever the matrix model was invariant under unitary conjugation. In this paper we present the fitting cumulants for random matrices whose law is invariant under orthogonal conjugation. The symplectic case could be carried out in a similar way.

A Hoeffding-Type Inequality for Ergodic Time Series

Abstract: In this paper, a Hoeffding-type inequality is presented for a class of ergodic time series. The inequality is then used to construct uniformly exponentially consistent tests, which are useful tools for studying Bayesian consistency.

Representation of Infinitely Divisible Distributions on Cones

Abstract: We investigate infinitely divisible distributions on cones in Frechet spaces. We show that every infinitely divisible distribution concentrated on a normal cone has the regular Levy–Khintchine representation if and only if the cone is regular. These results are relevant to the study of multidimensional subordination.

The Dirichlet Distribution and Process through Neutralities

Abstract: A new characterization of the Dirichlet distribution, based on the notion of complete neutrality and a regression version of neutrality, is derived. It unifies earlier characterizations by James and Mosimann (Ann. Stat. 8, 183–189, 1980) and by Seshadri and Wesolowski (Sankhyā, A 65, 248–291, 2003). Also new results on identification of the Dirichlet process in the class of neutral-to-the-right processes are obtained. The proof of the main result makes an extensive use of the method of moments.

An Almost Sure Functional Limit Theorem for the Domain of Geometric Partial Attraction of Semistable Laws

Abstract: An almost sure functional limit theorem is obtained for variables being in the domain of geometric partial attraction of a semistable law.

Logarithmic Level Comparison for Small Deviation Probabilities

Abstract: Log-level comparisons of the small deviation probabilities are studied in three different but related settings: Gaussian processes under the L2 norm, multiple sums motivated by tensor product of Gaussian processes, and various integrated fractional Brownian motions under the sup-norm.

Graph-Theoretic Approach to Stochastic Integrals with Clifford Algebras

Abstract: Given a fixed probability space (Ω,ℱ,ℙ) and m≥1, let X(t) be an L2(Ω) process satisfying necessary regularity conditions for existence of the mth iterated stochastic integral. For real-valued processes, these existence conditions are known from the work of D. Engel. Engel’s work is extended here to L2(Ω) processes defined on Clifford algebras of arbitrary signature (p,q), which reduce to the real case when p=q=0. These include as special cases processes on the complex numbers, quaternion algebra, finite fermion algebras, fermion Fock spaces, space-time algebra, the algebra of physical space, and the hypercube. Next, a graph-theoretic approach to stochastic integrals is developed in which the mth iterated stochastic integral corresponds to the limit in mean of a collection of weighted closed m-step walks on a growing sequence of graphs. Combinatorial properties of the Clifford geometric product are then used to create adjacency matrices for these graphs in which the appropriate weighted walks are recovered naturally from traces of matrix powers. Given real-valued L2(Ω) processes, Hermite and Poisson-Charlier polynomials are recovered in this manner.