Showing papers in "Journal of Theoretical Probability in 2004"

On exit and ergodicity of the spectrally one-sided Levy process reflected at its infimum

Abstract: Consider a spectrally one-sided Levy process X and reflect it at its past infimum I. Call this process Y. For spectrally positive X, Avram et al.(2) found an explicit expression for the law of the first time that Y=X−I crosses a finite positive level a. Here we determine the Laplace transform of this crossing time for Y, if X is spectrally negative. Subsequently, we find an expression for the resolvent measure for Y killed upon leaving [0,a]. We determine the exponential decay parameter ϱ for the transition probabilities of Y killed upon leaving [0,a], prove that this killed process is ϱ-positive and specify the ϱ-invariant function and measure. Restricting ourselves to the case where X has absolutely continuous transition probabilities, we also find the quasi-stationary distribution of this killed process. We construct then the process Y confined in [0,a] and prove some properties of this process.

Coupling for τ-Dependent Sequences and Applications

Abstract: Let X be a real-valued random variable and $$M$$ a σ-algebra. We show that the minimum $${\mathbb{L}}^1$$ -distance between X and a random variable distributed as X and independant of $$M$$ can be viewed as a dependence coefficient τ( $$M$$ ,X) whose definition is comparable (but different) to that of the usual β-mixing coefficient between $$M$$ and σ(X). We compare this new coefficient to other well known measures of dependence, and we show that it can be easily computed in various situations, such as causal Bernoulli shifts or stable Markov chains defined via iterative random maps. Next, we use coupling techniques to obtain Bennett and Rosenthal-type inequalities for partial sums of τ-dependent sequences. The former is used to prove a strong invariance principle for partial sums.

A new method for bounding rates of convergence of empirical spectral distributions

Abstract: The probabilistic properties of eigenvalues of random matrices whose dimension increases indefinitely has received considerable attention. One important aspect is the existence and identification of the limiting spectral distribution (LSD) of the empirical distribution of the eigenvalues. When the LSD exists, it is useful to know the rate at which the convergence holds. The main method to establish such rates is the use of Stieltjes transform. In this article we introduce a new technique of bounding the rates of convergence to the LSD. We show how our results apply to specific cases such as the Wigner matrix and the Sample Covariance matrix.

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

Abstract: The fractional Brownian density process is a continuous centered Gaussian $$S$$ ′(ℝ 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. ( $$S$$ ′(ℝ d ) is the space of tempered distributions). The main result proved in the paper is that if the intensity measure μ of the (initial) Poisson random measure on ℝ 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 α-stable Levy 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.

Exact L 2 Small Balls of Gaussian Processes

Abstract: We prove a comparison theorem extending Li(6) and develop a complex-analytic approach to treat L 2 small ball probabilities of Gaussian processes. We demonstrate the techniques for the m-times integrated Brownian motions and in examples where one can not apply Li comparison theorem.

An extension of the Kolmogorov-Feller weak law of large numbers with an application to the St. Petersburg game

Abstract: The Kolmogorov–Feller weak law of large numbers for i.i.d. random variables without finite mean is extended to a larger class of distributions, requiring regularly varying normalizing sequences. As an application we show that the weak law of large numbers for the St. Petersburg game is an immediate consequence of our result.

Wishart Distributions on Homogeneous Cones

Abstract: The classical family of Wishart distributions on a cone of positive definite matrices and its fundamental features are extended to a family of generalized Wishart distributions on a homogeneous cone using the theory of exponential families. The generalized Wishart distributions include all known families of Wishart distributions as special cases. The relationships to graphical models and Bayesian statistics are indicated.

Necessary and Sufficient Condition for the Functional Central Limit Theorem in Hölder Spaces

Abstract: Let (X i ) i≥1 be an i.i.d. sequence of random elements in the Banach space B, S n ≔X 1+⋅⋅⋅+X n and ξ n be the random polygonal line with vertices (k/n,S k ), k=0,1,...,n. Put ρ(h)=h α L(1/h), 0≤h≤1 with 0<α≤1/2 and L slowly varying at infinity. Let H ρ 0 (B) be the Holder space of functions x:[0,1]↦B, such that ∥x(t+h)−x(t)∥=o(ρ(h)), uniformly in t. We characterize the weak convergence in H ρ 0 (B) of n −1/2 ξ n to a Brownian motion. In the special case where B=ℝ and ρ(h)=h α , our necessary and sufficient conditions for such convergence are E X 1=0 and P(|X 1|>t)=o(t −p(α)) where p(α)=1/(1/2−α). This completes Lamperti (1962) invariance principle.

A Multivariate CLT for Decomposable Random Vectors with Finite Second Moments

Abstract: Stein's method is used to derive a CLT for dependent random vectors possessing the dependence structure from Barbour et al. J. Combin. Theory Ser. B47, 125–145, but under the assumption of second moments only. This allows us to derive Lindeberg–Feller type theorems for sums of random vectors with certain dependence structures. We apply the main theorem to the study of three problems: local dependence, random graph degree statistics and finite population statistics. In particular, we consider U-statistics of independent observations as well as of observations drawn without replacement.

On the Heyde Theorem for Finite Abelian Groups

Abstract: It is well-known Heyde's characterization theorem for the Gaussian distribution on the real line: if ξ j are independent random variables, α j , β j are nonzero constants such that β i α ±β j α−1 j ≠ 0 for all i≠ j and the conditional distribution of L 2=β1 ξ1 + ··· + β n ξ n given L 1=α1 ξ1 + ··· +α n ξ n is symmetric, then all random variables ξ j are Gaussian. We prove some analogs of this theorem, assuming that independent random variables take on values in a finite Abelian group X and the coefficients α j ,β j are automorphisms of X.

On a One-parameter Generalization of the Brownian Bridge and Associated Quadratic Functionals

Abstract: A one-parameter generalization of the Brownian bridge is studied. These processes are then used to compute the laws of some quadratic functionals of Brownian motion, and to obtain identities in law involving local time of modified Bessel processes up to their first hitting time.

On Symmetric Stable-Like Processes: Some Path Properties and Generators

Abstract: We derive some path properties of symmetric stable-like processes constructed via Dirichlet form theory and then sufficient conditions in order that the generators of the forms contain a nice functions space, are given.

An extension of the contraction principle

Abstract: The concept of quasi-continuity and the new concept of almost compactness for a function are the basis for the extension of the contraction principle in large deviations presented here. Important equivalences for quasi-continuity are proved in the case of metric spaces. The relation between the exponential tightness of a sequence of stochastic processes and the exponential tightness of its transform (via an almost compact function) is studied here in metric spaces. Counterexamples are given to the nonmetric case. Relations between almost compactness of a function and the goodness of a rate function are studied. Applications of the main theorem are given, including to an approximation of the stochastic integral.

SLLN for weighted independent identically distributed random variables

Abstract: For any sequence {a k } with sup $$\frac{1}{n}\sum {_{k = 1}^n \left| {a_k } \right|^q } < \infty $$ for some q>1, we prove that $$\frac{1}{n}\sum {_{k = 1}^n {\text{ }}a_k X_k } $$ converges to 0 a.s. for every {X n } i.i.d. with E(|X 1|)<∞ and E(X 1)=0; the result is no longer true for q=1, not even for the class of i.i.d. with X 1 bounded. We also show that if {a k } is a typical output of a strictly stationary sequence with finite absolute first moment, then for every i.i.d. sequence {X n { with finite absolute pth moment for some p> 1, $$\frac{1}{n}\sum {_{k = 1}^n {\text{ }}a_k X_k } $$ converges a.s.

Large Deviations View Points for Heavy-Tailed Random Walks

Abstract: Let X 1, X 2,... be a sequence of i.i.d. non-negative random variables with heavy tails. W e study logarithmic asymptotics for the distributions of the partial sums S n = X 1 + ··· + X n . Our main interest is in the crude estimates P(S n > n x ) ≈ n −αx + 1 for appropriate values of x where α is a specific parameter. The related conjecture proposed by Gantert (Stat. Probab. Lett. 49, 113–118) is investigated.

Infinite Dimensional Isoperimetric Inequalities in Product Spaces with the Supremum Distance

Abstract: We study the isoperimetric problem for product probability measures with respect to the uniform enlargement. We construct several examples of measures μ for which the isoperimetric function of μ coincides with the one of the infinite product μ∞. This completes earlier works by Bobkov and Houdre.

Some Results on Stochastic Differential Equations with Reflecting Boundary Conditions

Abstract: Some results related to stochastic differential equations with reflecting boundary conditions (SDER) are obtained. Existence and uniqueness of strong solution is ensured under the relaxation on the drift coefficient (instead of the Lipschitz character, a monotonicity condition is supposed).

Small Deviations of Stable Processes via Metric Entropy

Abstract: Let X=(X(t))t∈T be a symmetric α-stable, 0<α<2, process with paths in the dual E* of a certain Banach space E. Then there exists a (bounded, linear) operator u from E into some Lα(S,σ) generating X in a canonical way. The aim of this paper is to compare the degree of compactness of u with the small deviation (ball) behavior of \(\phi (\varepsilon ) = - \log \mathbb{P}(\left\| X \right\|_{E^* } < \varepsilon )\) as e→0. In particular, we prove that a lower bound for the metric entropy of u implies a lower bound for φ(e) under an additional assumption on E. As applications we obtain upper small deviation estimates for weighted α-stable Levy motions, linear fractional α-stable motions and d-dimensional α-stable sheets. Our results rest upon an integral representation of Lα-valued operators as well as on small deviation results for Gaussian processes due to Kuelbs and Li and to the authors.

Prokhorov Blocks and Strong Law of Large Numbers Under Rearrangements

Abstract: We find conditions on a sequence of random variables to satisfy the strong law of large numbers (SLLN) under a rearrangement. It turns out that these conditions are necessary and sufficient for the permutational SLLN (PSLLN). By PSLLN we mean that the SLLN holds under almost all simple permutations within blocks the lengths of which grow exponentially (Prokhorov blocks). In the case of orthogonal random variables it is shown that Kolmogorov's condition, that is known not to be sufficient for SLLN, is actually sufficient for PSLLN. It is also shown that PSLLN holds for sequences that are strictly stationary with finite first moments. In the case of weakly stationary sequences a Gaposhkin result implies that SLLN and PSLLN are equivalent. Finally we consider the case of general norming and generalization of the Nikishin theorem. The methods of proof uses on the one hand the idea of Prokhorov blocks and Garsia's construction of product measure on the space of simple permutations, and on the other hand, a maximal inequality for permutations.

On Gaussian Processes Equivalent in Law to Fractional Brownian Motion

Abstract: We consider Gaussian processes that are equivalent in law to the fractional Brownian motion and their canonical representations. We prove a Hitsuda type representation theorem for the fractional Brownian motion with Hurst index H≤1/2. For the case H>1/2 we show that such a representation cannot hold. We also consider briefly the connection between Hitsuda and Girsanov representations. Using the Hitsuda representation we consider a certain special kind of Gaussian stochastic equation with fractional Brownian motion as noise.

On Sums of Products of Bernoulli Variables and Random Permutations

Abstract: Let {X k } k≥1 be independent Bernoulli random variables with parameters p k . We study the distribution of the number or runs of length 2: that is $$S_n = \sum {_{k = 1}^n {\text{ }}X_k X_{k + 1}}$$ . Let S=lim n→∞ S n . For the particular case p k =1/(k+B), B being given, we show that the distribution of S is a Beta mixture of Poisson distributions. When B=0 this is a Poisson(1) distribution. For the particular case p k =p for all k we obtain the generating function of S n and the limiting distribution of S n for $$p = \sqrt {\lambda h} + o(1/\sqrt n )$$ .

Exact Asymptotics in log log Laws for Random Fields

Abstract: Let $$\{ X,X_k ,k \in {\mathbb{N}}^r \}$$ be i.i.d. random variables, and set S n =∑ k ≤ n X k . We exhibit a method able to provide exact loglog rates. The typical result is that $${\mathop {\lim }\limits_{\varepsilon \searrow \sigma \sqrt {2r}} } \sqrt {\varepsilon ^2 - 2r\sigma ^2 } \sum\limits_n {\frac{1}{{|\,n\,|}}P(|S_n \geqslant \varepsilon \sqrt {|\,n\,|\log \log |\,n\,|} ) = \frac{{\sigma \sqrt {2r} }}{{r!}},}$$ whenever EX=0,EX 2=σ2 and E[X 2(log+ | X |) r-1] < ∞. To get this and other related precise asymptotics, we derive some general estimates concerning the Dirichlet divisor problem, of interest in their own right.

Superprocesses with Coalescing Brownian Spatial Motion as Large-Scale Limits

Abstract: A Superprocess with coalescing spatial motion is constructed in terms of one-dimensional excursions. Based on this construction, it is proved that the superprocess is purely atomic and arises as scaling limit of a special form of the superprocess with dependent spatial motion studied in Dawson et al. (Refs. 5, 19–20).

Dilated Fractional Stable Motions

Abstract: Dilated fractional stable motions are stable, self-similar, stationary increments random processes which are associated with dissipative flows. Self-similarity implies that their finite-dimensional distributions are invariant under scaling. In the Gaussian case, when the stability exponent equals 2, dilated fractional stable motions reduce to fractional Brownian motion. We suppose here that the stability exponent is less than 2. This implies that the dilated fractional stable motions have infinite variance and hence they cannot be characterised by a covariance function. These dilated fractional stable motions are defined through an integral representation involving a nonrandom kernel. This kernel plays a fundamental role. In this work, we study the space of kernels for which the dilated processes are well-defined, indicate connections to Sobolev spaces, discuss uniqueness questions and relate dilated fractional stable motions to other self-similar processes. We show that a number of processes that have been obtained in the literature, are in fact dilated fractional stable motions, for example, the telecom process obtained as limit of renewal reward processes, the Takenaka processes and the so-called “random wavelet expansion” processes.

Small Ball Asymptotics for the Stochastic Wave Equation

Abstract: We examine the small ball asymptotics for the weak solution X of the stochastic wave equation $$\frac{{\partial ^2 X}}{{\partial t^2 }}(t,x) - \frac{{\partial ^2 X}}{{\partial x^2 }}(t,x) = g(X(t,x)) + f(t,x)dW(t,x)$$ on the real line with deterministic initial conditions.

Small Ball Estimates in p-Variation for Stable Processes

Abstract: Let Zt, t ≥ 0 be a strictly stable process on \({\mathbb{R}}\) with index α ∈ (0, 2]. We prove that for every p > α, there exists γ = γα, p and \(K = K_{\alpha ,p} \in (0, + \infty)\) such that $${\mathop {\lim }\limits_{\varepsilon \downarrow 0}} \varepsilon ^\gamma \log \;{\mathbb{P}}[||Z||_p \leqslant \varepsilon ] = - K,$$ where || Z|| p stands for the strong p-variation of Z on [0,1]. The critical exponent γα p, takes a different shape according as | Z| is a subordinator and p > 1, or not. The small ball constant \(K_{\alpha ,p}\) is explicitly computed when p > 1, and a lower bound on \(K_{\alpha ,p}\) is easily obtained in the general case. In the symmetric case and when p > 2, we can also give an upper bound on \(K_{\alpha ,p}\) in terms of the Brownian small ball constant under the (1/p)-Hoder semi-norm. Along the way, we remark that the positive random variable \(||Z||_p^p\) is not necessarily stable when p > 1, which gives a negative answer to an old question of P. E. Greenwood.10

On the Existence of φ -Moments of the Limit of a Normalized Supercritical Galton–Watson Process

Abstract: Let (Z n ) n≥ 0 be a supercritical Galton–Watson process with finite re-production mean μ and normalized limit W=lim n → ∞μ−n Z n . Let further φ: [0,∞) → [0,∞) be a convex differentiable function with φ(0)=φ′(0)=0 and such that φ( $$x^{1/2^n}$$ ) is convex with concave derivative for some n ≥ 0. By using convex function inequalities due to Topchii and Vatutin, and Burkholder, Davis and Gundy, we prove that 0 < E φ (W) < ∞ if, and only if, $$E{\mathbb{L}}\phi (Z_1 ) < \infty$$ , where $${\mathbb{L}}\phi (x){\mathop = \limits^{def}} \int_0^x {\int_0^s {\tfrac{{\phi '(r)}}{r}drds,\;\;\;x \geqslant 0.} }$$ We further show that functions φ(x)=x αL(x) which are regularly varying of order α ≤ 1 at ∞ are covered by this result if α ∉ {2 n : n ≥ 0 } and under an additional condition also if α=2 n for some n≥0. This was obtained in a slightly weaker form and analytically by Bingham and Doney. If α > 1, then $${\mathbb{L}}\phi (x)$$ grows at the same order of magnitude as φ(x) so that $$E{\mathbb{L}}\phi (Z_1 ) < \infty$$ and E φ(Z 1)< ∞ are equivalent. However, α=1 implies $$\lim _{x \to \infty } {\mathbb{L}}\phi (x)/\phi (x) = \infty$$ and hence that $$E{\mathbb{L}}\phi (Z_1 ) < \infty$$ is a strictly stronger condition than E φ(Z 1) < ∞. If φ(x)=x log p x for some p > 0 it can be shown that $${\mathbb{L}}\phi (x)$$ grows like x log p+1 x, as x→∞. For this special case the result is due to Athreya. As a by-product we also provide a new proof of the Kesten–Stigum result that E Z 1 log Z 1 0 are equivalent.

Löwner's equation from a noncommutative probability perspective

Abstract: Using concepts of noncommutative probability we show that the Lowner's evolution equation can be viewed as providing a map from paths of measures to paths of probability measures. We show that the fixed point of the Lowner map is the convolution semigroup of the semicircle law in the chordal case, and its multiplicative analogue in the radial case. We further show that the Lowner evolution “spreads out” the distribution and that it gives rise to a Markov process.

On Strongly Petrovskii's Parabolic SPDEs in Arbitrary Dimension and Application to the Stochastic Cahn–Hilliard Equation

Abstract: In this paper we show that the Cahn–Hilliard stochastic PDE has a function valued solution in dimension 4 and 5 when the perturbation is driven by a space-correlated Gaussian noise. We study the regularity of the trajectories of the solution and the absolute continuity of its law at some given time and position. This is done by showing a priori estimates which heavily depend on the specific equation, and by proving general results on stochastic and deterministic integrals involving general operators on smooth domains of ℝd which are parabolic in the sense of Petrovskii, and do not necessarily define a semi-group of operators. These last estimates might be used in a more general framework.

Factorization of Markov Chains

Abstract: Existence of following factorization is proved: $$I - A = \left( {I - B} \right)\left( {I - C} \right).{\text{ }}\left( F \right)$$ Here A is a stochastic or semi-stochastic (substohastic) d×d matrix (d≤∞); I is the unit matrix; B and C are nonnegative, upper and lower triangular matrices. B is a semistochastic matrix; the diagonal entries of C are ≤1. An exact information on properties of matrices B and C are obtained in particular cases. Some results on existence of invariant distribution x for Markov chains in the cases of absence or presence of sources g of walking particles are obtained using the factorization (F). These problems described by homogeneous or nonhomogeneous equation (I−A)x=g.