Central limit theorem for a class of one-dimensional kinetic equations

We introduce a class of kinetic-type equations on the real line, which constitute extensions of the classical Kac caricature. The collisional gain operators are defined by smoothing transformations with rather general properties. By establishing a connection to the central limit problem, we are able to prove long-time convergence of the equation??s solutions toward a limit distribution. For example, we prove that if the initial condition belongs to the domain of normal attraction of a certain stable law ?? _??, then the limit is a scale mixture of ?? _??. Under some additional assumptions, explicit exponential rates for the convergence to equilibrium in Wasserstein metrics are calculated, and strong convergence of the probability densities is shown.     

Weak convergence to multifractional Brownian motion of Riemann-Liouville type in Besov spaces

We study the weak convergence of the family of processes {V _n (t)} _n??? defined by $$V_n(t)=\int_{0}^t(t-u)^{H(t)-\frac{1}{2}}\theta_n(u)du,$$ where {?? _n (u)} _n??? is a family of processes converging in law to a Brownian motion, as n????. We consider two cases of {?? _n }. First, we construct ?? _n based on the well-known Donsker??s theorem and show that {V _n (t)} _n??? converges in law to a multifractional Brownian motion of Riemann-Liouville type, as n????. Second, we construct ?? _n based on a Poisson process, and then show that a multifractional Brownian motion of Riemann-Liouville type can be approximated in law by {V _n (t)} _n???.     

Functional central limit theorems for sums of nearly nonstationary processes*

We study some Holderian functional central limit theorems for the polygonal partial-sum processes built on a first-order autoregressive process y _n,k ?=??? _n y _n,k?1?+??? _k with ? _n converging to 1 and i.i.d. centered square-integrable innovations. In the case where ? _n ?=?e ^??/n with a negative constant ??, we prove that the limiting process is an integrated Ornstein?CUhlenbeck one. In the case where ? _n ?=?1? ?? _n /n, with ?? _n tending to infinity slower than n, the convergence to Brownian motion is established in Holder space in terms of the rate of ?? _n and the integrability of the ?? _k s.     

On the convergence to the multiple Wiener-Itô integral

We study the convergence to the multiple Wiener-Itô integral from processes with absolutely continuous paths. More precisely, consider a family of processes, with paths in the Cameron-Martin space, that converges weakly to a standard Brownian motion in C₀([0,T]). Using these processes, we construct a family that converges weakly, in the sense of the finite dimensional distributions, to the multiple Wiener-Itô integral process of a function f∈L²(n[0,T]). We prove also the weak convergence in the space C₀([0,T]) to the second-order integral for two important families of processes that converge to a standard Brownian motion.     

Applications of the continuous-time ballot theorem to Brownian motion and related processes

Motivated by questions related to a fragmentation process which has been studied by Aldous, Pitman, and Bertoin, we use the continuous-time ballot theorem to establish some results regarding the lengths of the excursions of Brownian motion and related processes. We show that the distribution of the lengths of the excursions below the maximum for Brownian motion conditioned to first hit λ>0 at time t is not affected by conditioning the Brownian motion to stay below a line segment from (0,c) to (t,λ). We extend a result of Bertoin by showing that the length of the first excursion below the maximum for a negative Brownian excursion plus drift is a size-biased pick from all of the excursion lengths, and we describe the law of a negative Brownian excursion plus drift after this first excursion. We then use the same methods to prove similar results for the excursions of more general Markov processes.     

On the distribution of multiples of real numbers

We introduce new series (of the variable ??) that enable to measure the irregularity of distribution of the sequence of fractional parts {n??}. A detailed analysis of the convergence and divergence of these series is done, depending mainly on the convergents of ??. As a by product, we obtain new Fourier series of square integrable functions that converge almost everywhere but at no rational number.     

Regularity of local times of random fields

In this paper, we study the fractional smoothness of local times of general processes starting from the occupation time formula, and obtain the quasi-sure existence of local times in the sense of the Malliavin calculus. This general result is then applied to the local times of N-parameter d-dimensional Brownian motions, fractional Brownian motions and the self-intersection local time of the 2-dimensional Brownian motion, as well as smooth semimartingales.     

“Trees under attack”: a Ray–Knight representation of Feller’s branching diffusion with logistic growth

We obtain a representation of Feller’s branching diffusion with logistic growth in terms of the local times of a reflected Brownian motion H with a drift that is affine linear in the local time accumulated by H at its current level. As in the classical Ray–Knight representation, the excursions of H are the exploration paths of the trees of descendants of the ancestors at time t = 0, and the local time of H at height t measures the population size at time t. We cope with the dependence in the reproduction by introducing a pecking order of individuals: an individual explored at time s and living at time t = H _s is prone to be killed by any of its contemporaneans that have been explored so far. The proof of our main result relies on approximating H with a sequence of Harris paths H ^N which figure in a Ray–Knight representation of the total mass of a branching particle system. We obtain a suitable joint convergence of H ^N together with its local times and with the Girsanov densities that introduce the dependence in the reproduction.     

On A-invariant mean and A-almost convergence

The idea of A-invariant mean and A-almost convergence is due to J. P. Duran [8], which is a generalization of the usual notion of Banach limit and almost convergence. In this paper, we discuss some important properties of this method and prove that the space F(A) of A-almost convergent sequences is a BK space with ?? · ??_??, and also show that it is a nonseparable closed subspace of the space l _?? of bounded sequences.     

Limit theorems for random processes with random time substitution

In this paper we prove a theorem on sufficient conditions for the convergence in the Skorokhod space D[0, 1] of a sequence of random processes with random time substitution. We obtain almost sure versions of this theorem.     

Weak convergence to fractional Brownian motion in Brownian scenery

 Kesten and Spitzer have shown that certain random walks in random sceneries converge to stable processes in random sceneries. In this paper, we consider certain random walks in sceneries defined using stationary Gaussian sequence, and show their convergence towards a certain self-similar process that we call fractional Brownian motion in Brownian scenery. Received: 17 April 2002 / Revised version: 11 October 2002 / Published online: 15 April 2003 Research supported by NSFC (10131040). Mathematics Subject Classification (2002): 60J55, 60J15, 60J65 Key words or phrases: Weak convergence – Random walk in random scenery – Local time – Fractional Brownian motion in Brownian scenery     

Self-intersection local times and collision local times of bifractional Brownian motions

In this paper, we consider the local time and the self-intersection local time for a bifractional Brownian motion, and the collision local time for two independent bifractional Brownian motions. We mainly prove the existence and smoothness of the self-intersection local time and the collision local time, through the strong local nondeterminism of bifractional Brownian motion, L2 convergence and Chaos expansion.     

Statistical lacunary summability and a Korovkin type approximation theorem

In this paper, we introduce statistical lacunary summability and strongly ?? _q -convergence (0 < q < ??) and establish some relations between lacunary statistical convergence, statistical lacunary summability, and strongly ?? _q -convergence. We further apply our new notion of summability to prove a Korovkin type approximation theorem.     

Adjoint methods for static Hamilton�CJacobi equations

We use the adjoint methods to study the static Hamilton?CJacobi equations and to prove the speed of convergence for those equations. The main new ideas are to introduce adjoint equations corresponding to the formal linearizations of regularized equations of vanishing viscosity type, and from the solutions ?? ^?? of those we can get the properties of the solutions u of the Hamilton?CJacobi equations. We classify the static equations into two types and present two new ways to deal with each type. The methods can be applied to various static problems and point out the new ways to look at those PDE.     

On the character of convergence to Brownian local time. I

Summary Many results are known about the convergence of some processes to Brownian local time. Among such processes are the process of occupation times of Brownian motion, the number of downcrossings of Brownian motion over smaller and smaller intervals before timet, the number of visits of the recurrent integer-valued random walk to some point duringn steps and others. In this paper we consider the asymptotic behaviour of the differences between Brownian local time and some of the processes which converge to it.     

Level sets of the Takagi function: local level sets

The Takagi function ??: [0,1] ?? [0, 1] is a continuous non-differentiable function constructed by Takagi in 1903. The level sets L(y)?=?{x : ??(x)?=?y} of the Takagi function ??(x) are studied by introducing a notion of local level set into which level sets are partitioned. Local level sets are simple to analyze, reducing questions to understanding the relation of level sets to local level sets, which is more complicated. It is known that for a ??generic?? full Lebesgue measure set of ordinates y, the level sets are finite sets. In contrast, here it is shown for a ??generic?? full Lebesgue measure set of abscissas x, the level set L(??(x)) is uncountable. An interesting singular monotone function is constructed associated to local level sets, and is used to show the expected number of local level sets at a random level y is exactly ${\frac{3}{2}}$ .     

Wasserstein convergence rates for random bit approximations of continuous Markov processes

We determine the convergence speed of a numerical scheme for approximating one-dimensional continuous strong Markov processes. The scheme is based on the construction of certain Markov chains whose laws can be embedded into the process with a sequence of stopping times. Under a mild condition on the process' speed measure we prove that the approximating Markov chains converge at fixed times at the rate of 1/4 with respect to every p-th Wasserstein distance. For the convergence of paths, we prove any rate strictly smaller than 1/4. Our results apply, in particular, to processes with irregular behavior such as solutions of SDEs with irregular coefficients and processes with sticky points.     

Iterating Brownian Motions,Ad Libitum

Let B ₁,B ₂,… be independent one-dimensional Brownian motions parameterized by the whole real line such that B _i (0)=0 for every i≥1. We consider the nth iterated Brownian motion W _n (t)=B _n (B _n?1(?(B ₂(B ₁(t)))?)). Although the sequence of processes (W _n ) _n≥1 does not converge in a functional sense, we prove that the finite-dimensional marginals converge. As a consequence, we deduce that the random occupation measures of W _n converge to a random probability measure μ _∞. We then prove that μ _∞ almost surely has a continuous density which should be thought of as the local time process of the infinite iteration W _∞ of independent Brownian motions. We also prove that the collection of random variables (W _∞(t),t∈??{0}) is exchangeable with directing measure μ _∞.     

Strong approximations of three-dimensional Wiener sausages

We prove that the centered three-dimensional Wiener sausage can be strongly approximated by a one-dimensional Brownian motion running at a suitable time clock. The strong approximation gives all possible laws of iterated logarithm as well as the convergence in law in terms of process for the normalized Wiener sausage. The proof relies on Le Gall [10]șs fine L ²-norm estimates between the Wiener sausage and the Brownian intersection local times. Research supported by the Hungarian National Foundation for Scientific Research, Grants T 037886, T 043037 and K 61052.     

Complex series for 1/π

Many series for 1/?? were discovered since the appearance of S. Ramanujan??s famous paper ??Modular equations and approximation to ???? published in 1914. Almost all these series involve only real numbers. Recently, in an attempt to prove a series for 1/?? discovered by Z.-W.?Sun, the authors found that a series for 1/?? involving complex numbers is needed. In this article, we illustrate a method that would allow us to prove series of this type.