A Change-of-Variable Formula with Local Time on Curves

Let be a continuous semimartingale and let be a continuous function of bounded variation. Setting and suppose that a continuous function is given such that F is C^1,2 on and F is on . Then the following change-of-variable formula holds: where is the local time of X at the curve b given by and refers to the integration with respect to . A version of the same formula derived for an Itô diffusion X under weaker conditions on F has found applications in free-boundary problems of optimal stopping.     

A Continuous Super-Brownian Motion in a Super-Brownian Medium

A continuous super-Brownian motion is constructed in which branching occurs only in the presence of catalysts which evolve themselves as a continuous super-Brownian motion . More precisely, the collision local time (in the sense of Barlow et al. ⁽¹⁾) of an underlying Brownian motion path W with the catalytic mass process goerns the branching (in the sense of Dynkin's additive functional approach). In the one-dimensional case, a new type of limit behavior is encountered: The total mass process converges to a limit without loss of expectation mass (persistence) and with a nonzero limiting variance, whereas starting with a Lebesgue measure , stochastic convergence to occurs.     

Rates of Decay and h-Processes for One Dimensional Diffusions Conditioned on Non-Absorption

Let (X _t ) be a one dimensional diffusion corresponding to the operator , starting from x>0 and T ₀ be the hitting time of 0. Consider the family of positive solutions of the equation with (0, ), where . We show that the distribution of the h-process induced by any such is , for a suitable sequence of stopping times (S _M : M0) related to which converges to with M. We also give analytical conditions for , where is the smallest point of increase of the spectral measure associated to .     

A new bound for an extension of Mason’s theorem for functions of several variables

In this paper, using the generalized Wronskian, we obtain a new sharp bound for the generalized Masons theorem [1] for functions of several variables. We also show that the Diophantine equation (The generalized Fermat-Catalan equation) where , such that k out of the n-polynomials are constant, and under certain conditions for has no non-constant solution. Received: 20 March 2003     

Approximation to Probabilities Through Uniform Laws on Convex Sets

Let P be a probability distribution on ^d and let be the family of the uniform probabilities defined on compact convex sets of ^d with interior non-empty. We prove that there exists a best approximation to P in , based on the L ₂-Wasserstein distance. The approximation can be considered as the best representation of P by a convex set in the minimum squares setting, improving on other existent representations for the shape of a distribution. As a by-product we obtain properties related to the limit behavior and marginals of uniform distributions on convex sets which can be of independent interest.     

Divergence of a Random Walk Through Deterministic and Random Subsequences

Let {S _n} _n0 be a random walk on the line. We give criteria for the existence of a nonrandom sequence n _i for which respectively We thereby obtain conditions for to be a strong limit point of {S _n} or {S _n /n}. The first of these properties is shown to be equivalent to for some sequence a _i , where T(a) is the exit time from the interval [–a,a]. We also obtain a general equivalence between and for an increasing function fand suitable sequences n _i and a _i. These sorts of properties are of interest in sequential analysis. Known conditions for and (divergence through the whole sequence n) are also simplified.     

Convolution Powers of Probabilities on Stochastic Matrices

For any probability on the space of d×d stochastic matrices we associate a probability ; on a finite group—a subgroup of the permutation group—related to the kernel of the semigroup generated by the support of . We show that ⁿ converges iff ⁿ converges.     

Refinement of Convergence Rates for Tail Probabilities

Let X₁, X₂,... be, i.i.d. random variables, and put . We find necessary and sufficient moment conditions for , where δ≥ 0 and q>0, and with a_n>0 and b_n is either or The series f(x) we deal with are classical series studied by Hsu and Robbins, Erdős, Spitzer, Baum and Katz, Davis, Lai, Gut, etc     

The Minimal Subgroup of a Random Walk

It is proved that for each random walk (S _n ) _n0 on ^d there exists a smallest measurable subgroup of ^d , called minimal subgroup of (S _n ) _n0, such that P(S _n )=1 for all n1. can be defined as the set of all x ^d for which the difference of the time averages n ^{–1 n} _k=1 P(S _k ) and n ^{–1 n} _k=1 P(S _k +x) converges to 0 in total variation norm as n. The related subgroup * consisting of all x ^d for which lim _n P(S _n )–P(S _n +x)=0 is also considered and shown to be the minimal subgroup of the symmetrization of (S _n ) _n0. In the final section we consider quasi-invariance and admissible shifts of probability measures on ^d . The main result shows that, up to regular linear transformations, the only subgroups of ^d admitting a quasi-invariant measure are those of the form ₁×...× _k × ^l–k ×{0} ^d–l , 0kld, with ₁,..., _k being countable subgroups of . The proof is based on a result recently proved by Kharazishvili⁽³⁾ which states no uncountable proper subgroup of admits a quasi-invariant measure.     

Characterization of Normal Traces on Von Neumann Algebras by Inequalities for the Modulus

It is proved that if a normal semifinite weight on a von Neumann algebra satisfies the inequality for any selfadjoint operators in , then this weight is a trace. Several similar characterizations of traces among the normal semifinite weights are proved. In particular, Gardner's result on the characterization of traces by the inequality is refined and reinforced.     

The Law of the Iterated Logarithm for the Total Length of the Nearest Neighbor Graph

Let be the Poisson point process with intensity 1 in [–n,n] ^d . We prove the law of the iterated logarithm for the total length of the nearest neighbor graph on .     

On the Weak Limiting Behavior of Almost Surely Convergent Row Sums from Infinite Arrays of Rowwise Independent Random Elements in Banach Spaces

For an array {V _nk ,k1,n1} of rowwise independent random elements in a real separable Banach space with almost surely convergent row sums , we provide criteria for S _n –A _n to be stochastically bounded or for the weak law of large numbers to hold where {A _n ,n1} is a (nonrandom) sequence in .     

The Law of the Logarithm for Weighted Sums of Independent Random Variables

Let X,X _n ;n1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2<1. Set b _n =B(n),n1. If

Sharp Conditions for the CLT of Linear Processes in a Hilbert Space

In this paper we study the behavior of sums of a linear process associated to a strictly stationary sequence with values in a real separable Hilbert space and are linear operators from H to H. One of the results is that satisfies the CLT provided are i.i.d. centered having finite second moments and . We shall provide an example which shows that the condition on the operators is essentially sharp. Extensions of this result are given for sequences of weak dependent random variables under minimal conditions.     

The Metric of Large Deviation Convergence

We construct a metric space of set functions ( , d) such that a sequence {P _n} of Borel probability measures on a metric space ( , d*) satisfies the full Large Deviation Principle (LDP) with speed {a _n} and good rate function I if and only if the sequence converges in ( , d) to the set function e ^–I . Weak convergence of probability measures is another special case of convergence in ( , d). Properties related to the LDP and to weak convergence are then characterized in terms of ( , d).     

Large Deviations of the Finite Cluster Shape for Two-Dimensional Percolation in the Hausdorff and L1 Metric

We consider supercritical two-dimensional Bernoulli percolation. Conditionally on the event that the open cluster C containing the origin is finite, we prove that: the laws of C/N satisfy a large deviations principle with respect to the Hausdorff metric; let f(N) be a function from to such that f(N)/ln N+ and f(N)/N0 as N goes to the laws of {x ² : d(x, C)f(N)}/N satisfy a large deviations principle with respect to the L ¹ metric associated to the planer Lebesgue measure. We link the second large deviations principle with the Wulff construction.     

Packing Measure and Dimension of Random Fractals

We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals , their almost sure Hausdorff dimension. We show that some almost deterministic conditions known to ensure that the Hausdorff measure satisfies also imply that the packing measure satisfies 0< . When these conditions are not satisfied, it is known . Correspondingly, we show that in this case , provided a random strong open set condition is satisfied. We also find gauge functions (t) so that the -packing measure is finite.     

Interpolation orbits in couples of Lebesgue spaces

The paper deals with interpolation orbits for linear operators acting from a arbitrary couple { (U ₀), (U ₁)} of weighted L _p spaces into an arbitrary couple { (V ₀), (V ₁)} of such spaces, where 1 p ₀,p ₁,q ₀,q ₁ . Here L _p (U) is the space of measurable functions f on a measure space such that fU L _p , equipped with the norm . The paper describes the orbits of arbitrary elements a (U ₀) + (U ₁). It contains proofs of the results announced in C. R. Acad. Sci. Paris, Ser. I, 334, 881–884 (2002).__________Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 39, No. 1, pp. 56–68, 2005Original Russian Text Copyright © by V. I. OvchinnikovTranslated by V. I. Ovchinnikov     

The Functional Law of the Iterated Logarithm for the Empirical Process Based on Sample Means

Consider a double array of i.i.d. random variables with mean and variance and set . Let denote the empirical distribution function of Z_{1, n} ,..., Z _{N, n} and let be the standard normal distribution function. The main result establishes a functional law of the iterated logarithm for , where n=n(N) as N. For the proof, some lemmas are derived which may be of independent interest. Some corollaries of the main result are also presented.