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.     

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 .     

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.     

Selective Limit Theorems for Random Walks on Parabolic Biangle and Triangle Hypergroups

Let K be respectively the parabolic biangle and the triangle in and be a sequence in [0, +[ such that lim_p (p)=+. According to Koornwinder and Schwartz,⁽⁷⁾ for each there exist a convolution structure (*_(p)) such that (K, *_(p)) is a commutative hypergroup. Consider now a random walk on (K, *_(p)), assume that this random walk is stopped after j(p) steps. Then under certain conditions given below we prove that the random variables on K admit a selective limit theorems. The proofs depend on limit relations between the characters of these hypergroups and Laguerre polynomials that we give in this work.     

On the Harris Recurrence of Iterated Random Lipschitz Functions and Related Convergence Rate Results

A result by Elton⁽⁶⁾ states that an iterated function system

Weak Variation of Gaussian Processes

Let X(t) (tR) be a real-valued centered Gaussian process with stationary increments. We assume that there exist positive constants ₀, C ₁, and c ₂ such that for any tR and hR with |h|₀ and for any 0r<min{|t|, ₀} where is regularly varying at zero of order (0 < < 1). Let be an inverse function of near zero such that (s)=(s) log log(1/s) is increasing near zero. We obtain exact estimates for the weak -variation of X(t) on [0,a].     

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.     

Renewal Theorems for Singular Differential Operators

Let * be the convolution on M( ₊) associated with a second order singular differential operator L on ]0, +[. If is a probability measure on ₊ with suitable moment conditions, we study how to normalize the measures * ⁿ ; n } (resp. ) in order to get vague convergence if n+ (resp. x+). The results depend on the asymptotic drift of the operator L and on a precise study of the asymptotic behaviour of its eigenfunctions.     

Multiple Wick Product Chaos Processes

Let u(x) xR ^q be a symmetric nonnegative definite function which is bounded outside of all neighborhoods of zero but which may have u(0)=. Let p _x, (·) be the density of an R ^q valued canonical normal random variable with mean x and variance and let {G _x, ; (x, )R ^q ×[0,1 ]} be the mean zero Gaussian process with covariance

A Generalization of the Hausdorff-Young Theorem

Considering mixed-norm sequence spaces lp,q, p, q 1, C. N. Kellogg proved the following theorem: if 1 < p 2 then lp,2 and lp,2 , where 1/p + 1/p = 1. This result extends the Hausdorff-Young Theorem.We introduce here multiple mixed-norm sequence spaces , examine their properties and characterize the multipliers of spaces of the form lp,[s;n],q, with the index s repeated n times. By an interpolation-type argument we prove that (l,[2;n],2, lp,[1;n],1) for 1 < p 2. Using these results we obtain a further generalization of the Hausdorff-Young Theorem: if 1 < p 2 then lp,[2;n] and lp,[2;n] for each n = 0, 1, 2, ¨. The spaces lp,[2;n] decrease and lp,[2;n] increase properly with n for 1 < p < 2 and 1/p + 1/p = 1. We also extend a theorem of J. H. Hedlund on multiplers of Hardy spaces and deduce other results.     

Nontrivial Phase Transition in a Continuum Mirror Model

We consider a Poisson point process on with intensity , and at each Poisson point we place a two sided mirror of random length and orientation. The length and orientation of a mirror is taken from a fixed distribution, and is independent of the lengths and orientations of the other mirrors. We ask if light shone from the origin will remain in a bounded region. We find that there exists a with 0 < < for which, if < , light leaving the origin in all but a countable number of directions will travel arbitrariliy far from the origin with positive probability. Also, if > , light from the origin will almost surely remain in a bounded region.     

The Invariants of the Clifford Groups

The automorphism group of the Barnes-Wall lattice L _m in dimension 2 ^m (m ; 3) is a subgroup of index 2 in a certain Clifford group of structure 2 ₊ ^1+2m . O ⁺(2m,2). This group and its complex analogue of structure .Sp(2m, 2) have arisen in recent years in connection with the construction of orthogonal spreads, Kerdock sets, packings in Grassmannian spaces, quantum codes, Siegel modular forms and spherical designs. In this paper we give a simpler proof of Runge@apos;s 1996 result that the space of invariants for of degree 2k is spanned by the complete weight enumerators of the codes , where C ranges over all binary self-dual codes of length 2k; these are a basis if m k - 1. We also give new constructions for L _m and : let M be the -lattice with Gram matrix . Then L _m is the rational part of M ^m, and = Aut(M^m). Also, if C is a binary self-dual code not generated by vectors of weight 2, then is precisely the automorphism group of the complete weight enumerator of . There are analogues of all these results for the complex group , with doubly-even self-dual code instead of self-dual code.     

Small Ball Estimates in <Emphasis Type="Italic">p</Emphasis>-Variation for Stable Processes

Let Z _t , t 0 be a strictly stable process on with index (0, 2]. We prove that for every p > , there exists = _{, p} and such that

Markov Processes with Equal Capacities

Let X and be transient standard Markov processes in weak duality with respect to a -finite measure m. Let (Y, , ) be a second dual pair with the same state space E as (X, , m). Let Cap ^X and Cap ^Y be the 0-order capacities associated with (X, , m) and (Y, , ), and let V and denote the potential kernels for Y and . Assume that singletons are polar with respect to both X and Y, and that semipolar sets are of capacity zero for both dual pairs. We show that if Cap ^X (B)=Cap ^Y (B) for every Borel subset of E then there is a strictly increasing continuous additive functional D=(D _t) _t0 of (X, , m) such that

On Codes Identifying Sets of Vertices in Hamming Spaces

A code is called (t, 2)-identifying if for all the words x, y(x y) and the sets (B _t (x) B _t (y)) C and are nonempty and different. Constructions of such codes and a lower bound on the cardinality of these codes are given. The lower bound is shown to be sharp in some cases. We also discuss a more general notion of -identifying codes and introduce weakly identifying codes.     

A Large Deviations Principle Related to the Strong Arc-Sine Law

We show a large deviations principle for the family of random variables when t+, where B=(B _u ,u0) is a standard linear Brownian motion.     

Phase Retrieval for Probability Distributions on Quantum Groups and Braided Groups

For nilpotent quantum groups [as introduced by Franz et al. ⁽⁷⁾], we show that (in sharp contrast to the classical case) the symmetrization of a probability distribution and the first moments of together determine uniquely the original distribution .     

The Expected Number of Local Maxima of a Random Algebraic Polynomial

In this work we obtain an asymptotic estimate for the expected number of maxima of the random algebraic polynomial , where a _j (j=0, 1,...,n–1) are independent, normally distributed random variables with mean and variance one. It is shown that for nonzero , the expected number of maxima is asymptotic to log n, when n is large.     

Entropy for Random Partitions and Its Applications

Asymptotic properties of partitions of the unit interval are studied through the entropy for random partition