Limit theorems for free multiplicative convolutions

We determine the distributional behavior for products of free random variables in a general infinitesimal triangular array. The main theorems in this paper extend a result for measures supported on the positive half-line, and provide a new limit theorem for measures on the unit circle with nonzero first moment.

     

Limit theorems for large deviations probabilities of certain quadratic forms

Let (ξ _k ,F _k ) be a martingale difference sequence. The paper concerns the tail behavior of the quadratic formS _n = ∑ _k=1 ⁿ ∑ _j=1 ^k−1 β _n ^k−j χ _k χ _j , where β _n asn→∞. The main conclusions aboutP}n ⁻¹ S _n >x _n }, wherex _n →∞, asn→∞, are obtained using the tail behavior of a martingale with values in a certain Hilbert space. Vilnius University, Naugarduko 24; Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 37, No. 4, pp. 532–549, October–December, 1997.     

Idempotent probabilities on semigroups

Summary In this paper, idempotent probability measures have been considered on semigroups which are locally compact or metric and satisfy: (*) A ^–1 B and Ax ^–1 are compact whenever A and B are so, for every x in the semigroup. Such semigroups are more general than compact semigroups which do admit of such measures. On such semigroups we can construct such measures by the usual process if there is a compact sub-semigroup. It is shown in this paper that if such a measure exists in such semigroups, then it must be such an extension measure. Some related results concerning the conditions (*) are also discussed here.     

Local limit theorems for multiplicative free convolutions

This paper describes the quality of convergence to an infinitely divisible law relative to free multiplicative convolution. We show that convergence in distribution for products of identically distributed and infinitesimal free random variables implies superconvergence of their probability densities to the density of the limit law. Superconvergence to the marginal law of free multiplicative Brownian motion at a specified time is also studied. In the unitary case, the superconvergence to free Brownian motion and that to the Haar measure are shown to be uniform over the entire unit circle, implying further a free entropic limit theorem and a universality result for unitary free Lévy processes. Finally, the method of proofs on the positive half-line gives rise to a new multiplicative Boolean to free Bercovici–Pata bijection.     

Limit theorems for semigroups with perturbed generators,with applications to multiscaled random evolutions

For parameters η, let {B(η)} denote infinitesimal operators of strongly continuous semigroups, with resolvents R(λ; B(η)) satisfying λR(λ; B(η)) = P(η) + λV(η) + o(λ). For parameters α, let {A(α)} denote possibly unbounded, linear operators for which {A(α) + B(η)} are infinitesimal operators of strongly continuous semigroups {U_α·η(t)}. For α, η converging simultaneously, we show strong convergence of the semigroups U_α·η(t) to a strongly continuous semigroup U(t), with limiting infinitesimal operator characterized by lim_α·η ∑_jP(η) A(α) × (V(η) A(α))ⁱf. We give applications of the abstract perturbation theorems to limit theorems of random evolutions and associated abstract Cauchy problems, in which multiscaling occurs in the convergence.     

Perturbation theorems for holomorphic semigroups

The concept of the gap function is used to give new perturbation results for generators of holomorphic semigroups. In particular, we show that if A is the generator of a holomorphic semigroup on a Banach space and , then every closed linear operator C such that for some and

Commutativity theorems for cancellative semigroups

Psomopoulos has proved that \([x^n, y] = [x, y^{n+1}]\) for a positive integer n implies commutativity in groups. Here we show that cancellative semigroups admitting commutators and satisfying the identity \([x^n, y] = [x, y^{n+k}]\) implies that the element \(y^k\) is central. The special case of \(k=1\) yields the above mentioned commutativity theorem. To accommodate negative exponents, we consider the functional equation \([f(x), y] = [x, g(y)f(y)] \) where f and g are unary functions satisfying certain formal syntactic rules and prove that in cancellative semigroups admitting commutators, the functional equation \([f(x), y] = [x, g(y)f(y)]\) implies that the element g(y) is central i.e. \(xg(y) = g(y)x\) for all x and y. By the way, these results are new even in group theory.     

Limit theorems for mergesort

Central and local limit theorems (including large deviations) are established for the number of comparisons used by the standard top-down recursive mergesort under the uniform permutation model. The method of proof utilizes Dirichlet series, Mellin transforms, and standard analytic methods in probability theory. © 1996 John Wiley & Sons, Inc.     

Limit laws for normed and weighted Boolean convolutions

We prove a central limit theorem and a weak law of large numbers for normed Boolean convolutions with weighted components. Our technical assumptions on normalizations and weights cover many interesting outcomes, and both limit laws hold under minimal moment conditions.     

Mean ergodic theorems for bi-continuous semigroups

In this paper we study the main properties of the Cesàro means of bi-continuous semigroups, introduced and studied by Kühnemund (Semigroup Forum 67:205–225, 2003). We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in C _b (ℝ ^N ) and to evolution operators associated with nonautonomous second-order differential operators in C _b (ℝ ^N ) with time-periodic coefficients.     

Multiparameter ratio ergodic theorems for semigroups

After one-parameter treatment of ratio ergodic theorems for semigroups, we formulate the Sucheston a.e. convergence principle of continuous parameter type. This principle plays an effective role in proving some multiparameter generalizations of Chacon?s type continuous ratio ergodic theorems for semigroups and of Jacobs? type continuous random ratio ergodic theorems for quasi-semigroups. In addition, a continuous analogue of the Brunel–Dunford–Schwartz ergodic theorem is given of sectorially restricted averages for a commutative family of semigroups. We also formulate a local a.e. convergence principle of Sucheston?s type. The local convergence principle is effective in proving multiparameter local ergodic theorems. In fact, a multiparameter generalization of Akcoglu–Chacon?s local ratio ergodic theorem for semigroups of positive linear contractions on _L1$L_1$ is proved. Moreover, some multiparameter martingale theorems are obtained as applications of convergence principles.     

On the residuality of mixing by convolutions probabilities

A probability measureμ on a locally compactσ — compact amenable Hausdorff groupG is called mixing by convolutions if for every pair of probabilitiesν ₁,ν ₂ onG we have: $$\mathop {\lim }\limits_{n \to \infty } \left\| {\left( {\nu _1 - \nu _2 } \right) \star \mu ^{ \star n} } \right\| = \mathop {\lim }\limits_{n \to \infty } \left\| {\left( {\nu _1 - \nu _2 } \right) \star \mu ^{ \star n} } \right\| = 0.$$ . It is proved that the set of all mixing by convolutions probabilities is a norm (variation) dense subset of the setP(G) of all probabilities onG. IfG is additionally second countable the mixing measures are residual inP(G).