On the Law of The Iterated Logarithm for Rapidly Increasing Subsequences

The usual law of the iterated logarithm states that the partial sums S_n of independent and identically distributed random variables can be normalized by the sequence a_n = √nlog log n, such that limsup_n→∞ S_n/a_n = √2 a.s. As has been pointed out by Gut (1986) the law fails if one considers the limsup along subsequences which increase faster than exponentially. In particular, for very rapidly increasing subsequences {n_k≥1} one has limsup_k→∞ S_nk/a_nk = 0 a.s. In these cases the normalizing constants a_nk have to be replaced by √nk log k to obtain a non-trivial limiting behaviour: limsup_k→∞ S_nk/ √nk log k = √2 a.s. We will present an intelligible argument for this structural change and apply it to related results.     

The law of iterated logarithm for logarithmic combinatorial assemblies

The strong convergence of dependent random variables is analyzed and the law of iterated logarithm for real additive functions defined on the class of combinatorial assemblies is obtained. Published in Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 532–547, October–December, 2006.     

A note on the Conley attractors of lower semicontinuous multifunctions

We discuss Conley-type approach to attractive sets for lower semicontinuous multifunctions. Since every iterated function system induces a Barnsley–Hutchinson multifunction which is l.s.c. in such a case it is much more natural to consider a multifunctions of that type then closed relations on compact spaces earlier considered by some authors. We use topological (Kuratowski’s) limit instead of commonly used Hausdorff metric.     

Superconvergence of interpolated collocation solutions for Hammerstein equations

In this article, we discuss the superconvergence of the interpolated collocation solutions for Hammerstein equations. Applying this new interpolation postprocessing to the collocation approximation x_h, we get a higher accuracy approximation I x_h, whose convergence order is the same as that of the iterated collocation method. Such an interpolation postprocessing method is much simpler. Also, numerical experiments are shown to demonstrate the efficiency of the interpolation postprocessing method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Understanding preservation theorems,II

We present an exposition of much of Sections VI.3 and XVIII.3 from Shelah's book Proper and Improper Forcing. This covers numerous preservation theorems for countable support iterations of proper forcing, including preservation of the property “no new random reals over V ”, the property “reals of the ground model form a non‐meager set”, the property “every dense open set contains a dense open set of the ground model”, and preservation theorems related to the weak bounding property, the weak ^ωω ‐bounding property, and the property “the set of reals of the ground model has positive outer measure” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A nonclassical LIL for geometrically weighted series in Banach space

Let {X, Xn ; n ≥ 0} be a sequence of independent and identically distributed random variables, taking values in a separable Banach space (B,||·||) with topological dual B* . Considering the geometrically weighted series ξ(β) =∑∞n=0βnXn for 0 β 1, and a sequence of positive constants {h(n), n ≥ 1}, which is monotonically approaching infinity and not asymptotically equivalent to log log n, a limit result for(1-β2)1/2||ξ(β)||/(2h(1/(1-β2)))1/2 is achieved.     

On the Iterated Biclique Operator

A biclique of a graph G is a maximal induced complete bipartite subgraph of G. The biclique graph of G, denoted by , is the intersection graph of the bicliques of G. We say that a graph G diverges (or converges or is periodic) under an operator F whenever ( for some m, or for some k and , respectively). Given a graph G, the iterated biclique graph of G, denoted by , is the graph obtained by applying the biclique operator k successive times to G. In this article, we study the iterated biclique graph of G. In particular, we classify the different behaviors of when the number of iterations k grows to infinity. That is, we prove that a graph either diverges or converges under the biclique operator. We give a forbidden structure characterization of convergent graphs, which yield a polynomial time algorithm to decide if a given graph diverges or converges. This is in sharp contrast with the situsation for the better known clique operator, where it is not even known if the corresponding problem is decidable. © 2012 Wiley Periodicals, Inc. J. Graph Theory 73: 181–190, 2013     

Generalized Law of the Iterated Logarithm and Its Convergence Rate

Abstract

This article considers the partial sums from a sequence of independent and identically distributed random variables. It is well-known that the Hartman-Wintner law of the iterated logarithm holds if and only if the second moment exists. This article studies the generalized law of the iterated logarithm for the partial sums when they are normalized by a sequence of constants that are regularly varying with index 1/2. As a result, two equivalent conditions for the law are obtained.