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Central Limit Theorem visualized in Excel

Hodaňová J., Zdráhal T.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

2014

Vol. 19

51--55

The Central Limit Theorem states that regardless of the underlying distribution, the distribution of the sample means approaches normality as the sample size increases. This paper describes the steps in MS Excel to help students’ better understanding of this theorem.

The speed of convergence of random products of sums of independent random variables

Krajka T., Rychlik Z.

Journal of Mathematics and Applications

2009

Vol. 31

51-66

On the almost sure central limit theorems in the joint version for the maxima and minima of some random variables

Dudziński M., Furmańczyk K.

Demonstratio Mathematica

2008

Vol. 41, nr 3

705-722

Let: {Xi} be a sequence of r.v.'s, and: Mn := max (X1,..., Xn), mn := min (X1,..., Xn). Our goal is to prove the almost sure central limit theorem for the properly normalized vector {Mn,mn}, provided: 1) {Xi} is an i.i.d. sequence, 2) {Xi} is a certain standardized stationary Gaussian sequence.

On the almost sure central limit theorems for the vectors of several large maxima and for some random permanents

Dudziński M.

Demonstratio Mathematica

2006

Vol. 39, nr 4

949-963

In our paper we prove two kinds of the so-called almost sure central limit theorem (ASCLT). The first one is the ASCLT for the vectors ((Mn(1) , . . . ,Mn(r)), where Mn(j) n - the j-th largest maximum of X1, . . . ,Xn and {Xi} is an i.i.d. sequence. Our second result is the ASCLT for some random permanents.

On the hyperbolic Hausdorff dimension of the boundary of a basin of attraction for a holomorphic map and of quasirepellers

Przytycki F.

Bulletin of the Polish Academy of Sciences. Mathematics

2006

Vol. 54, no 1

41-52

We prove that the hyperbolic Hausdorff dimension of Fr Ω, the boundary of the simply connected immediate basin of attraction Ω to an attracting periodic point of a rational mapping of the Riemann sphere, which is not a finite Blaschke product in some holomorphic coordinates, or a 2 : 1 factor of a Blaschke product, is larger than 1. We prove a "local version" of this theorem, for a boundary repelling to the side of the domain. The results extend an analogous fact for polynomials proved by A. Zdunik and relies on the theory elaborated by M. Urbanski, A. Zdunik and the author in the late 80-ties. To prove that the dimension is larger than 1, we use expanding repellers in δΩ constructed in [P2]. To reach our results, we deal with a quasi-repeller, i.e. the limit set for a geometric coding tree, and prove that the hyperbolic Hausdorff dimension of the limit set is larger than the Hausdorff dimension of the projection via the tree of any Gibbs measure for a Holder potential on the shift space, under a non-cohomology assumption. We also consider Gibbs measures for Holder potentials on Julia sets.