Most Power Series have Radius of Convergence 0 or 1*

1975 ◽  
Vol 18 (1) ◽  
pp. 39-40
Author(s):  
J. J. F. Fournier ◽  
P. M. Gauthier
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Complex Plane ◽  
Probability Space ◽  
Random Variables ◽  
Radius Of Convergence ◽  
Independent Random Variables ◽  
Natural Boundary ◽  
Random Power Series

Consider a random power series Σ0∞ cn zn, that is, with coefficients {cn}0∞ chosen independently at random from the complex plane. What is the radius of convergence of such a series likely to be?One approach to this question is to let the {cn}0∞ be independent random variables on some probability space. It turns out that, with probability one, the radius of convergence is constant. Moreover, if the cn are symmetric and have the same distribution, then the circle of convergence is almost surely a natural boundary for the analytic function given by the power series (See [1, Ch. IV, Section 3]). Our treatment of the question will be elementary and will not use these facts.

The distribution of the values of a random power series in the unit disk

1983 ◽  
Vol 94 (3-4) ◽  
pp. 251-263 ◽  
Author(s):  
Matthias Jakob ◽  
A. C. Offord
Keyword(s):  
Power Series ◽  
Unit Disk ◽  
Random Variables ◽  
Equal Probability ◽  
Independent Random Variables ◽  
Random Power Series ◽  
The Family ◽  
Unit Radius ◽  
Image Position ◽  
Almost All

SynopsisThis is a study of the family of power series where Σ αnZn has unit radius of convergence and the εn are independent random variables taking the values ±1 with equal probability. It is shown that ifthen almost all these power series take every complex value infinitely often in the unit disk.

scholarly journals Random Power Series in Q p , 0 Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Haiyin Li ◽  
Yan Wu
Keyword(s):  
Power Series ◽  
Probability Space ◽  
Random Variables ◽  
Random Power Series ◽  
Independent Identically Distributed

Aulaskari et al. proved if 0 < p < 1 and ε n is sequence of independent, identically distributed Rademacher random variables on a probability space, then the condition Σ n = 0 ∞ n 1 − p a n 2 < ∞ implies that the random power series R f z = ∑ n = 0 ∞ a n ε n z n ∈ Q p almost surely. In this paper, we improve this result showing that the condition Σ n = 0 ∞ n 1 − p a n 2 < ∞ actually implies R f ∈ Q p , 0 almost surely.

scholarly journals Some results on the span of families of Banach valued independent, random variables

1991 ◽  
Vol 14 (2) ◽  
pp. 381-384
Author(s):  
Rohan Hemasinha
Keyword(s):  
Banach Space ◽  
Probability Space ◽  
Linear Span ◽  
Random Variables ◽  
Isomorphic Copy ◽  
Independent Random Variables ◽  
Closed Linear Span

LetEbe a Banach space, and let(Ω,ℱ,P)be a probability space. IfL1(Ω)contains an isomorphic copy ofL1[0,1]then inLEP(Ω)(1≤P<∞), the closed linear span of every sequence of independent,Evalued mean zero random variables has infinite codimension. IfEis reflexive orB-convex and1<P<∞then the closed(in LEP(Ω))linear span of any family of independent,Evalued, mean zero random variables is super-reflexive.

Accurate power series for eigenvalues of spheroidal angle functions and their convergence radii

2011 ◽  
Vol 89 (11) ◽  
pp. 1083-1099
Author(s):  
Tam Do-Nhat
Keyword(s):  
Power Series ◽  
Least Squares ◽  
Complex Plane ◽  
Branch Point ◽  
Upper Bound ◽  
Least Squares Method ◽  
Radius Of Convergence ◽  
New Method ◽  
Recursion Relations

In this paper, the radius of convergence of the spheroidal power series associated with the eigenvalue is calculated without using the branch point approach. Studying the properties of the power series using the recursion relations among its coefficients in the new method offers some insights into the spheroidal power series and its associated eigenfunction. This study also used the least squares method to accurately compute the convergence radii to five or six significant digits. Within the circle of convergence in the complex plane of the parameter c = kF, where k is the wavenumber and F is the semifocal length of the spheroidal system, the extremely fast convergent spheroidal power series are computed with full precision. In addition, a formula for the magnitude of the upper bound of the error is obtained.

Power series methods and almost sure convergence

1993 ◽  
Vol 113 (1) ◽  
pp. 195-204 ◽  
Author(s):  
Rüdiger Kiesel
Keyword(s):  
Power Series ◽  
Probability Space ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Power Series Methods

Let (Ω, Σ, P) be a probability space and suppose that all random variables are defined on this space.

ON POLYNOMIAL SEQUENCES WITH RESTRICTED GROWTH NEAR INFINITY

2002 ◽  
Vol 34 (2) ◽  
pp. 189-199 ◽  
Author(s):  
J. MÜLLER ◽  
A. YAVRIAN
Keyword(s):  
Analytic Function ◽  
Power Series ◽  
Complex Plane ◽  
Partial Sums ◽  
Limit Function ◽  
Maximal Domain ◽  
Polynomial Sequences ◽  
Domain Of Existence ◽  
Similar Growth ◽  
Closed Plane

Let (Pn) be a sequence of polynomials which converges with a geometric rate on some arc in the complex plane to an analytic function. It is shown that if the sequence has restricted growth on a closed plane set E which is non-thin at ∞, then the limit function has a maximal domain of existence, and (Pn) converges with a locally geometric rate on this domain. If (snk) is a sequence of partial sums of a power series, a similar growth restriction on E forces the power series to have Ostrowski gaps. Moreover, the requirement of non-thinness of E at ∞ is necessary for these conclusions.

Crossings of Set-Partitions and Addition Crossings of Graded-Independent Random Variables

1997 ◽  
Vol 08 (05) ◽  
pp. 645-664 ◽  
Author(s):  
James A. Mingo ◽  
Alexandru Nica
Keyword(s):  
Power Series ◽  
Tensor Product ◽  
Linear Functional ◽  
Random Variables ◽  
Independent Random Variables ◽  
Set Partitions ◽  
Crossing Numbers ◽  
Exponential Generating Function ◽  
Graded Tensor Product ◽  
Formal Power

We consider a q-deformation, introduced in [7], of the cumulants associated with a linear functional on polynomials; combinatorially, the deformation is defined using crossing numbers of set-partitions. The paper is concerned with the case q = -1. We show that if the linear functional μ : C[X] → C is symmetric (i.e.μ(Xn) = 0 for n odd), then the exponential generating function of the even (-1)-cumulants of μ is equal to [Formula: see text], (as a formal power series in z). We discuss the connection of this fact with the form of independence for (non-commutative) random variables which corresponds to the operation of Z2-graded tensor product.

scholarly journals Max Weibull-G Power Series Distributions

2021 ◽  
pp. 15-29
Author(s):  
Munteanu Bogdan Gheorghe
Keyword(s):  
Probability Distribution ◽  
Power Series ◽  
Probability Distributions ◽  
Random Variables ◽  
Random Variable ◽  
Independent Random Variables ◽  
New Family ◽  
The Family ◽  
Distribution Family ◽  
Power Series Distributions

Based on the Weibull-G Power probability distribution family, we have proposed a new family of probability distributions, named by us the Max Weibull-G power series distributions, which may be applied in order to solve some reliability problems. This implies the fact that the Max Weibull-G power series is the distribution of a random variable max (X1 ,X2 ,...XN) where X1 ,X2 ,... are Weibull-G distributed independent random variables and N is a natural random variable the distribution of which belongs to the family of power series distribution. The main characteristics and properties of this distribution are analyzed.

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