scholarly journals Multifractal structure of Bernoulli convolutions

2011 ◽  
Vol 151 (3) ◽  
pp. 521-539 ◽  
Author(s):  
THOMAS JORDAN ◽  
PABLO SHMERKIN ◽  
BORIS SOLOMYAK
Keyword(s):  
Hausdorff Dimension ◽  
Random Variables ◽  
Multifractal Spectrum ◽  
Random Series ◽  
Absolutely Continuous ◽  
Bernoulli Convolutions ◽  
Image Position ◽  
Almost All

AbstractLet νpλbe the distribution of the random series$\sum_{n=1}^\infty i_n \lam^n$, whereinis a sequence of i.i.d. random variables taking the values 0, 1 with probabilitiesp, 1 −p. These measures are the well-known (biased) Bernoulli convolutions.In this paper we study the multifractal spectrum of νpλfor typical λ. Namely, we investigate the size of the setsOur main results highlight the fact that for almost all, and in some cases all, λ in an appropriate range, Δλ,p(α) is nonempty and, moreover, has positive Hausdorff dimension, for many values of α. This happens even in parameter regions for which νpλis typically absolutely continuous.

On a Class of Nonparametric Tests for Independence—Bivariate Case(1)

1973 ◽  
Vol 16 (3) ◽  
pp. 337-342 ◽  
Author(s):  
M. S. Srivastava
Keyword(s):  
Random Variables ◽  
Nonparametric Tests ◽  
Absolutely Continuous ◽  
Bivariate Case ◽  
Image Position ◽  
Tests For Independence ◽  
Mutually Independent

Let(X1, Y1), (X2, Y2),…, (Xn, Yn) be n mutually independent pairs of random variables with absolutely continuous (hereafter, a.c.) pdf given by(1)where f(ρ) denotes the conditional pdf of X given Y, g(y) the marginal pdf of Y, e(ρ)→ 1 and b(ρ)→0 as ρ→0 and,(2)We wish to test the hypothesis(3)against the alternative(4)For the two-sided alternative we take — ∞< b < ∞. A feature of the model (1) is that it covers both-sided alternatives which have not been considered in the literature so far.

A non-Markovian birth process with logarithmic growth

1975 ◽  
Vol 12 (04) ◽  
pp. 673-683
Author(s):  
G. R. Grimmett
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Birth Process ◽  
Image Position ◽  
Almost All ◽  
Increasing Function ◽  
Logarithmic Growth

I show that the sumof independent random variables converges in distribution when suitably normalised, so long as theXksatisfy the following two conditions:μ(n)= E |Xn|is comparable withE|Sn| for largen,andXk/μ(k) converges in distribution. Also I consider the associated birth processX(t) = max{n:Sn≦t} when eachXkis positive, and I show that there exists a continuous increasing functionv(t) such thatfor some variableYwith specified distribution, and for almost allu. The functionv, satisfiesv(t) =A(1 +o(t)) logt. The Markovian birth process with parameters λn= λn, where 0 &lt; λ &lt; 1, is an example of such a process.

A local limit theorem for attractions under a stable law

1980 ◽  
Vol 87 (1) ◽  
pp. 179-187 ◽  
Author(s):  
Sujit K. Basu ◽  
Makoto Maejima
Keyword(s):  
Limit Theorem ◽  
Local Limit Theorem ◽  
Random Variables ◽  
Local Limit ◽  
Independent Random Variables ◽  
Stable Law ◽  
Absolutely Continuous ◽  
Normal Attraction ◽  
Image Position ◽  
Domain Of Normal Attraction

AbstractLet {Xn} be a sequence of independent random variables each having a common d.f. V1. Suppose that V1 belongs to the domain of normal attraction of a stable d.f. V0 of index α 0 ≤ α ≤ 2. Here we prove that, if the c.f. of X1 is absolutely integrable in rth power for some integer r > 1, then for all large n the d.f. of the normalized sum Zn of X1, X2, …, Xn is absolutely continuous with a p.d.f. vn such thatas n → ∞, where v0 is the p.d.f. of Vo.

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.

Singular n-tuples and Hausdorff dimension

1977 ◽  
Vol 81 (3) ◽  
pp. 377-385 ◽  
Author(s):  
R. C. Baker
Keyword(s):  
Hausdorff Dimension ◽  
Euclidean Space ◽  
Measure Zero ◽  
Rational Numbers ◽  
Linearly Independent ◽  
Singular Numbers ◽  
Image Position ◽  
Almost All ◽  
Dimensional Lebesgue Measure

1. Introduction. Throughout the paper θ = (θ1, …, θn), φ = (φ1, …, φn), … denote points of Euclidean space Rn. We write Kn for the set of θ in Rn for which θl, …, θn, 1 are linearly independent over the rational numbers. We denote points of the set of integer n-tuples Zn by x, y, … We writeIf α is a real number, ∥α∥ denotes the distance from α to the nearest integer.Let θ ∈ Rn. By a theorem of Dirichlet ((2), chapter 1, theorem VI).for all X ≥ 1. We say that θ is singular ifSingular points form a set of n-dimensional Lebesgue measure zero. In fact, H. Davenport and W. M. Schmidt (3) showed thatfor almost all θ in Rn. Although there are no singular numbers in Kl ((2), p. 94) there are ‘highly singular’ n-tuples in Kn for n ≥ 2.

scholarly journals Absolute continuity of complex Bernoulli convolutions

2016 ◽  
Vol 161 (3) ◽  
pp. 435-453 ◽  
Author(s):  
PABLO SHMERKIN ◽  
BORIS SOLOMYAK
Keyword(s):  
Hausdorff Dimension ◽  
Complex Plane ◽  
Absolute Continuity ◽  
Exceptional Set ◽  
Parameter Region ◽  
Absolutely Continuous ◽  
Bernoulli Convolutions ◽  
Supercritical Parameter ◽  
Self Similar

AbstractWe prove that complex Bernoulli convolutions are absolutely continuous in the supercritical parameter region, outside of an exceptional set of parameters of zero Hausdorff dimension. Similar results are also obtained in the biased case, and for other parametrised families of self-similar sets and measures in the complex plane, extending earlier results.

Extensions of a renewal theorem

1955 ◽  
Vol 51 (4) ◽  
pp. 629-638 ◽  
Author(s):  
Walter L. Smith
Keyword(s):  
Random Variables ◽  
Frequency Function ◽  
Renewal Theorem ◽  
Absolutely Continuous ◽  
Image Position ◽  
First Moment ◽  
Independent Identically Distributed

Let X1, X2, X3,… be a sequence of independent, identically distributed, absolutely continuous random variables whose first moment is μ1. Let Sk = X1 + X2 + … + Xk, and let fk(x) be the frequency function of Sk, defined as the k-fold convolution of f1(x). When f1(x) has been defined for all x, fk(x) is uniquely defined for all k, x. Write

A non-Markovian birth process with logarithmic growth

1975 ◽  
Vol 12 (4) ◽  
pp. 673-683 ◽  
Author(s):  
G. R. Grimmett
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Birth Process ◽  
Image Position ◽  
Almost All ◽  
Increasing Function ◽  
Logarithmic Growth

I show that the sum of independent random variables converges in distribution when suitably normalised, so long as the Xk satisfy the following two conditions: μ(n)= E |Xn| is comparable with E |Sn| for large n, and Xk/μ(k) converges in distribution. Also I consider the associated birth process X(t) = max{n: Sn ≦ t} when each Xk is positive, and I show that there exists a continuous increasing function v(t) such that for some variable Y with specified distribution, and for almost all u. The function v, satisfies v (t) = A (1 + o (t)) log t. The Markovian birth process with parameters λn = λn, where 0 < λ < 1, is an example of such a process.

Asymptotics for geometric location problems over random samples

1999 ◽  
Vol 31 (3) ◽  
pp. 632-642 ◽  
Author(s):  
K. McGivney ◽  
J. E. Yukich
Keyword(s):  
Finite Number ◽  
Positive Constant ◽  
Complete Convergence ◽  
Location Problem ◽  
Random Variables ◽  
Location Problems ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Random Samples ◽  
Image Position

Consider the basic location problem in which k locations from among n given points X1,…,Xn are to be chosen so as to minimize the sum M(k; X1,…,Xn) of the distances of each point to the nearest location. It is assumed that no location can serve more than a fixed finite number D of points. When the Xi, i ≥ 1, are i.i.d. random variables with values in [0,1]d and when k = ⌈n/(D+1)⌉ we show that where α := α(D,d) is a positive constant, f is the density of the absolutely continuous part of the law of X1, and c.c. denotes complete convergence.

scholarly journals On the structure of the set of solutions of the Darboux problem for hyperbolic equations

1986 ◽  
Vol 29 (1) ◽  
pp. 7-14 ◽  
Author(s):  
F. S. de Blasi ◽  
J. Myjak
Keyword(s):  
Integrable Function ◽  
Hyperbolic Equations ◽  
Continuous Functions ◽  
Darboux Problem ◽  
Absolutely Continuous Functions ◽  
Absolutely Continuous ◽  
Image Position ◽  
Almost All

Consider the Darboux problemwhere φ,ψ:I→Rd (I=[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × Rd→Rd (Q = I × I) satisfies the following hypotheses:(A1) f(.,.,z) is measurable for every z ∈ Rd;(A2) f(x, y,.) is continuous for a.a. (almost all) (x, y) ∈ Q;(A3) there exists an integrable function α:Q →[0, + ∞) such that |f(x, y, z)|≦α(x, y) for every (x, y, z)∈ Q × Rd.

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