scholarly journals The realization of positive random variables via absolutely continuous transformations of measure on Wiener space

2006 ◽  
Vol 3 (0) ◽  
pp. 170-205 ◽  
Author(s):  
D. Feyel ◽  
A. S. Üstünel ◽  
M. Zakai
Keyword(s):  
Random Variables ◽  
Wiener Space ◽  
Absolutely Continuous

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (4) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi, , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n–1Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

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.

On the number of leaves of a euclidean minimal spanning tree

1987 ◽  
Vol 24 (04) ◽  
pp. 809-826 ◽  
Author(s):  
J. Michael Steele ◽  
Lawrence A. Shepp ◽  
William F. Eddy
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Random Variables ◽  
Minimal Spanning Tree ◽  
Absolutely Continuous ◽  
Mean And Variance ◽  
Number Of Leaves ◽  
The Mean ◽  
Vertex Degrees ◽  
Minimal Spanning Trees

Let Vk,n be the number of vertices of degree k in the Euclidean minimal spanning tree of Xi , , where the Xi are independent, absolutely continuous random variables with values in Rd. It is proved that n –1 Vk,n converges with probability 1 to a constant α k,d. Intermediate results provide information about how the vertex degrees of a minimal spanning tree change as points are added or deleted, about the decomposition of minimal spanning trees into probabilistically similar trees, and about the mean and variance of Vk,n.

A d-DEFORMATION OF FREE GAUSSIAN RANDOM VARIABLES

2005 ◽  
Vol 08 (04) ◽  
pp. 669-680 ◽  
Author(s):  
JANUSZ WYSOCZAŃSKI
Keyword(s):  
Tensor Product ◽  
Hypergeometric Series ◽  
Fock Space ◽  
Recurrence Formula ◽  
Random Variables ◽  
Product Formula ◽  
Continuous Part ◽  
Absolutely Continuous ◽  
Gaussian Random Variables ◽  
Special Case

We define a deformation of free creations (and annihilations), given by operators on the full Fock space, acting nontrivially only between the vacuum subspace ℂΩ and the twofold tensor product [Formula: see text]. Then we study the distribution of the deformed free gaussian operators, with the deformation containing also a real parameter d. The recurrence formula for moments is shown, and the Cauchy transform of the distribution measure is computed. This yields the description of the measure: absolutely continuous part and the atomic part. The existence of atoms depends on the parameter d. The special case d =1 is studied with all details, with the formula for moments is given as values of the hypergeometric series. Finally we show the formula for computing the mixed moments of the deformed operators.

Linear processes are nearly Gaussian

1967 ◽  
Vol 4 (2) ◽  
pp. 313-329 ◽  
Author(s):  
C. L. Mallows
Keyword(s):  
Distribution Function ◽  
Continuous Distribution ◽  
Random Variables ◽  
Linear Process ◽  
Real Numbers ◽  
Linear Processes ◽  
Absolutely Continuous ◽  
Restricted Sense ◽  
Finite Set ◽  
Third Moment

Let U denote the set of all integers, and suppose that Y = {Yu; u ∈ U} is a process of standardized, independent and identically distributed random variables with finite third moment and with a common absolutely continuous distribution function (d.f.) G (·). Let a = {au; u ∈ U} be a sequence of real numbers with Σuau2 = 1. Then Xu = ΣwawYu–w defines a stationary linear process X = {Xu; u ɛ U} with E(Xu) = 0, E(Xu2) = 1 for u ∊ U. Let F(·) be the d.f. of X0. We prove that if maxu |au| is small, then (i) for each w, Xw is close to Gaussian in the sense that ∫∞−∞(F(y) − Φ(y))2dy ≦ g maxu |au | where Φ(·) is the standard Gaussian d.f., and g depends only on G(·); (ii) for each finite set (w1, … wn), (Xw1, … Xwn) is close to Gaussian in a similar sense; (iii) the process X is close to Gaussian in a somewhat restricted sense. Several properties of the measures of distance from Gaussianity employed are investigated, and the relation of maxu|au| to the bandwidth of the filter a is studied.

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