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.

The average number of real zeros of a random trigonometric polynomial

1968 ◽  
Vol 64 (3) ◽  
pp. 721-730 ◽  
Author(s):  
Minaketan Das
Keyword(s):  
Trigonometric Polynomial ◽  
Mathematical Expectation ◽  
Random Variables ◽  
Trigonometric Polynomials ◽  
Real Zeros ◽  
Number Of Zeros ◽  
Number Of Real Zeros ◽  
Random Trigonometric Polynomial ◽  
Image Position ◽  
Mutually Independent

AbstractLet a1, a2,… be a sequence of mutually independent, normally distributed, random variables with mathematical expectation zero and variance unity; let b1, b2,… be a set of positive constants. In this work, we obtain the average number of zeros in the interval (0, 2π) of trigonometric polynomials of the formfor large n. The case when bk = kσ (σ > − 3/2;) is studied in detail. Here the required average is (2σ + 1/2σ + 3)½.2n + o(n) for σ ≥ − ½ and of order n3/2; + σ in the remaining cases.

A generalization of McDiarmid's theorem for mixed Bernoulli percolation

1980 ◽  
Vol 88 (1) ◽  
pp. 167-170 ◽  
Author(s):  
J. M. Hammersley
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Finite Sequence ◽  
Random Set ◽  
Percolation Probability ◽  
Open Path ◽  
Finite Set ◽  
Image Position ◽  
Infinite Set ◽  
Mutually Independent

Let G be an infinite partially directed graph of finite outgoing degree. Thus G consists of an infinite set of vertices, together with a set of edges between certain prescribed pairs of vertices. Each edge may be directed or undirected, and the number of edges from (but not necessarily to) any given vertex is always finite (though possibly unbounded). A path on G from a vertex V1 to a vertex Vn (if such a path exists) is a finite sequence of alternate edges and vertices of the form E12, V2, E23, V3, …, En − 1, n, Vn such that Ei, i + 1 is an edge connecting Vi and Vi + 1 (and in the direction from Vi to Vi + 1 if that edge happens to be directed). In mixed Bernoulli percolation, each vertex Vi carries a random variable di, and each edge Eij carries a random variable dij. All these random variables di and dij are mutually independent, and take only the values 0 or 1; the di take the value 1 with probability p, while the dij take the value 1 with probability p. A path is said to be open if and only if all the random variables carried by all its edges and all its vertices assume the value 1. Let S be a given finite set of vertices, called the source set; and let T be the set of all vertices such that there exists at least one open path from some vertex of S to each vertex of T. (We imagine that fluid, supplied to all the source vertices, can flow along any open path; and thus T is the random set of vertices eventually wetted by the fluid). The percolation probabilityis defined to be the probability that T is an infinite set.

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 average number of maxima of a random algebraic curve

1969 ◽  
Vol 65 (3) ◽  
pp. 741-753 ◽  
Author(s):  
Minaketan Das
Keyword(s):  
Mathematical Expectation ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Random Variables ◽  
Image Position ◽  
Mutually Independent ◽  
Normally Distributed

AbstractLet g0, gl, g2,…be a sequence of mutually independent, normally distributed random variables with mathematical expectation zero and variance unity. In this work, we obtain the average number of maxima (minima) of the random algebraic curves with the equationsThis average is (½(3½ + 1)) log N + O((log N)⅔ (log log N)½), when N is large.

A note on random walks

1971 ◽  
Vol 8 (01) ◽  
pp. 198-201 ◽  
Author(s):  
R. M. Phatarfod ◽  
T. P. Speed ◽  
A. M. Walker
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Variables ◽  
Image Position ◽  
Mutually Independent ◽  
Reflecting Barriers ◽  
Independent And Identically Distributed

Let {Xn } be a random walk between reflecting barriers at 0 and a &gt; 0 with jumps {Zn }. By we mean the random walk between absorbing barriers at — a and 0+ with the same jumps {Zn }. It has been known for some time that when {Zn } is a sequence of mutually independent and identically distributed random variables, and 0 ≦x &lt;a, we have for all n:

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

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.

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.

A note on random walks

1971 ◽  
Vol 8 (1) ◽  
pp. 198-201 ◽  
Author(s):  
R. M. Phatarfod ◽  
T. P. Speed ◽  
A. M. Walker
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Random Variables ◽  
Image Position ◽  
Mutually Independent ◽  
Reflecting Barriers ◽  
Independent And Identically Distributed

Let {Xn} be a random walk between reflecting barriers at 0 and a > 0 with jumps {Zn}. By we mean the random walk between absorbing barriers at — a and 0+ with the same jumps {Zn}. It has been known for some time that when {Zn} is a sequence of mutually independent and identically distributed random variables, and 0 ≦x <a, we have for all n:

Asymptotics for geometric location problems over random samples

1999 ◽  
Vol 31 (03) ◽  
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 X 1,…,X n are to be chosen so as to minimize the sum M(k; X 1,…,X n ) 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 X i , 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 X 1, and c.c. denotes complete convergence.

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