Distances on circles, toruses and spheres

1978 ◽  
Vol 15 (1) ◽  
pp. 136-143 ◽  
Author(s):  
B. W. Silverman
Keyword(s):  
Weak Convergence ◽  
Empirical Distribution ◽  
Random Variables ◽  
Suitable Function ◽  
Distribution Process

Families of heavily dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj) for some suitable function g of two arguments and independent uniformly distributed random variables Y1, Y2, ··· on the circle, the torus or the sphere. The weak convergence of the empirical distribution process is discussed. The particular case of distances between pairs of observations on the circle is considered in greater detail.

Distances on circles, toruses and spheres

1978 ◽  
Vol 15 (01) ◽  
pp. 136-143 ◽  
Author(s):  
B. W. Silverman
Keyword(s):  
Weak Convergence ◽  
Empirical Distribution ◽  
Random Variables ◽  
Suitable Function ◽  
Distribution Process

Families of heavily dissociated random variables are defined and discussed. These include families of the form g(Yi , Yj ) for some suitable function g of two arguments and independent uniformly distributed random variables Y 1, Y 2, ··· on the circle, the torus or the sphere. The weak convergence of the empirical distribution process is discussed. The particular case of distances between pairs of observations on the circle is considered in greater detail.

Limit theorems for dissociated random variables

1976 ◽  
Vol 8 (04) ◽  
pp. 806-819 ◽  
Author(s):  
B. W. Silverman
Keyword(s):  
Data Analysis ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Limit Theorems ◽  
Normal Approximation ◽  
Empirical Distribution ◽  
Random Variables ◽  
Distribution Process ◽  
Suitable Space

Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj , …, Yz ) for some function g of m arguments and some sequence Yn of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed.

Limit theorems for dissociated random variables

1976 ◽  
Vol 8 (4) ◽  
pp. 806-819 ◽  
Author(s):  
B. W. Silverman
Keyword(s):  
Data Analysis ◽  
Central Limit Theorem ◽  
Limit Theorem ◽  
Weak Convergence ◽  
Limit Theorems ◽  
Normal Approximation ◽  
Empirical Distribution ◽  
Random Variables ◽  
Distribution Process ◽  
Suitable Space

Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj, …, Yz) for some function g of m arguments and some sequence Yn of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed.

Beyond the heuristic approach to Kolmogorov-Smirnov theorems

1982 ◽  
Vol 19 (A) ◽  
pp. 359-365 ◽  
Author(s):  
David Pollard
Keyword(s):  
Weak Convergence ◽  
Goodness Of Fit ◽  
Empirical Distribution ◽  
Distribution Functions ◽  
Heuristic Approach ◽  
Convergence Theory ◽  
Simple Method ◽  
Starting Point ◽  
Kolmogorov Smirnov ◽  
The Subject

The theory of weak convergence has developed into an extensive and useful, but technical, subject. One of its most important applications is in the study of empirical distribution functions: the explication of the asymptotic behavior of the Kolmogorov goodness-of-fit statistic is one of its greatest successes. In this article a simple method for understanding this aspect of the subject is sketched. The starting point is Doob's heuristic approach to the Kolmogorov-Smirnov theorems, and the rigorous justification of that approach offered by Donsker. The ideas can be carried over to other applications of weak convergence theory.

scholarly journals Limiting behavior of the perturbed empirical distribution functions evaluated at U-statistics for strongly mixing sequences of random variables

1997 ◽  
Vol 10 (1) ◽  
pp. 3-20 ◽  
Author(s):  
Shan Sun ◽  
Ching-Yuan Chiang
Keyword(s):  
Empirical Distribution Function ◽  
Empirical Distribution ◽  
Random Variables ◽  
Distribution Functions ◽  
Limiting Behavior ◽  
U Statistics ◽  
Strongly Mixing ◽  
Almost Sure Representation ◽  
Mixing Sequences ◽  
Empirical Distribution Functions

We prove the almost sure representation, a law of the iterated logarithm and an invariance principle for the statistic Fˆn(Un) for a class of strongly mixing sequences of random variables {Xi,i≥1}. Stationarity is not assumed. Here Fˆn is the perturbed empirical distribution function and Un is a U-statistic based on X1,…,Xn.

Some Functional Stable Limit Theorems

1973 ◽  
Vol 16 (2) ◽  
pp. 173-177 ◽  
Author(s):  
D. R. Beuerman
Keyword(s):  
Weak Convergence ◽  
Limit Theorems ◽  
Stable Process ◽  
Random Variables ◽  
Domain Of Attraction ◽  
Stable Processes ◽  
Stable Limit ◽  
Stable Law ◽  
Partial Sum Process ◽  
Image Position

Let Xl,X2,X3, … be a sequence of independent and identically distributed (i.i.d.) random variables which belong to the domain of attraction of a stable law of index α≠1. That is,1whereandwhere L(n) is a function of slow variation; also take S0=0, B0=l.In §2, we are concerned with the weak convergence of the partial sum process to a stable process and the question of centering for stable laws and drift for stable processes.

Weak convergence of randomly indexed sequences of random variables

1978 ◽  
Vol 83 (1) ◽  
pp. 117-126 ◽  
Author(s):  
D. J. Aldous
Keyword(s):  
Positive Integer ◽  
Weak Convergence ◽  
Metric Space ◽  
Random Variables ◽  
Random Elements ◽  
Image Position

Let Yn ⇒ Y∞ be a sequence of random variables converging in distribution, or more generally a sequenceof random elements of a suitable metric space whose distributions are converging weakly. Let τn → ∞ be positive integer-valued random variables. If {τn} and {Yn} are independent, it is trivial that

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