Weak Convergence of Stochastic Processes Whose Trajectories Have No Discontinuities of the Second Kind and the “Heuristic” Approach to the Kolmogorov-Smirnov Tests

1956 ◽  
Vol 1 (1) ◽  
pp. 140-144 ◽  
Author(s):  
N. N. Chentsov
Keyword(s):  
Weak Convergence ◽  
Stochastic Processes ◽  
Heuristic Approach ◽  
Kolmogorov Smirnov

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.

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.

Weak Convergence to Stochastic Integrals

2021 ◽  
pp. 724-756
Author(s):  
James Davidson
Keyword(s):  
Brownian Motion ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Final Section ◽  
Stochastic Integrals ◽  
Martingale Difference ◽  
Mixing Processes ◽  
Linear Processes ◽  
Special Cases

The main object of this chapter is to prove the convergence of the covariances of stochastic processes with their increments to stochastic integrals with respect to Brownian motion. Some preliminary theory is given relating to random functionals on C, stochastic integrals, and the important Itô isometry. The main result is first proved for the tractable special cases of martingale difference increments and linear processes. The final section is devoted to proving the more difficult general case, of NED functions of mixing processes.

scholarly journals On the weak convergence of stochastic processes with embedded point processes

1988 ◽  
Vol 20 (2) ◽  
pp. 473-475 ◽  
Author(s):  
Panagiotis Konstantopoulos ◽  
Jean Walrand
Keyword(s):  
Stochastic Process ◽  
Weak Convergence ◽  
Stochastic Processes ◽  
Point Process ◽  
Real Line ◽  
Continuous Time ◽  
Point Processes ◽  
Mixing Condition ◽  
The Real ◽  
Palm Distribution

We consider a stochastic process in continuous time and two point processes on the real line, all jointly stationary. We show that under a certain mixing condition the values of the process at the points of the second point process converge weakly under the Palm distribution with respect to the first point process, and we identify the limit. This result is a supplement to two other known results which are mentioned below.

Some applications of weak convergence to statistics

1988 ◽  
Vol 25 (A) ◽  
pp. 201-211
Author(s):  
R. M. Loynes
Keyword(s):  
Weak Convergence ◽  
Stochastic Processes ◽  
Asymptotic Properties ◽  
Convergence Results

Results showing the weak convergence of certain stochastic processes are used to derive both known and new (asymptotic) properties of signs of residuals from regression; other weak convergence results are derived, and used to determine the behaviour of runs of residuals.

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