Empirical Process
In empirical process theory, we study the function-valued central limit theorem. The empirical process is defined by the stochastic process using finite random variables.
In empirical process theory, we study the function-valued central limit theorem. The empirical process is defined by the stochastic process using finite random variables.
For each finite {wk}, the central limit theorem holds. However, its uniform convergence over w is not trivial.
For each finite {wk}, the central limit theorem holds. However, its uniform convergence over w is not trivial.
There exists the unique Gaussian process that has the same expectation and variance as the empirical process. Our purpose is to prove the uniform convergence over w of the empirical process to the unique Gaussian process.
There exists the unique Gaussian process that has the same expectation and variance as the empirical process. Our purpose is to prove the uniform convergence over w of the empirical process to the unique Gaussian process.
Prohorov theorem plays the important role in empirical process theory. To employ this theorem, the function space should be separable and complete. If a sequence of empirical process is tight, then it converges in distribution to the unique Gaussian process uniformly over w.
Prohorov theorem plays the important role in empirical process theory. To employ this theorem, the function space should be separable and complete. If a sequence of empirical process is tight, then it converges in distribution to the unique Gaussian process uniformly over w.
A sufficient condition for the tightness of the empirical process is called a property of Donsker.
A sufficient condition for the tightness of the empirical process is called a property of Donsker.
There exist several sufficient conditions for Donsker class.
There exist several sufficient conditions for Donsker class.
This sufficient condition for Donsker class is often employed in statistical learning theory.
This sufficient condition for Donsker class is often employed in statistical learning theory.
If a sequence of random variables converges in distribution, and if it is uniformly integrable, then the sequence of their averages converges to the average of the limit random variable.
If a sequence of random variables converges in distribution, and if it is uniformly integrable, then the sequence of their averages converges to the average of the limit random variable.
This is a simple example of an empirical process. It is easy to make another examples, hence computer simulations are recommended.
This is a simple example of an empirical process. It is easy to make another examples, hence computer simulations are recommended.
This is an example of the empirical process (30 paths).
This is an example of the empirical process (30 paths).
This is an another example (1000 paths).
This is an another example (1000 paths).