Re: WHY IS SOUND to get C.V.´s from Mont Carlo

What UNIFORM and POITNWISE CONVERGENCES are?

(from MATHWORLD)

The GLIVENKO-CANTELLY Theorem states:
As n -- > infinity,
____sup | fn(x) - f(x) | --- > 0_______x in R
that is
with probability 1, fn(x) --- > F(x) uniformly in x

1)_________UNIFORM CONVERGENCE

[General concept] A sequence {fn} of functions converges UNIFORMLY to a limiting function f if the speed of convergence fn(x) to f(x) DOESN`T depend on x.
*** DEFINITION
_Suppose S is a set and fn: S --- >R are real-valued functions for any natural number n. Then, we say that the sequence {fn} is uniformly convergent with the limit f: S--- >R if for every e* >0, there exist a natural number N such that for all x in S and all n>=N,
_________________|fn(x) -f(x)| < e*.

Note that
____The sequence {fn}converges pointwise with limit f: S--- > R if and only if there exist a natural number N such that for all x in S and all n>=N,
_________________|fn(x) -f(x)| < e*.
with the proviso that N may depend on e* and (attention!) on x.

[ CONCLUSION : pointwise convergence is weaker than uniform convergence, in other words an uniform convergence is always a pointwise convergence: he converse is not true].

AND In order to have critical values from a EMPIRICAL DISTRIBUTION it was enough to have pointwise convergence, but the uniform one (assured by the Glivenko-Cantelli theorem (so-called the FUNDAMENTAL THEOREM OF STATISTICS) guarantees that I proved
___WHY IS SOUND to get C.V.´s from Monte Carlo___________

HOWEVER, and against ALL EVIDENCE,
Jack Tomsky (Ph. D, Stanford ,1974) uses all sarcasm he could join, all verbal violence, all defamatory panoply, all ad hominem weapons to deny the worth of a so (at least) intuitively and fully used *intensive computer* technique since 4 decades.

_______licas (Luis A. Afonso)
.

Relevant Pages

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