Previous posting had a working out error, although the final result
remains the same.

Replace all "1 / [ (m-n)^L + 1]" with "1 / [ (m-n) + 1]^L"

SCHOENFELD THEOREM:
An infinite bounded sequence of random numbers contains all finite
sequences of bounded numbers.

P(S occurs in R)
= 1 - lim j-+inf (1 - 1 / [(m-n) + 1]^L)^j
= 1 - 0
= 1

where
R : random sequence of numbers where all elements Ri are n = Ri
= m
m : largest number in R
n : smallest number in R
S : arbitrary sequence of numbers where all elements Si are n =
Si = m
L : length of S


JS

Le 8 Dec 2003 03:42:26 -0800,
John Schoenfeld grava à la saucisse et au marteau:

Previous posting had a working out error, although the final result
remains the same.

Replace all "1 / [ (m-n)^L + 1]" with "1 / [ (m-n) + 1]^L"

SCHOENFELD THEOREM:
An infinite bounded sequence of random numbers contains all finite
sequences of bounded numbers.

It's false

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 ... ad vitam aeternam is an infinite
bounded sequence of random numbers.

What you want to say is that the probability of a given sequence to
appear in your infinite sequence tends towards 1 as the number of terms
goes to infinity, which is not the same thing.

--
Nicolas

On 8 Dec 2003 03:42:26 -0800, (John
Schoenfeld) wrote:

Previous posting had a working out error, although the final result
remains the same.

Replace all "1 / [ (m-n)^L + 1]" with "1 / [ (m-n) + 1]^L"

SCHOENFELD THEOREM:
An infinite bounded sequence of random numbers contains all finite
sequences of bounded numbers.

What you mean by this needs to be clarified. As stated it's clearly
false. For example, there are uncountably many bounded finite
sequences, but a given sequence contains only countably many
of them.

P(S occurs in R)
= 1 - lim j-+inf (1 - 1 / [(m-n) + 1]^L)^j
= 1 - 0
= 1

where
R : random sequence of numbers where all elements Ri are n = Ri
= m
m : largest number in R
n : smallest number in R
S : arbitrary sequence of numbers where all elements Si are n =
Si = m
L : length of S


JS

************************

David C. Ullrich

John Schoenfeld wrote:

SCHOENFELD THEOREM:
An infinite bounded sequence of random numbers contains all finite
sequences of bounded numbers.


I assume you meant to say "...finite sequences of numbers bounded by
bounds contained within the bounds of the infinite sequence..."

This seems pretty obvious -- in fact, it also seems pretty obvious that
each finite subsequence must occur in the infinite sequence an infinite
number of times. In fact, I think that's even mentioned in various
texts as an example of a corollary to some theorem. Don't recall which,
tho.

I also don't see the relevance of this to relativity, which is the
newsgroup in which I saw it. Why was it cross posted there?

In article ,
John Schoenfeld wrote:

SCHOENFELD THEOREM:
An infinite bounded sequence of random numbers contains all finite
sequences of bounded numbers.

At the very least you should phrase your theorem carefully. There is
no such thing as a "random number". I think what you mean is something
like a "random number generator", where "random" modifies "generator",
not "number". Better: "A randomly selected bounded sequence of numbers..."

But then of course the result is nonsense: if you select a bounded
sequence at random, you might very well select (0,0,0,0,0,...).

See how clear things become when you try to state things with precision?

(As you try to do so, you will eventually find that the word "random"
almost never belongs in a carefully-worded mathematical sentence!
"For all..." "There exists..." "...measure..." -- thoses are the phrases
you probably want to use instead.)

dave

SCHOENFELD THEOREM:
An infinite bounded sequence of random numbers contains all finite
sequences of bounded numbers.

At the very least you should phrase your theorem carefully. There is
no such thing as a "random number". I think what you mean is something
like a "random number generator", where "random" modifies "generator",
not "number". Better: "A randomly selected bounded sequence of numbers..."

Your arguments are of little value. Whether or not I chose to disguise
simple concepts with excessive academic pedantry is irrespective of
the validity of the Schoenfeld Theorem.

But then of course the result is nonsense: if you select a bounded
sequence at random, you might very well select (0,0,0,0,0,...).

If the infinite random sequence R contains all 0's, then it is bound
by [0,0], and the Schoenfeld Theorem remains true.

Also, the Schoenfeld Theorem says "sequence of random numbers" not
"random sequence of numbers".


See how clear things become when you try to state things with precision?

(As you try to do so, you will eventually find that the word "random"
almost never belongs in a carefully-worded mathematical sentence!
"For all..." "There exists..." "...measure..." -- thoses are the phrases
you probably want to use instead.)

Your disproof of the Schoenfeld Theorem [Copyright(c) John
Schoenfeld] amounts to a set of befuddled attacks on my mathematical
ettiquette. Try again, champ.

JS

In message , John
Schoenfeld writes
SCHOENFELD THEOREM:
An infinite bounded sequence of random numbers contains all finite
sequences of bounded numbers.

At the very least you should phrase your theorem carefully. There is
no such thing as a "random number". I think what you mean is something
like a "random number generator", where "random" modifies "generator",
not "number". Better: "A randomly selected bounded sequence of numbers..."

Your arguments are of little value.

Why not counter them, then? All you have to do is to define what you
mean by "sequence of random numbers".

Whether or not I chose to disguise
simple concepts with excessive academic pedantry

In mathematics, that's usually phrased "rigorous proof". Do it, and
you're unassailable. Don't, and you're open to onslaught by a host of
academic pedants. And one or two people who wonder what you think a
"random number" might be.

is irrespective of
the validity of the Schoenfeld Theorem.

That much is true. Your "simple concepts" say nothing about its
validity.

But then of course the result is nonsense: if you select a bounded
sequence at random, you might very well select (0,0,0,0,0,...).

If the infinite random sequence R contains all 0's, then it is bound

Bounded.

by [0,0], and the Schoenfeld Theorem remains true.

Also, the Schoenfeld Theorem says "sequence of random numbers" not
"random sequence of numbers".

That's worse, until you define "random numbers"..

See how clear things become when you try to state things with precision?

(As you try to do so, you will eventually find that the word "random"
almost never belongs in a carefully-worded mathematical sentence!
"For all..." "There exists..." "...measure..." -- thoses are the phrases
you probably want to use instead.)

Your disproof of the Schoenfeld Theorem [Copyright(c) John
Schoenfeld] amounts to a set of befuddled attacks on my mathematical
ettiquette. Try again, champ.

--
Richard Herring

(John Schoenfeld) wrote in message om...
SCHOENFELD THEOREM:
An infinite bounded sequence of random numbers contains all finite
sequences of bounded numbers.

At the very least you should phrase your theorem carefully. There is
no such thing as a "random number". I think what you mean is something
like a "random number generator", where "random" modifies "generator",
not "number". Better: "A randomly selected bounded sequence of numbers..."

Your arguments are of little value. Whether or not I chose to disguise
simple concepts with excessive academic pedantry is irrespective of
the validity of the Schoenfeld Theorem.

But then of course the result is nonsense: if you select a bounded
sequence at random, you might very well select (0,0,0,0,0,...).

If the infinite random sequence R contains all 0's, then it is bound
by [0,0], and the Schoenfeld Theorem remains true.

Also, the Schoenfeld Theorem says "sequence of random numbers" not
"random sequence of numbers".


See how clear things become when you try to state things with precision?

(As you try to do so, you will eventually find that the word "random"
almost never belongs in a carefully-worded mathematical sentence!
"For all..." "There exists..." "...measure..." -- thoses are the phrases
you probably want to use instead.)

Your disproof of the Schoenfeld Theorem [Copyright(c) John
Schoenfeld] amounts to a set of befuddled attacks on my mathematical
ettiquette. Try again, champ.

JS

At the risk of repeating an argument made in a previous post, the
sequence {s_i} where s_i = 2 if i is odd and s_i = 12 if i is even
might occur randomly, just like an infinite sequence of coin tosses
might be all heads. Now we certainly don't see every possible
sequence of integers between 2 and 12 occurring. You have to go back
to David Ullrich's addition of "with probability 1" to your result.
In particular you should study probability theory as applied to
infinite sample spaces (in this casee the set of all sequences of
integers between upper and lower bounds.

By the way, you can't copyright a theorem.