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[math-learn] Strong Law of Large Numbers and the Proximity Function - Doctorow
Posted:
Jan 28, 2001 2:11 PM
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From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Sunday Jan. 21, 2001
11:07AM
Writing pn(x1,x2,...,xn; y) = 1 + y - Sn (the + of the last paper should
have been minus, and y < = xi for i = 1, 2, ..., n), we can even have random
coordinates and forces, in which case the simplest strong law of large
numbers (SLLN) in probability says that if the xi are a sequence of
uncorrelated random variables and their second moments have a single upper
bound, then Sn - E(Sn) --> 0 almost everywhere (a.e.). If they are
identically distributed, then Sn --> u or mu a.e., the population mean of
the random variables xi. In the literature of the SLLN, Xi rather than xi
are usually used for random variables, with xi their respective values, and
Sn is usually used for x1 + ... + xn instead of for this sum divided by n.
So in this case pn --> 1 + y - u where u is the common mean of the xi.
Certainly y < = u so the limit is between 0 and 1. This has an interesting
interpretation, namely, that if an infinite number of random identically
distributed uncorrelated forces or coordinates with uniform upper bound
influence something y (y would usually be a force or coordinate, but could
be something else), then the total influence is 1 + y - u where each
force/coordinate has population mean u. If they are not identically
distributed, the influence is 1 + y - lim E(Sn) where lim E(Sn) is the limit
as n approach infinity of the population mean of Sn, if it exists. If they
are not uncorrelated with uniformly bounded second moment (roughly, not
uniformly bounded variances), then various generalizations are often
possible. Since quantum theory deals largely with infinite dimensional
spaces, the infinite dimensional force/coordinate scenario is quite
plausible. It might even tie in somehow with the 10-11 dimensions of string
theory in physics.
Osher Doctorow
Ventura College, West Los Angeles College, etc.
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