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[mathlearn] Strong Law of Large Numbers and the Proximity Function  Doctorow
Posted:
Jan 28, 2001 2:11 PM


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 1011 dimensions of string theory in physics.
Osher Doctorow Ventura College, West Los Angeles College, etc.
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