From Osher Doctorow

Neumaier (2003) rounds off his Section 4 (Ensembles) with the Weak Law
of Large Numbers (WLLN), which is well known in mathematical
probability-statistics, but which is entirely an Aggregate/Ensemble law
of almost no relevance for the Individual.

In fact, all the Laws of Large Numbers ever do in mathematical
probability-statistics is tell us how sample means converge to
population means (namely, whether they converge almost everywhere,
everywhere, converge in probability, converge in Lp, etc.). The WLLN
tells us conditions when they converge in probability, but this has
nothing to do with Individuals - the relationships between/among
convergence in probability (---p for short) and convergence almost
everywhere (--a.e.) and convergence in Lp norm (---Lp) and
convergence in distrubution (---d) a

A. ---a.e. implies ---p
B. ---Lp to 0 implies ---p to 0.
C. ---p to 0 and uniform boundedness by an element of Lp implies
---Lp.
D. {Xn} ---p 0 iff E(/Xn/ divided by (1 + /Xn/) -- 0.
E. ---a.e. does not imply ---Lp
F. ---p implies ---D

More concisely for most of this:

G. ---a.e. implies ---p implies ---D
H. ---Lp (to 0) implies ---p (to 0) implies ---D (to 0)
I. ---p with uniform boundedness by an element of Lp implies ---Lp.

See Jau Kau Chung (Stanford U.), A Course in Probability Theory,
Harcourt, Brace & World: N.Y., 1968 for these.

The "mystique" that surrounds the "Laws of Large Numbers" is
essentially nothing but the mystique that surrounds convergence to a
constant, which we already have seen in regard to c and h in previous
postings.

Osher Doctorow

From Osher Doctorow

I meant to type Kai Lai Chung rather than Jau Kau Chung - the latter
comes from typing one key to the left of K, one key to the left of L,
and one key to the left of i, which can happen due to a shift of
position if one's cat is jumping on the keyboard while one is typing.

Osher Doctorow

From Osher Doctorow

Is there any really Local probability? Yes. I discuss these below.

Let's look at statistical dependence, which is a mainstream field in
mathematical probability-statistics. The expression:

1) F(x, y) - FX(x)FY(y)

is the main object of study in this field, where F(x, y) is the joint
cumulative distribution function of random variables X and Y at (x, y)
and where FX(x) is the (marginal) cumulative distribution function of X
at x. These are all local.

Another mainstream subfield of mathematical probability-statistics is
the study of probability distributions of random variables, which
includes: (a) probability density (pdfs) or mass functions, (b)
cumulative distribution functions (cdfs), (c) characteristic functions
(cfs), (d) probability generating functions, (e) moment generating
functions, (f) moments, (g) comparing different distributions with
regard to efficiency, sufficiency, unbiasedness, etc. These are all
local.

Maximum likelihood methods in statistics involve pdfs entirely and are
very important in statistical hypothesis testing. These are local.

Goodness of fit methods (including chi-square) are a major subfield of
probability-statistics entirely concerned with pdfs or cdfs which are
of course local.

Reliability in engineering is entirely local and involves cdfs and 1 -
cdf expressions.

Stochastic inequalities and majorization are a branch of
probability-statistics and are local.

Probable Influence (PI) is entirely local.

Large Deviations are usually local unless emphasis is on the large
deviations of means/aggregates.

Quality control is mostly local and involves confidence intervals which
are also local.

Estimation of population parameters (constants) by sample statistics is
local except when expectations (mean) and similar types of quantities
are involved. Estimation is a large branch of mathematical
probability-statistics.

Order statistics (maxima, minima, medians, etc.) are a branch of
mathematical probability-statistics which are mostly studied locally,
and are an important branch.

Osher Doctorow

From Osher Doctorow

In fact, it is rather amusing to note that for nonnegative random
variables, research into expectations and moments (which are also types
of expectations) is largely local and Individual because of convergence
properties of integrals being related to inequalities of integrands for
nonnegative random variables and even nonnegative functions. It is
only the resulting expectations/moments which are "aggregate" and
non-Individual.

Estimation and hypothesis testing for Individual scores/values is a
branch of mathematical probability-statistics and is Local and
Individual.

Osher Doctorow

From Osher Doctorow

Oops! I made an error in saying that estimation and hypothesis
testing are local and Individual. The latter is only true when they
are estimating or testing Individual scores, although they are always
local in reference to either the center of the confidence interval or
the boundary of the rejection region or even the null hypothesis or
alternative hypothesis.

Statistical Outliers are a recent very important topic in mathematical
probability-statistics as well as applied branches of various fields,
and they are both Individual and Local; in fact, "think Exception"
should be the password of science and mathematics. While capitalism
and socialism have produced ridiculously high levels of inbreeding in
political leaders and wealthy families, capitalism is currently very
Exception-oriented, while socialism of the Russian and Chinese types
suffers from way too much Aggregation. To some extent, there is a
tradeoff between inbred stupidity of leaders and Exception-orientation
in capitalism too, and one never knows which trend will win or survive.
Big numbers tend to hypnotize both socialists and capitalists, who
both tend to support larger populations compared to smaller populations
and thereby counteract their Exception-orientation tendencies if any.
If you have noticed it by now, PI has a "law of Little Numbers" since
the Rare Event/Process is the more influential one, like Creative
Genius itself.

Osher Doctorow