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