  The objective of this paper is to investigate the layered structure of
topological complexity in the tail of a probability distribution. We establish
the functional strong law of large numbers for Betti numbers, a basic
quantifier of algebraic topology, of a geometric complex outside an open ball
of radius $R_n$, such that $R_n\to\infty$ as the sample size $n$ increases. The
nature of the obtained law of large numbers is determined by the decay rate of
a probability density. It especially depends on whether the tail of a density
decays at a regularly varying rate or an exponentially decaying rate. The
nature of the limit theorem depends also on how rapidly $R_n$ diverges. In
particular, if $R_n$ diverges sufficiently slowly, the limiting function in the
law of large numbers is crucially affected by the emergence of arbitrarily
large connected components supporting topological cycles in the limit.
