  The main objective of this paper is to look from the unique point of view at
some phenomena arising in different areas of probability theory and
mathematical statistics. We will try to understand what is common between
classical probabilistic results, such as the law of iterated logarithm for
example, and well-known problem in adaptive estimation called price to pay for
adaptation. Why exists two different kinds of this price? What relates
exponential inequalities for M-estimators, so-called uniform-in-bandwidth
consistency in density or regression model and the bounds for modulus of
continuity of gaussian random functions defined on a metric space equipped with
doubling measure? It turned out that all these and many others problems can be
reduced to finding upper functions for a collection real- valued random
variables. Each variable is the value of a given sub-additive positive
functional of continuous random mapping defined on a totally bounded subset of
a metric space.
