The Fundamental Conjectures

Abstract

In Chapter 12 we outlay a long range research program. We believe that it might well be true in considerable generality that “chaining explains all the part of the boundedness which is due to cancellation”. Even if this conjecture is true, there would remain to describe the “part of the boundedness which owes nothing to cancellation”, and for this part also we propose sweeping conjectures. At the heuristic level, the underlying idea of these conjectures is that ultimately, a bound for a stochastic process always arises from the use of the ‘union bound’ P(∪ _n A _n )≤∑ _n P(A _n ) in a simple situation, and the use of basic principles such as linearity and positivity, or combinations of these.