A Modern Theory of Random Variation: With Applications in Stochastic Calculus, Financial Mathematics, and Feynman Integration

Chapter 1

Prologue

1.1    About This Book

This is a self-contained study of a Riemann sum approach to the theory of random variation, assuming only some familiarity with probability or statistical analysis, basic Riemann integration, and mathematical proof. The primary idea of the book, and the reason why it is different from other treatments of random variation, is its use of non-absolute convergence. The series diverges to infinity. On the other hand, the oscillating series converges—but only on condition that the terms are added up in the order in which they are written, without rearranging them. This convergence is called conditional or non-absolute.

What has this got to do with the theory of random variation? Any conception or understanding of the random variation phenomenon hinges on the notions of probability and its mathematical representation in the form of probability distribution functions. The central, recurring theme of this book is that, provided a non-absolute method of summation is used, every finitely additive function of disjoint intervals is integrable. In other words, every distribution function is integrable.

In contrast, more traditional methods in probability theory exclude significant classes of such functions whose integrability cannot be established whenever ...