Theory of Probability

8Random Processes with Independent Increments

8.1 Introduction

8.1.1. In Chapter 7, we saw that viewing Heads and Tails as a random process enabled us to present certain problems (laws of large numbers, the central limit theorem) in a more expressive form – as well as giving us insights into their solution. It was, essentially, a question of obtaining a deeper understanding of the problems by looking at them from an appropriate dynamic viewpoint.

This same dynamic viewpoint lends itself, in a natural way, to the study of a number of other problems. Not only does it serve as an aid to one’s intuition, but also, and more importantly, it reveals connections between topics and problems that otherwise appear unconnected (a common circumstance, which results in solutions being discovered twice over, and hence not appearing in their true perspective); in so doing, it provides us with a unified overall view.

We have already seen, in the case of Heads and Tails, how the representation as a process enabled us to derive, in an elegant manner, results which could then easily be extended to more general cases. We now proceed by following up this idea in two different, but related, directions:

• making precise the kind of process to be considered as a first development of the case of Heads and Tails;
• considering (first for the case of Heads and Tails, and then in the wider ambit mentioned above) problems of a more complicated nature than those studied so far.

8.1.2. The random processes ...