# 5.4 Geometric Random Walks

The essential idea underlying the random walk for real processes is the assumption of mutually independent increments of the order of magnitude for each point of time. However, economic time series in particular do not satisfy the latter assumption. Seasonal fluctuations of monthly sales figures for example are in absolute terms significantly greater if the yearly average sales figure is high. By contrast, the relative or percentage changes are stable over time and do not depend on the current level of . Analogously to the random walk with i.i.d. absolute increments , a geometric random walk is assumed to have i.i.d. relative increments

For example, a geometric binomial random walk is given by

 (5.5)

where are mutually independent and for

Given the independence assumption and it follows from equation (4.5) that increases or decreases exponentially as the case may be or :

If the process is on average stable, which is the case for

For a recombining process, i.e.  this relationship simplifies to

Taking logarithms in equation (4.5) yields:

The process is itself an ordinary binomial process with starting value and increments for which hold:

For large, is approximately normally distributed, i.e.  is approximately lognormally distributed.