Emergence of patterns in random processes.

Sixty years ago, it was observed that any independent and identically distributed (i.i.d.) random variable would produce a pattern of peak-to-peak sequences with, on average, three events per sequence. This outcome was employed to show that randomness could yield, as a null hypothesis for animal populations, an explanation for their apparent 3-year cycles. We show how we can explicitly obtain a universal distribution of the lengths of peak-to-peak sequences in time series and that this can be employed for long data sets as a test of their i.i.d. character. We illustrate the validity of our analysis utilizing the peak-to-peak statistics of a Gaussian white noise. We also consider the nearest-neighbor cluster statistics of point processes in time. If the time intervals are random, we show that cluster size statistics are identical to the peak-to-peak sequence statistics of time series. In order to study the influence of correlations in a time series, we determine the peak-to-peak sequence statistics for the Langevin equation of kinetic theory leading to Brownian motion. To test our methodology, we consider a variety of applications. Using a global catalog of earthquakes, we obtain the peak-to-peak statistics of earthquake magnitudes and the nearest neighbor interoccurrence time statistics. In both cases, we find good agreement with the i.i.d. theory. We also consider the interval statistics of the Old Faithful geyser in Yellowstone National Park. In this case, we find a significant deviation from the i.i.d. theory which we attribute to antipersistence. We consider the interval statistics using the AL index of geomagnetic substorms. We again find a significant deviation from i.i.d. behavior that we attribute to mild persistence. Finally, we examine the behavior of Standard and Poor's 500 stock index's daily returns from 1928-2011 and show that, while it is close to being i.i.d., there is, again, significant persistence. We expect that there will be many other applications of our methodology both to interoccurrence statistics and to time series.