Probability Distribution of Distances between Local Extrema of Random Number Series

There is a sequence of random numbers x1, x2 . . . , xn and so on. Numbers are independent of each other, but all numbers are from the same continuous distribution. If xi−1 < xi > xi+1, then xi is a local maximum. Here, we show that the probability mass function (PMF) fm(d) of distances d between local maxima is non-parametric and the same for any probability distribution of random numbers in the sequence, and that the average distance is exactly 3. We present a method of computation of this PMF and its table for distances betwen 2 and 29. This PMF is confirmed to match distance distributions of sample random number sequences, which were created by pseudo-random number generators or obtained from ”true” random number sources. 1. Average distnace between local maxima Let’s take any number in the sequence and find out the probability that it’s a local maximum. Definition 1. A number xi is a local maximum, if the following condition is true xi−1 < xi > xi+1. First, we’ll use a combinatorial approach. Consider the following sequence (Figure 1) of pseudo-random numbers generated by MS Excel RAND() function: (1) 0.935536495 (2) 0.191531578 (3) 0.429049655 (4) 0.308968021 (5) 0.179540986 (6) 0.401329789 (7) 0.71581906 (8) 0.604617962 (9) 0.877254876 (10) 0.973280207 (11) 0.489033299 (12) 0.912367351 (13) 0.604552972 (14) 0.039395302 (15) 0.3780448 Date: August 12, 2006. 1991 Mathematics Subject Classification. Primary 60G70.