title: Why do simulated arrival times from a Poisson distribution seem to show periodicity?
body: <p>I am experimenting with simulated arrival times drawn from a Poisson distribution. To construct the arrival times, I am randomly drawing inter-arrival times from the inverse CDF, which is exponentially distributed.</p>
  <p>The formula for the inter-arrival times is:</p>
  <p>X = -(1/r) * ln(1 - p), where X is the inter-arrival time, r is the Poisson rate, and p is a randomly drawn probability between [0, 1].</p>
  <p>A series of arrival times can be constructed as follows (in python), for r = 0.1:</p>
  <pre><code>import random
  import numpy as np
  
  r = 0.1
  
  num_arrival_times = 40
  
  arrival_time_list = []
  
  arrival_time = 0.0
  print(&quot;%.20f&quot; % arrival_time)
  arrival_time_list.append(arrival_time)
  
  for i in np.arange(0, num_arrival_times, 1):
      p = random.random()
      inter_arrival_time = -np.log(1.0 - p) / r
      arrival_time = arrival_time + inter_arrival_time
      print(&quot;%.20f&quot; % arrival_time)
      arrival_time_list.append(arrival_time)
  </code></pre>
  <p>Here is my question: When I take these arrival times and do a periodicity analysis on them (e.g., using Lomb-Scargle), the periodogram shows significant peaks. This is surprising to me. Is there some explanation for why the inter-arrivals drawn from a exponential distribution would appear periodic/quasi-periodic?</p>