Why do simulated arrival times from a Poisson distribution seem to show periodicity?

Question

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.

The formula for the inter-arrival times is:

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].

A series of arrival times can be constructed as follows (in python), for r = 0.1:

import random
import numpy as np

r = 0.1

num_arrival_times = 40

arrival_time_list = []

arrival_time = 0.0
print("%.20f" % 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("%.20f" % arrival_time)
    arrival_time_list.append(arrival_time)

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?