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My question is related to this question on Math.SE where it was shown that there exist discrete distributions with probability mass functions that "oscillate" in a non-monotonic way over an infinite support. However, as far as I know, all "known" infinite discrete distributions, (e.g., those listed on Wikipedia) are non-increasing, starting from some index --- i.e. $p_{i+1} \leq p_{i}$, starting from some index $i < \infty$.

Can you give an example of an application of an "alternating" discrete distribution in practice? Are there such distributions, which are named after someone?

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First of all, it is a common misconception that a probability distribution only "exists" if it is of one of the forms found in the families of distributions that receive attention in academic literature and are therefore listed on resources like Wikipedia. Every discrete distribution obeying the rules of probability exists, regardless of whether it is part of one of these listed families of distributions or not. The only properties imposed by the rules of probability is that the probabilities in the discrete distribution must be non-negative and must add up to one over the whole range of the random variable (which may be a countably infinite set).

In practical applications, alternating distributions (where the probability mass oscillates) sometimes arise in trigonometric problems when you are looking at circular behaviour, where you can get damped oscillations in probability. Another place they can arise is when you "mix" together two different distributions in a way that creates an alternating distribution from two previously monotonic distributions.

As a simple toy example, suppose you have a fair six-sided die and an unfair coin (which flips heads with probability $3/5$). You flip the coin once and you roll the die over and over again until you get an even number. Let $R$ be the number of rolls of the die and let $H$ be the indicator that the coin came up heads (i.e., $H=1$ if it was heads and $H=0$ if it was tails). Now look at the distribution of the number:

$$T \equiv 2R-H.$$

If you compute the probability mass function for the values $T=1,2,3,...$, I think you will find that you get the oscillating probabilities:

$$p = \frac{3}{10}, \frac{2}{10}, \frac{3}{20}, \frac{2}{20}, \frac{3}{40}, \frac{2}{40}, \frac{3}{80}, \frac{2}{80}, \cdots$$

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  • $\begingroup$ How can mixing two monotone distribution provide an oscillating distribution? Thank you. $\endgroup$
    – LrM
    Oct 20, 2020 at 14:55
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    $\begingroup$ The above is essentially an example of this. The monotonic distribution here is the geometric distribution for $R$. The example "mixes" two copies of this distribution over the odd and even numbers, with different probability multiples, leading to the oscillating pattern you see. $\endgroup$
    – Ben
    Oct 20, 2020 at 14:59

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