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The standard Pólya's urn process can be stated as follows:

You have an urn with red and green balls. At any time unit you choose one ball at random, note the colour, and give the ball back. At the same time, you add another ball of the same colour.

It is natural to study the behaviour of the colour distribution over time, and in particular the corresponding concentration statements. While this problem has been well studied in the literature, we are interested in very similar problem, where we assume that the creation of a ball requires a certain amount of time units described by a given distribution $\mathcal D$.

Given a discrete distribution $\mathcal D$ on positive integers, suppose $X_i$ $(i>0)$ are indep. r. v. distributed according to $\mathcal D$. The delayed Pólya's urn process can be stated as follows:

You have an urn with red and green balls. At $i$-th time unit, you choose one ball at random, note the colour, and give the ball back. After $X_i$ time units, you add another ball of the same colour.

The analysis of this delayed process seems hard even in a seemingly easy case when $\mathcal D$ is degenerate distribution, for instance, if the creation of a ball always takes a fixed constant number of time units. Any insights or comments, in particular regarding obtaining tight concentration of the colour ratio over time, are most welcome.

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  • $\begingroup$ What kind of results are you hoping for? In the classic case (I think) the quotient $Q_n:={R_n \over G_n}$ tends a.s to a random variable $Q$ which is distributed as ${Z \over 1-Z}$, with $Z$ being $\mathrm{Beta}(r,g)$ distributed - how do you interpret "tight concentration" here? $\endgroup$ – esg Dec 2 at 17:19

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