Exact and limiting distributions in diagonal Pólya processes

Abstract

The original version of the history unfortunately contained a mistake. The correct approval history is given here.

The continuum Pólya-like random walk

“…Remark 3.2. The proof of Theorem 3.1 is a generalization of the proof in Balaji and Mahmoud [2]. However, it needed some new techniques.…”

“…However, it needed some new techniques. In [2], there is a conditional argument that uses a sum on the number of balls of a color, given that number. Of course, the number of balls is an integer and such a sum can be carried out.…”

“…The inverse transform-to translate results in the continuous domain back to results in the discrete domain-is fraught with difficulty. Some authors developed interest in the continuous-time Pólya process for its own sake (see [2,3,4,10]). In the Pólya process, each ball carries an internal clock that rings in Exp(1) time (a random amount of time, according to an exponential random variable with mean 1), independently of the behavior of all other clocks.…”

Polya’s bees: A model of decentralized decision-making

“…The evolution of the number of balls of a given color is independent of the evolution of other colors (until the threshold is hit). Thus, Lemma 3.1 of ( 51 ) directly applies.…”

“…We precisely characterize how the composition of balls in the urn evolves over time. Lemma 3.1 of ( 51 ) gives us the distribution of balls of each color at any time t (in the absence of a threshold for stopping the process).…”