Abstract
We compute the tail asymptotics of the product of a beta random variable and a generalized gamma random variable which are independent and have general parameters. A special case of these asymptotics were proved and used in a recent work of Bubeck, Mossel, and Racz in order to determine the tail asymptotics of the maximum degree of the preferential attachment tree. The proof presented here is simpler and highlights why these asymptotics hold.
Highlights
There has been a lot of recent interest in various urn schemes due to their appearance in many graph growth models
The limiting distributions arising in these urn schemes are often related to the beta and gamma distributions
We believe that the simple proof we present here is useful in highlighting why these asymptotics hold
Summary
There has been a lot of recent interest in various urn schemes due to their appearance in many graph growth models (see, e.g., [15, 2, 3, 16, 4, 17, 5, 9, 18]). The computation of various statistics in random graph models often boils down to using algebraic properties of these distributions, commonly referred to as the beta-gamma algebra [6]. Involving beta and generalized gamma random variables. There has been lots of work on understanding the distribution and tail asymptotics of products of random variables; see, e.g., [19] for a paper from nearly half a century ago, and [10, 11] and references therein for recent developments. In a very interesting recent work, Peköz, Röllin, and Ross [18] showed that generalized gamma random variables with p being an integer greater than 2 arise as limits in time inhomogeneous Pólya-type urn schemes. It would be interesting to find connections to urn schemes and random graph models for general values of p
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