On the age of a randomly picked individual in a linear birth-and-death process

Abstract

We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution.

Convergence rates for the degree distribution in a dynamic network model

“…Thus, also given Y T alone, the difference A l (T ) − A l+1 (T ) is stochastically dominated by Z l . By the same argument as for (10), it follows that (11) is smaller than or equal to…”

“…Furthermore, let A J∞ = Z, where Z is the random variable from Corollary A.10. From the proof of this corollary (see [11], proof of Corollary 2.4), we obtain that we may write…”

“…A further important ingredient is that the reciprocal of the node process (Y t ) t≥0 conditioned on survival is a supermartingale (see Lemma A.15 in the appendix), which makes it easy to deal with the expected value of its maximum. Finally, we also use that the age A J T (T ) of the randomly picked individual converges quickly to A J∞ , which was recently proven in [11] (see Corollary A.10 in the appendix for the precise result we apply).…”

“…. In order to find an upper bound for the second conditional expectation on the right-hand side of (40), we apply a recent result from [11] about the age distribution in a linear birth and death process. This result is stated in Corollary A.10 in the appendix and implies that this conditional expectation is bounded from above by…”

“…From [11] we know furthermore an expression and an upper bound for the conditional expectation of 1/Y t given Y t > 0 that is essentially of the anticipated order e −(λ−µ)t as t → ∞.…”