Foss, Sergey; Miyazawa, Masakiyo Tails in a fixed-point problem for a branching process with state-independent immigration. (English) Zbl 1465.60078 Markov Process. Relat. Fields 26, No. 4, 613-635 (2020). Consider a non-negative integer-valued random variable \(X\) solution a distributional fixed point equation such that \(X\) has the same law as \(A + \sum_{i=1}^X B_i\), where \(A\) and \(B_i\) are independent non-negative random variables independent of \(X\), and \(B_i\) are i.i.d. random variables. This fixed point equation appears as the invariant measure of a branching process with immigration \(A\) at each generation and reproduction law \(B\). It also appears in the analysis of single-server queue processes with feedback and permanent customers.It is well known that given the law of \(A\) and \(B_1\) and some mild integrability conditions, there exists a unique solution to this fixed point equation. However, determining the law of \(X\) from the law of \(A\) and \(B_1\) turns out to be complicate. In this article, the authors work under the assumptions that \(A\) and/or \(B\) have heavy tails (for example belonging to the class of intermediate or extended regularly varying distribution) and compute under these conditions an equivalent from the tail of \(X\). Under some assumptions on the tail behaviour of \(A\) and \(B\), they show that \(\mathbb{P}(x < X \leq x/\mathbb{E}(b)) \sim c (\mathbb{P}(A > x) + \mathbb{P}(B>x))\) as \(x \to \infty\) for some explicit \(c > 0\).Two generalizations of the fixed point equation are also considered. First a continuous analogue \(X =_\text{st} A + \int_0^X \mathrm{d} B_s\), where \(A\) is a non-negative real-valued random variable and \(B\) an independent subordinator, both being further independent of \(X\), which satisfies the same type of tail estimate as the discrete case. Then a generalized fixed point equation corresponding to a branching process with immigration in which particles reproduce for two generations before dying out. For this model as well, the tail behaviour of \(X\) is related to the tail behaviour of the immigration variable and the reproduction law. Reviewer: Bastien Mallein (Paris) Cited in 1 Document MSC: 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60J05 Discrete-time Markov processes on general state spaces 60K25 Queueing theory (aspects of probability theory) 60E99 Distribution theory Keywords:heavy tail asymptotics; branching process; state-independent immigration; fixed-point equation; single-server feedback queue; long tail; dominantly varying tail; (intermediate) regularly varying tail PDFBibTeX XMLCite \textit{S. Foss} and \textit{M. Miyazawa}, Markov Process. Relat. Fields 26, No. 4, 613--635 (2020; Zbl 1465.60078) Full Text: arXiv Link