On the existence of \(\varphi\)-moments of the limit of a normalized supercritical Galton-Watson process
Let \((\mathbb{Z})_{n\geq 0}\) be a supercritical Galton-Watson process with offspring distribution \((p_j)_{j\geq 0}\) and finite mean offspring \(\mu\). The normalized process \(W_n\overset{\text{def}} = \mu^{-n}\mathbb{Z}_n\), \(n\geq 0\), is a nonnegative and thus a.s. convergent martingale with limit \(W\). The Kesten-Stigum theorem states that in order for \(W\) to be positive on the set of non-extinction it is necessary and sufficient that \(EZ_1\log Z_1= \sum_{j\geq 1}p_j\log j<\infty\). K. B. Athreya (1971) showed for any \(p\geq 0\) that \(0< EW|\log W|^p<\infty\) holds if and only if \(EZ_1\log Z_1< \infty\). Also, he proved this equivalence to be true for any integrable solution \(W\) of the stochastic equation \[ W\overset {\text{d}}={1\over \mu} \sum^{Z_1}_{k=1} W(k).\tag{1} \] This article extends these results by providing a necessary and sufficient moment condition on \((p_j)_{j\geq 0}\) for the existence of \(E\varphi(W)\) for an even larger class of functions \(\varphi\). Given any nondecreasing convex function \(\varphi:[0,\infty)\to [0,\infty)\), the authors define the operator \(\mathbb{L}\) through \[ \mathbb{L}\varphi(x)\overset{\text{def}} = \int^s_0\int^s_0 {\varphi'(r)\over r} dr\,ds,\quad x\geq 0.\tag{2} \] Theorem 1.1. Let \((\mathbb{Z}_n)_{n\geq 0}\) be a supercritical Galton-Watson process with offspring distribution \((p_j)_{j\geq 0}\); finite mean offspring \(\mu\) and normalized limit \(W= \lim_{n\to\infty} \mu^{-n}\mathbb{Z}_n\). Then for each \(\varphi\in C^*\) \[ 0< E\varphi(W)<\infty\quad\text{iff}\quad E(\varphi(Z_1))< \infty \] holds true with \(\mathbb{L}\varphi\) as in (2). Theorem 1.2. In the situation of Theorem 1.1 the following assertions are equivalent: \[ 0< E\varphi(W)<\infty;\quad \sup_{n\geq 0} E\varphi(W_n)< \infty, \] \((\varphi(W_n))_{n\geq 0}\) is uniformly integrable, \[ \lim_{n\to\infty} E|\varphi(W_n)- \varphi(W)|< \infty,\;E\varphi\Biggl(\sup_{n\geq 0} W_n\Biggr)< \infty. \] Section 2 contains a discussion of the main result in the context of regularity varying functions and shows in particular that it implies the related \(\varphi\)-moment result of \textit{N. H. Bingham} and \textit{R. A. Doney} [Adv. Appl. Probab. 6, 711--731 (1974; Zbl 0297.60044)]. Some general facts on functions \(\varphi\) from the classes \(C\) and \(C^*\) and the associated \(\mathbb{L}\varphi\) are provided in Section 3 and Section 4 contains various inequalities for the \(\varphi\)-moments of \(W\). Section 5 contains the proof of Theorem 1.1.
scientific article; zbMATH DE number 2153672