1. Let {Z(t), t ≧ 0} be an age-dependent branching process with offspring generating function and life-time distribution function G(t). Denote by N(t) the progeny of the process, that is the total number of objects which have been born in [0, t], counting the ancestor. (See Section 2 for definitions.) Then in the Galton-Watson process (i.e., when G(t) = 0 for t ≦ 1, G(t) = 1 for t > 1) we have the simple relation Nn = Z0 + Z1 + ··· + Zn, so that the asymptotic behaviour of Nn as n → ∞ follows from a knowledge of the asymptotic behaviour of Zn. In particular, if 1 < m = h'(1) < ∞ and Zn(ω)/E(Zn) → Z(ω) > 0 then also Nn(ω)/E(Nn) → Z(ω) > 0; since E(Zn)/E(Nn) → 1 – m–1 this means that

Consider a queueing system of the following type: random arrivals enter the system, where they are assigned to one of r service channels. Each arrival is then serviced and discharged from the system. The classical results for this class of problems are based upon systems where customers join a single queue and then are randomly assigned to an available server from this queue. (See, for instance, Takács (1962).)

Let D = D[0, ∞) be the space of all real-valued right-continuous functions on [0, ∞) with limits from the left. For any stochastic process X in D, let the associated supremum process be S(X), wherefor any x ∊ D. It is easy to verify that S: D → D is continuous in any of Skorohod's (1956) topologies extended from D[0,1] to D[0, ∞) (cf. Stone (1963) and Whitt (1970a, c)). Hence, weak convergence Xn ⇒ X in D implies weak convergence S(Xn) ⇒ S(X) in D by virtue of the continuous mapping theorem (cf. Theorem 5.1 of Billingsley (1968)).

1. Let the (honest) Markov process with transition functions (pij(0)) have transition rates (qij) and suppose that, for some M, so that the matrix Q = (qij) determines a bounded operator on the Banach space l1 by right-multiplication. Then in the terminology of [8], (pp. 12 and 19) Q will be bounded and ΩF will be a closed restriction of Q with dense domain, so that ΩF = Q; that is, we shall have a process whose associated semigroup has a bounded generator. In these circumstances Theorem 10.3.2 of [2] applies and the matrix Pt = (pij(t)) is given by where exp{·} denotes the function defined by the exponential power-series. We shall be interested here (as in [5] and [9]) in the determination of the limit matrix P∞ = (limt→∞pij(t)).

In [3] Heyde proposed an extended version of the well-known Hájek-Rényi inequality to include random variables without finite moments. The purpose of this note is to point out that the theorem in [3] is in error, and to prove the following theorem in its stead:

Theorem. Let X1, X2, ··· be any sequence of random variables, and define be independent random variables, each with mean zero and finite variance, and define Zk = Xk - Yk for k ≧ 1. Let c1, c2, ··· be a non-increasing sequence of positive numbers.