We show that the sequence of moments of order less than 1 of averages of i.i.d. positive random variables is log-concave. For moments of order at least 1, we conjecture that the sequence is log-convex and show that this holds eventually for integer moments (after neglecting the first $p^2$ terms of the sequence).

It is proved that the best constant factor in the Rademacher-Gaussian tail comparison is between two explicitly defined absolute constants c 1 and c 2 such that c 2 ≈1.01 c 1.A discussion of relative merits of this result versus limit theorems is given.

We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

A new necessary and sufficient condition for a distribution of unbounded support to be in a domain of partial attraction is given. This relates the recent work of [5] and [6].

Let {Zn} be a finite mean supercritical Bienaymé– Galton–Watson process. It is known that there exist norming constants {Cn} such that {Zn/Cn} converges almost surely to a limit W. Also there is a whole literature concerning properties of {Cn} and W. We attempt a new approach to the limit theory of {Zn} by relating it to the theory of sums of independent and identically distributed random variables.

I show that the sum of independent random variables converges in distribution when suitably normalised, so long as the Xk satisfy the following two conditions: μ(n)= E |Xn| is comparable with E |Sn| for large n, and Xk/μ(k) converges in distribution. Also I consider the associated birth process X(t) = max{n: Sn ≦ t} when each Xk is positive, and I show that there exists a continuous increasing function v(t) such that for some variable Y with specified distribution, and for almost all u. The function v, satisfies v (t) = A (1 + o (t)) log t. The Markovian birth process with parameters λn = λn, where 0 < λ < 1, is an example of such a process.

The strong law of large numbers for independent and identically distributed random variables Xi, i = 1,2,3, …, with finite mean µ can be stated as, for any ∊ > 0, the number of integers n such that |n−1 Σi=1nXi − μ| > ∊, N(∊), is finite a.s. It is known, furthermore, that EN(∊) < ∞ if and only if EX12 < ∞. Here it is shown that if EX12 < ∞ then ∊2EN(∊)→ var X1 as ∊ → 0.