We study the asymptotic behaviour of the trace (the sum of the diagonal parts) τn = τn(ω) of a plane partition ω of the positive integer n, assuming that ω is chosen uniformly at random from the set of all such partitions. We prove that (τn − c0n2/3)/c1n1/3 log1/2n converges weakly, as n → ∞, to the standard normal distribution, where c0 = ζ(2)/ [2ζ(3)]2/3, c1 = √(1/3/) [2ζ(3)]1/3 and ζ(s) = Σj=1∞j−s.
In a one-parameter model for evolution of random trees strong law of large numbers and central limit theorem are proved for the number of vertices with low degree. The proof is based on elementary martingale theory.
By applying the Skorohod martingale embedding method, a strong approximation theorem for partial sums of asymptotically negatively
dependent (AND) Gaussian sequences, under polynomial decay rates, is established. As applications, the law of the iterated
logarithm, the Chung-type law of the iterated logarithm and the almost sure central limit theorem for AND Gaussian sequences
Let Ys,n denote the number of part sizes ≧ s in a random and uniform partition of the positive integer n that are counted without multiplicity. For s = λ(6n)1/2/π + o(n1/4), 0 ≦ λ < ∞, as n → ∞, we establish the weak convergence of Ys,n to a Gaussian distribution in the form of a central limit theorem. The mean and the standard deviation are also asymptotically