We improve, first, a strong invariance principle from Kifer (2013) [10] for nonconventional sums of the form ∑n=1[Nt]F(X(n),X(2n),…,X(ℓn)) (normalized by 1/N) where X(n),n≥0’s is a sufficiently fast mixing vector process with some moment conditions and stationarity properties and F...

Lower and upper bounds in Dobrushin’s inequality for the variance of sums of functionals defined on a non-homogeneous Markov chain together with some related probability results are analyzed.

Two rumour models are considered, the first due to Daley and Kendall (1965) and the second due to Maki and Thompson (1973). The size of a rumour, defined as the number of individuals in the population eventually hearing the rumour, is investigated using both deterministic and stochastic...