The paper examines those continuous time Markov processes Z(·) on the positive integers which have the ‘skip free upwards’ property, with regard to their asymptotic behaviour in the event of Z(t) tending to infinity. The behaviour is characterised in terms of the convergence or divergence of an appropriate function of Z(t), and the description is improved by central limit and iterated logarithm theorems. The conditions of the theorems are expressed entirely in terms of the matrix Q of instantaneous transition rates for Z(·). The method is applied, by way of example, to the super-critical linear birth and death process.

The inverse of an extremal process {Y(t), t ≧ 0} is an additive process whose Lévy measure can be computed. This measure controls among other things the Poisson number of jumps of Y while Y is in the vertical window (c, d]. A simple transformation of the inverse of the extremal process governed by Λ (x) = exp{– e–x} is also extremal-Λ (x) and this fact enables one to relate behavior of Y-Λ at t = ∞ to behavior near t = 0. Some extensions of these ideas to sample sequences of maxima of i.i.d. random variables are carried out.

{Xn, n ≧ 1} are i.i.d. unbounded random variables with continuous d.f. F(x) =1 —e –R(x). Xj is a record value of this sequence if Xj >max {X1, …, Xj-1} The almost sure behavior of the sequence of record values {XLn} is studied. Sufficient conditions are given for lim supn→∞XLn/R–l(n)=ec, lim inf n → ∞XLn/R−1 (n) = e−c, a.s., 0 ≦ c ≦ ∞, and also for lim supn→∞ (XLn—R–1(n))/an =1, lim infn→∞ (XLn—R–1(n))/an = − 1, a.s., for suitably chosen constants an. The a.s. behavior of {XLn} is compared to that of the sequence {Mn}, where Mn = max {X1, …, Xn}. The method is to translate results for the case where the Xn's are exponential to the general case by means of an extended theory of regular variation.

An iterated logarithm type theorem is obtained for maxima of nonstationary discrete-parameter Gaussian processes.