Let ξt, t ∈ [0, T ], be a strong Markov process with values in a complete separable metric space (X, ρ) and with transition probability functionIt is shown that a certain growth condition on α(h, a), as a ↓ 0 and h stays fixed, implies the almost sure boundedness of the p-variation of ξt, where p depends on the rate of growth.

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Let {ξ1, ξ2, . . .} be a sequence of independent random variables, and η be a counting random variable independent of this sequence. In addition, let S0 := 0 and Sn := ξ1 + ξ2 + · · · + ξn for n 1. We consider conditions for random variables {ξ1, ξ2, . . .} and η under which the distribution functions of the random maximum ξ (η) := max{0, ξ1, ξ2, . . . , ξη} and of the random maximum of sums S (η) := max{S0, S1, S2, . . . , Sη} belong to the class of consistently varying distributions. In our consideration the random variables {ξ1, ξ2, . . .} are not necessarily identically distributed.

We provide two upper bounds on the Clayton copula Cθ(u1,...,un) if θ > 0 and n ≥ 2 and a lower bound in the case θ ∈ [-1,0) and n ≥ 2. The obtained bounds provide a nice probabilistic interpretation related to some negative dependence structures and also allow defining three new two-dimensional copulas which tighten the classical Fréchet–Hoeffding bounds for the Clayton copula when n = 2.

Using generalized Blumenthal-Getoor indices, we obtain criteria for the finiteness of the p-variation of Lévy-type processes. This class of stochastic processes includes solutions of Skorokhod-type stochastic differential equations (SDEs), certain Feller processes and solutions of Lévy driven SDEs. The class of processes is wider than in earlier contributions and using fine continuity we are able to handle general measurable subsets of R d as state spaces. Furthermore, in contrast to previous contributions on the subject, we introduce a local index in order to complement the upper index. This local index yields a sufficient condition for the infiniteness of the p-variation. We discuss various examples in order to demonstrate the applicability of the method.