On the range of a regenerative sequence

For a given countable partition of the range of a regenerative sequence {X_n: n ? 0}, let R_n be the number of distinct sets in the partition visited by X up to time n. We study convergence issues associated with the range sequence {R_n: n ? 0}. As an application, we generalize a theorem of Chosid and Isaac to Harris recurrent Markov chains.     

The minimal sublinear expectations and their related properties

In this paper, we prove that for a sublinear expectation ɛ[·] defined on L ²(Ω,), the following statements are equivalent:

A characterization of minimal valid inequalities for mixed integer programs

A simple condition on the underlying subadditive function is shown to characterize minimal valid inequalities. This result is proved in a very general master problem framework and completes the characterization there. We explain the condition also in the context of value functions and finally give some related, unresolved questions.     

The representations of two types of functionals on L~∞(Ω,F) and L~∞(Ω,F,P)

This article gives the representations of two types of real functionals on Z∞(Ω,F) or L∞(Ω,F,P) in terms of Choquet integrals. These functionals are comonotonically subadditive and comonotonically convex, respectively.     

Lp-intensities of random measures

Given a random measure η and a fixed number p>1, the L_p-intensity 6η6_p of ηis defined as the total variation measure of the subadditive set function 6η(·)6_p. It is shown that 6η6_p can exist (be locally finite) only if the usual intensity measure Eη exists and $\text{η 《}\text{E}\text{η}$ a.s, and that in this case $\text{6η6}{}_{\mathrm{p}}\text{B}\text{=?}{}_{\mathrm{<mtext>B</mtext>}}\text{6}\text{d}\text{η}\text{d}\text{E}\text{η}\text{6}{}_{\mathrm{p}}\text{d}\text{E}\text{η}$. If η is the conditional intensity of a simple point process ξ, then 6η6_p equals the total variation of the subadditive set functions 6P[ξB = 1|B^cξ]6_p and 6E[ξB|B^cξ]6_p. Some applications to stochastic geometry and particle systems are discussed briefly.     

Thermodynamic formalism for subadditive sequence of discontinuous functions

The topological pressure for subadditive sequence of discontinuous functions is defined on any invariant subset having a nested family of subsets in the compact metric space.Two subadditive variational principles associated with two different relatively weak conditions are developed for the defined topological pressure.As an application,we give an example on systems with nonzero Lyapunov exponents.     

On the range of recurrent Markov chains

Let {X_n}₀^∞ be an irreducible recurrent Markov Chain on the nonnegative integers. A result of Chosid and Isaac (1978) gives a sufficient condition for n^?1R_n → 0 w.p.1. where R_n is the range of the chain. We give an alternative proof using Kingman's subadditive ergodic theorem (Kingman, 1973). Some examples are also given.     

Divergence rates for the number of rare numbers

Suppose thatX ₁,X ₂, ... is a sequence of i.i.d. random variables taking value inZ ⁺. Consider the random sequenceA(X)(X ₁,X ₂,...). LetY _n be the number of integers which appear exactly once in the firstn terms ofA(X). We investigate the limit behavior ofY _n /E[Y _n ] and establish conditions under which we have almost sure convergence to 1. We also find conditions under which we dtermine the rate of growth ofE[Y _n ]. These results extend earlier work by the author.     

超凸空间中的选择定理及弱外超凸集的性质

本文利用半序方法,研究具有外超凸集值的带次可加模的集值映象的保不动点集,带相同次可加模的单值选择,获得了这类单值选择的存在性及唯一性条件.且研究了弱外超凸集的一些相交性质.