Remarks on a Markov chain example of Kolmogorov

Summary It is shown that the stochastic transition matrix P(t), in Kolmogorov's example of a process with an instantaneous state, is uniquely determined by the derivative matrix Q=P(0), and the most general such substochastic P(t) is also found. The example is used to show that, if 0 is an instantaneous state, then 1-p ₀₀(t) can tend to 0 arbitrarily slowly and on the other hand (1-p ₀₀ (t))/t can tend to +t8 arbitrarily slowly.     

A class of power indices for voting games

A class of power indices is discussed which includes not only the usual power indices but also the class of semivalues and the indices introduced byDeegan/Packel [1978]. Sufficient conditions for members of this class to be symmetric, support-independent, and self-dual are given. We then discuss various conditions under which members of this class coincide with the ShapleyShubik and Banzhaf indices.     

Stationary regenerative phenomena

Summary Corresponding to anyp-functionp, a stationary version of the associated regenerative phenomenon is constructed for which the underlying probability measure may have infinite total mass (though it will always be -finite ifp is standard). As a trivial consequence,p is a positive-definite function. The construction is generalised to quasi-Markov chains.     

On a class of special flows

Summary A special flow, in which the basic automorphism is a Bernoulli automorphism and the ceiling function depends only on the present position of the Bernoulli sequence and is not lattice distributed, is a K-flow.This paper represents the result, in a revised form, which was presented at the Symposium on Ergodic Theory held at Mathematisches Forschungsinstitut Oberwolfach from August 4 to 10, 1968.