Two probabilistic hit-and-run algorithms are presented to detect nonredundant constraints in a full dimensional system of linear inequalities. The algorithms proceed by generating a random sequence of interior points whose limiting distribution is uniform, and by searching for a nonredundant constraint in the direction of a random vector from each point in the sequence. In the hypersphere directions algorithm tile direction vector is drawn from a uniform distribution on a hypersphere. In tile computalionalb superior coordinate directions algorithm a search is carried out along one of the coordinate vectors. The algorithms are terminated through the use of a Bayesian stopping rule. Computational experience with the algorithms and the stopping rule will be reported.

Summary. Iffor a process (en):'=_ 00 the conditional distribution of i;n given the past does not depend on n for e.g. n;?; 0, then the process may be called a chain with infinite connections. Under a well-known continuity condition on this conditional distribution the process is shown to be distributed as an instantaneous function of a countable state Markov chain. Also under a certain weaker continuity condition uniqueness of the distributions of the stationary chains is obtained.

Summary. Speed of convergence is studied for a Marcinkiewicz-Zygmund strong law for partial sums of bounded dependent random variables under conditions on their mixing rate. Though o:-mixing is also considered, the most interesting result concerns absolutely regular sequences. The results are applied to renewal theory to show that some of the estimates obtained by other authors on coupling are best possible. Another application sharpens a result for averaging a function along a random walk.

No abstract

For a stationary ergodic process it is proved that the dependence coefficient associated with absolute regularity has a limit connected with a periodicity concept. Similar results can then be obtained for stronger dependence coefficients. The periodicity concept is studied separately and it is seen that the double tail a-field can be trivial while the period is 2. The paper imbeds renewal theory in ergodic theory. The total variation metric is used.