No abstract

A diffusion process X(·) in the infinite-dimensional ordered simplex is characterized in terms of the generator defined on an appropriate domain. It is shown that X(·) is the limit in distribution of several sequences of discrete stochastic models of the infinitely-many-neutral-alleles type. It is further shown that X(·) has a unique stationary distribution and is reversible and ergodic. Kingman's limit theorem for the descending order statistics of the symmetric Dirichlet distribution is obtained as a corollary.

Communicated by Zoltan GinglToral introduced so-called cooperative Parrondo games, in which there are N ≥ 3 players arranged in a circle. At each turn one player is randomly chosen to play. He plays either game A or game B. Game A results in a win or loss of one unit based on the toss of a fair coin. Game B results in a win or loss of one unit based on the toss of a biased coin, with the amount of the bias depending on whether none, one, or two of the player's two nearest neighbors have won their most recent games. Game A is fair, so the games are said to exhibit the Parrondo effect if game B is losing or fair and the random mixture (1/2)(A + B) is winning. With the parameter space being the unit cube, we investigate the region in which the Parrondo effect appears. Explicit formulas can be found if 3 ≤ N ≤ 6 and exact computations can be carried out if 7 ≤ N ≤ 19, at least. We provide numerical evidence suggesting that the Parrondo region has nonzero volume in the limit as N → ∞.

No abstract

No abstract