This paper deals with three major types of convergence of probability measures on metric spaces: weak convergence, setwise convergence, and convergence in total variation. First, it describes and compares necessary and sufficient conditions for these types of convergence, some of which are well-known, in terms of convergence of probabilities of open and closed sets and, for the probabilities on...

We consider a class of discrete distribution functions whose characteristic are separated from zero, i.e., their absolute values larger than certain positive constant on the entire real line. This is rather wide: it contains infinitely divisible functions, lattice distributions have no zeroes line, and also with jump greater 1/2. Recently, we demonstrated in our study that corresponding to elem...

The problem of the nonparametric estimation of a probability distribution is considered from three viewpoints: the consistency in total variation, the consistency in information divergence, and consistency in reversed order information divergence. These types of consistencies are relatively strong criteria of convergence, and a probability distribution cannot he consistently estimated in either...

We prove weak ergodicity of the inhomogeneous Markov process generated by the generalized transition probability of Tsallis and Stariolo under power-law decay of the temperature. We thus have a mathematical foundation to conjecture convergence of simulated annealing processes with the generalized transition probability to the minimum of the cost function. An explicitly solvable example in one d...

We derive some results on contrarian and one-sided strategies by Skeptic for the fair-coin game in the framework of the game-theoretic probability of Shafer and Vovk [8]. In particular, concerning the rate of convergence of the strong law of large numbers (SLLN), we prove that Skeptic can force that the convergence has to be slower than or equal to O(n−1/2). This is achieved by a very simple co...

We aim at clarifying the relationship between laws of large numbers for fuzzy sets or possibility distributions and laws of large numbers for fuzzy or possi-bilistic variables. We contend that these two frameworks are different and present the relationships between them that explain why this fact was unrecognized so far. A classical result in Probability Theory is the law of large numbers. If ξ...

The Fréchet mean generalizes the idea of averaging in spaces where pairwise addition is not well-defined. In general metric spaces, however, the Fréchet sample mean is not a consistent estimator of the theoretical Fréchet mean. For non-trivial examples, sequences of Fréchet sample mean sets may fail to converge in a set-analytical sense. Hence, it becomes necessary to consider other types of co...