Martingale convergence theorem for a conditional intuitionistic fuzzy mean value

The aim of this contribution is to show a representation of a conditional intuitionistic fuzzy mean value of intuitionistic fuzzy observables by a conditional mean value of random variables. We formulate a martingale convergence theorem for a conditional intuitionistic fuzzy mean value, too.

Martingale Convergence Theorem for the Conditional Intuitionistic Fuzzy Probability

For the first time, the concept of conditional probability on intuitionistic fuzzy sets was introduced by K. Lendelová. She defined the conditional intuitionistic fuzzy probability using a separating intuitionistic fuzzy probability. Later in 2009, V. Valenčáková generalized this result and defined the conditional probability for the MV-algebra of inuitionistic fuzzy sets using the state and probability on this MV-algebra. She also proved the properties of conditional intuitionistic fuzzy probability on this MV-algebra. B. Riečan formulated the notion of conditional probability for intuitionistic fuzzy sets using an intuitionistic fuzzy state. We use this definition in our paper. Since the convergence theorems play an important role in classical theory of probability and statistics, we study the martingale convergence theorem for the conditional intuitionistic fuzzy probability. The aim of this contribution is to formulate a version of the martingale convergence theorem for a conditional intuitionistic fuzzy probability induced by an intuitionistic fuzzy state m. We work in the family of intuitionistic fuzzy sets introduced by K. T. Atanassov as an extension of fuzzy sets introduced by L. Zadeh. We proved the properties of the conditional intuitionistic fuzzy probability.

Conditional intuitionistic fuzzy probability and martingale convergence theorem using IF-probability

The aim of this paper is to formulate the conditional intuitionistic fuzzy probability and a version of martingale convergence theorem with respect an intuitionistic fuzzy probability. Since the intuitionistic fuzzy probability can be decomposed to two intuitionistic fuzzy states, we can use the results holding for intuitionistic fuzzy states.

On two classes of interacting stochastic processes arising in cancer modeling

The processes X and Y are said to interact if the laws governing the changes of either of them at time t depend on the values of the other process at times up to t. For bivariate interacting Markov processes, their limiting behavior is analysed by means of an approximation suggested by Fuhrmann, consisting of discretizing time, and assuming that in each time interval the processes develop independently, according to the laws obtained by fixing the value of the other process at its level attained at the beginning of the interval.In this way the conditions for almost sure extinction, escape to ∞ with positive probability, etc., are obtained (by using the martingale convergence theorem) for state-dependent branching processes studied by Roi, and for bivariate processes with one component piecewise determined.