On Convergence in Necessity and Its Laws of Large Numbers

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 is a variable, defined on a probability space (Ω, A, P) and with mean µ, and {ξ n } n is a sequence of independent, identically distributed variables, then the following holds: 1 n n i=1 ξ i → µ. If convergence is almost sure, i.e. ξ n → ξ if ξ n (ω) → ξ(ω) except for at most those ω in a set of probability 0, it is called the Strong LLN. If convergence is in probability, i.e. P (|ξ n − ξ| > ε) → 0 for every ε > 0, then it is called the Weak LLN. Almost sure convergence is stronger than convergence in probability, hence the names. If we replace the probability measure P by a possibility measure Π, the variable is usually called a fuzzy or possibilistic variable. One can then ask whether similar theorems hold. The notion of almost sure convergence applies verbatim and convergence in probability has an immediate analog: ξ n → ξ in necessity if Π(|ξ n − ξ| > ε) → 0 for every ε > 0. The name comes from the dual expression N ec(|ξ n − ξ| ≤ ε) → 1, with N ec being the dual measure to Π, a necessity measure. The notion of independence is, however, more involved, as discussed elsewhere [6]. Here we will replace the notion of independence by-relatedness, without inquiring whether-related variables are 'independent' in the intuitive sense or not. Let be a triangular norm, then two variables ξ and η are