A random collision process with transition probabilities belonging to the same type of distribution is considered. It was proved that if the characteristic function of the initial distribution has a positive radius of convergence, then the sequence {Fn(x)} converges weakly to a distribution G(x) [9]. We define a metric e1[Fn;G], which is analogous to the functional e[f] introduced by Tanaka to Kac's model of Maxwellian gas [10]. We prove that e1[Fn; G] is monotone non-increasing as n → ∞, and also the convergence of the sequence {Fn(x)} under the weaker assumption that for some a > 1 the initial distribution has an ath moment.

By analogy with statistical mechanics we consider a random collision process with discrete time wand continuous states x ∈ [0, ∞). We assume three conditions (i), (ii) and (iii), which can be applied to Kac's model of a Maxwellian gas, and show that the sequence of probability distributions converges to a probability distribution using their moments.

By analogy with statistical mechanics random collision processes are considered. In two cases we derive their limiting distributions using discrimination information. In one case the limiting distribution is unique and in the other one the family of limiting distributions is an exponential family.