An inequality for a functional of probability distributions and its application to Kac's one-dimensional model of a Maxwellian gas

where the infimum is taken over all pairs of random variables X and Y defined on (f2, P) and distributed according to f and g respectively; here g is the Gaussian distribution with mean 0 and variance a 2 =~2 (f)e I-f] is sometimes denoted by e IX] when X is a random variable with distribution f. It should be noticed that the value of e [ f ] does not depend upon a choice of the probability space (f2, P). The purpose of this paper is to present some basic properties of e (especially, the inequality (2.2)) together with an application to the central limit theorem and then to show that the functional e is monotone decreasing along Boltzmann solutions of Kac's one-dimensional model of a Maxwellian gas. Some of our results can be generalized to the case of R 3; for example, the functional e similarly defined in R 3 decreases along solutions of Boltzmann's problem for the 3-dimensional Maxwellian gas, but this will be discussed in another occasion.