For a plane domain the circle transform assigns to each circle its length of intersection with the domain. The problem is to determine the geometry of the domain given the mean of these intersection lengths as the circles' centers vary and given how that mean varies with circle radius. We also consider the mean square and higher moments of the intersection lengths — as a function of the circle radius. The discussion includes an identification of the geometric content of the mean when centers are inside of a disk and of the second moment when the centers are arbitrary points in the plane. The latter is equivalent to the distribution of distance between pairs of points of the domain.

Families of exchangeably dissociated random variables are defined and discussed. These include families of the form g(Yi, Yj, …, Yz) for some function g of m arguments and some sequence Yn of i.i.d. random variables on any suitable space. A central limit theorem for exchangeably dissociated random variables is proved and some remarks on the closeness of the normal approximation are made. The weak convergence of the empirical distribution process to a Gaussian process is proved. Some applications to data analysis are discussed.