Department of Statistics, UCLA

Let X₁ , X₂ , ..., be stationary and ergodic random variables with values in a metric space M with distance d, let P(A) = P(X_n in A) and let S(x,r)) > 0 if r > 0, and suppose also that for x in M₀ , P(S(s,r)) is continuous in x and is differentiable in r for r >= 0, and with a positive derivative for all r in a neighborhood of 0. Consider the set M^* of pairs (x,y) such that both x and y are in M₀ and lim_r->0 P(S(x,y)) exists and is a finite positive number R(x,y). Then R(x,y) is called the relative density of P for the pair x,y.

The differentiability condition is essentially the same as required for P to have a positive density in the Euclidean case. Note there may be pairs of elements (x,y) such that lim_r->0 P(S(x,r))/P(S(y,r)) fails to exist, is zero, or is positive infinity. For example, if P on the square [0,1]X[0,1] concentrates a total probability of .5 uniformly on the line x = y, 0 <= x,y <= 1, and distributes probability uniformly on the square excepting this line, then the line of pairs x = y is in M^* and so is the square excepting the line. But a pair with one element on the line and the other gives a limit of 0 or positive infinity depending which element appears in the numerator (or denominator). This kinds of measures may be of considerable interest and examples where they arise can be given.