Rate of Convergence in the Central Limit Theorem for Generalized Convolutions

The theory of Generalized Convolutions represents an unifying approach to many limit schemes in the probability theory, like Central Limit Theorems for i.i.d. random variables, extreme value theory, Kingman convolution and others. The main idea is, that in contrast to the usual summation and maxima scheme, the operation between random variables may itself be random. This random effect of the operation first appeared in a seminal work of Kingman (1963). The structure of Kingman convolution attracted attention of many specialists, among them Urbanik (1964, 1973, 1988), Bingham (1971, 1984), Volkovich (1980, 1984). It was Urbanik (1964), who developed a theory of generalized convolutions studying binary operations on probability measures on the positive half-line, that possess analogues of the most important properties of ordinary convolution. Kingman’s work provides only an example of such an operation. We shall refer to generalized convolutions of probability measures on R + as to Urbanik convolutions. The theory of Urbanik convolutions is very extensive now. We summarize it in Section 2. The main open problem in this area is the rate of convergence in the Central Limit Theorem. In Section 3 we deal with sharp estimates of the rate of convergence for normalized Urbanik convolutions to generalized stable laws. In Section 4 we present an example of a generalized convolution of random vectors and provide an estimate of the rate of convergence of an n-fold generalized convolution to a generalized stable vector.