Introduction. It is known that the distribution theory of sums of iiidependent random variables can be developed from several points of view which are, however, in the main, equivalent. One, and possibly the most general, approach is represented by Kolmogoroff's axiomatic treatment1 of questions of probability distribution. This approach applies, in particular, to the problem of probable convergelnce of series of indepenident random variables, as first solved by Khintcbhiie and Kolmogoroff.2 It also implies the treatment based on the Lebesgue measure theory of infinite product spaces, as developed by Steinhaus, Littlewood, Paley and Zygmund, Jessen, anid others.3 It is known 4 that the main result of Khintchine and Kolmogoroff, which is based on the notion of equivalent series, can be formulated also in terms of infinite convolutions. The results of the present paper concern infinite convolutions and imply, among other things, certain facts which are equivalent to theorems concerning the divergenice problem of series of independent random variables. In particular, the results imply essential refinements of certain facts indicated by Levy.5 Since Levy's statements will not be used, the following considerations imply detailed proofs for them. Theorem 1, which applies not only to convolution sequences, seems to have an independent interest. Theorems 3 and 5 delimit all possibilities which canl