Fractionally stable distributions 179 THEOREM 2. Under the above assumptions concerning the r.v.'s T\ and Χι, there exists a finite positive constant CQ = co(ct, β) such thatThus, we shall name distribution Q(x;a,ß, 1) and Q(t\ß,a, 1) for«,/? e (0, 1] and θ = 1 coupled relative to space-time.Going over from finite dimensional distributions of process Σ (?) to the consideration of weak convergence of this process in Skorokhod space D([0, oo), M), we note that in [23] it has been proved that under conditions (6) and (7) the continuous-time random walk process weakly converge to the superposition Z{t) = A(B(t)) with corresponding normalisation. Here A (t) is the self-similar random process with stationary independent increments (Levy) process with Hurst exponent 1 /a, B(t) is the self-similar random process with Hurst exponent 1/β. Moreover, processes A(t) and Β(t) are independent andIt follows that increments of B(t) for β < 1 is non stationary. So B(t) is not a process with stationary increments. The process Z(t) is a self-similar one, but not a process with stationary increments.

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