Fractionally stable distributions

Abstract

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.

Nonlinear Markov Semigroups and Interacting Lévy Type Processes

“…. , x l ) = g(x 1 ) + · · · + g(x l ) (the weak form (8) being introduced in [39], see also Sections 5 and 7 for details). Notice that though the original interacting particle systems leading to (5), (7) (mean field interaction preserving the number of particles) and to (8) (including k-ary jumps with possible fragmentation or coagulation) are quite different, equation (8) can be written in form (5).…”

“…Notice that though the original interacting particle systems leading to (5), (7) (mean field interaction preserving the number of particles) and to (8) (including k-ary jumps with possible fragmentation or coagulation) are quite different, equation (8) can be written in form (5). Moreover, it is easy to understand that equation (8) describing the limit of systems with interactions not preserving the number of particles, can be also written in form (6). In fact, if all m ≥ k, then (due to the symmetry of transition kernels) …”

“…As the latter are of special interest among Lévy type processes, due to their applications in a wide variety of models from plasma physics to finances (see e.g. [3], [8], [4], [58]), we shall concentrate further on stable type processes ending Section 3 with a simple illustration of Theorem 3.1 in the case of non-degenerate second order part of a pseudo-differential generator.…”

“…Section 7 is devoted to the interaction changing the number of particles, whose law of large number is described by equation (8). As we noted above this equation can be also written in form (10) with conditionally positive operators B k in C(X k ), which allows to consider them as particular cases of (10).…”

Дробно-Устойчивая Статистика Экспрессии Генов В Экспериментальных Данных Секвенирования Нового Поколения

“…The fractional-stable laws are limit distributions of sums of independent identical distributed random variables (see [15]). At working with these distributions the main difficulty consists in absence of explicit expressions for densities.…”