The structure of self-similar stable mixed moving averages

• 成果类型: Article
• 署名作者: Pipiras, V; Taqqu, MS
• 署名单位: Boston University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2002
• 页码: 898-932
• 关键词: construction

摘要

Let alpha is an element of (1, 2) and X-alpha be a symmetric alpha-stable (SalphaS) process with stationary increments given by the mixed moving average X-alpha(t) = integral(X) integral(R) (G(x, t + u) - G(x, u)) Malpha(dx, du), t is an element of R, where (X, chi, mu) is a standard Lebesgue space, G: X x R --> R is some measurable function and M-alpha is a SalphaS random measure on X x R with the control measure m(dx, du) = mu(dx)du. Assume, in addition, that X-alpha is self-similar with exponent H is an element of (0, 1). In this work, we obtain a unique in distribution decomposition of a process X-alpha into three independent processes X-alpha =(d) X-alpha((1)) + X-alpha((2)) + X-alpha((3)). We characterize X-alpha((1)) and X-alpha((2)) and provide examples of X-alpha((3)). The first process X-alpha((1)) can be represented as integral(Y) integral(R) integral(R) e(-kappas)(F(y, e(s)(t + u)) - F(y, e(s)u))M-alpha(dy, ds, du), where kappa = H - 1/alpha, (Y, Y, nu) is a standard Lebesgue space and M-alpha is a SalphaS random measure on Y x R x R with the control measure m(dy, ds, du) = nu(dy) ds du. Particular cases include the limit of renewal reward processes, the so-called random wavelet expansion and Takenaka process. The second process X-alpha((2)) has the representation integral(Z) integral(R) (G(1)(z)((t + u)(+)(kappa) - u(+)(kappa)) + G(2)(z)((t + u)(-)(kappa) - u(-)(kappa)))Malpha(dz, du), if kappa not equal 0, integral(Z) integral(R) (G(1)(z)(ln \t + u\ - ln \u\) + G(2)(z)(l((0, infinity))(t + u) - l((0, infinity))(u)))M-alpha(dz, du), if kappa = 0, where (Z, Z, nu) is a standard Lebesgue space and M-alpha is a SalphaS random measure on Z x R with the control measure m(dz, du) = nu(dz)du. Particular cases include linear fractional stable motions, log-fractional stable motion and SalphaS Levy motion. An example of a process X-alpha((3)) is integral(0)(1) integral(R) ((t + u)(+)(kappa) l([0, 1/2))({x + ln \t + u\}) - u(+)(kappa) l([0, 1/2))({x + ln \u\}))M-alpha(dx, du), where M-alpha is a SalphaS random measure on [0, 1) x R with the control measure m(dx, du) = dxdu and {.} is the fractional part function.