Wavelet expansion of some stable random processes

JUAN MIGUEL MEDINA; BRUNO CERNUSCHI FRÍAS

Tandil

Congreso; Reunión anual de la Unión Matemática Argentina; 2010

Unión Matemática Argentina

For a symmetric -stable, 2 [1; 2] random measure M over R, with nite absolutely continuous control measure d = w dx, given f 2 L(R; dx), we de ne the process2 [1; 2] random measure M over R, with nite absolutely continuous control measure d = w dx, given f 2 L(R; dx), we de ne the processd = w dx, given f 2 L(R; dx), we de ne the process Y (x) =(x) = Z R I(f)(x �� t)dM(t) :(f)(x �� t)dM(t) : Where I(:) is the Riesz fractional integration operator of order , 0 < < 1. Let G(Y )  R2 be the graph of Y . We prove that, for appropriate and , the Hausdordimension of G(Y ), veri es dimH(G(Y ))  1 + 2I(:) is the Riesz fractional integration operator of order , 0 < < 1. Let G(Y )  R2 be the graph of Y . We prove that, for appropriate and , the Hausdordimension of G(Y ), veri es dimH(G(Y ))  1 + 2G(Y )  R2 be the graph of Y . We prove that, for appropriate and , the Hausdordimension of G(Y ), veri es dimH(G(Y ))  1 + 2G(Y ), veri es dimH(G(Y ))  1 + 2 �� a.s.. Moreover this process has the following local behaviour: Let B a ball for which, (B) > 0, and there exists a constant c > 0 such that f > c a.e. over B, then the restricted graph veri es: dimH(G(Y jB)) = 1 + 2�� a.s.. Moreover this process has the following local behaviour: Let B a ball for which, (B) > 0, and there exists a constant c > 0 such that f > c a.e. over B, then the restricted graph veri es: dimH(G(Y jB)) = 1 + 2B a ball for which, (B) > 0, and there exists a constant c > 0 such that f > c a.e. over B, then the restricted graph veri es: dimH(G(Y jB)) = 1 + 2c > 0 such that f > c a.e. over B, then the restricted graph veri es: dimH(G(Y jB)) = 1 + 2H(G(Y jB)) = 1 + 2 �� . For this processes, we study the convergence of its wavelet and multirresolution expansions. More precisely, we prove that if the wavelets are associated to a MRA, with scale functions - such that there exists a bounded integrable radial function�� . For this processes, we study the convergence of its wavelet and multirresolution expansions. More precisely, we prove that if the wavelets are associated to a MRA, with scale functions - such that there exists a bounded integrable radial function- such that there exists a bounded integrable radial function , such that j-j   a.e. Then if Z(x) =, such that j-j   a.e. Then if Z(x) = R R f(x��t)dM(t), the seriesf(x��t)dM(t), the series P k2Z2Z I(-k j)(x) and(-k j)(x) and P mjj P k2Z2Z I( k m)(x) converges in the mean to Y as j ! 1. Under suitable conditions this convergences is also, with probability one.( k m)(x) converges in the mean to Y as j ! 1. Under suitable conditions this convergences is also, with probability one.