Convergence in mean of some random Fourier series

For a symmetric stable process X(t,ω) with index α ∈ (1, 2], f ∈ Lp[0, 2π], p α, an = 1 2π 2π 0 e−intf (t)dt and An(ω) = 1 2π 2π 0 e−int dX(t,ω), we establish that the random Fourier–Stieltjes (RFS) series ∞ n=−∞ anAn(ω)eint (in)β converges in the mean to the stochastic integral 1 2π 2π 0 fβ(t −u) dX(u,ω), where fβ is the fractional integral of order β of the function f for 1 p <β < 1 + 1 p . Further it is proved that the RFS series ∞ n=−∞ anAn(ω)eint (in)β is Abel summable to 1 2π 2π 0 fβ(t − u) dX(u,ω). Also we define fractional derivative of the sum ∞n=−∞ anAn(ω)eint of order β for an, An(ω) as above and 1 p < 1−β <1+ 1 p. We have shown that the formal fractional derivative of the series ∞n=−∞ anAn(ω)eint of order β exists in the sense of mean. © 2007 Elsevier Inc. All rights reserved.