Random Fourier-Stieltjes series associated with stable process (Q1098493)

Let X(t,\(\omega)\) be a symmetric stable process of index \(\alpha\), \(1\leq \alpha \leq 2\), \(a\leq t\leq b\). The authors define the integral \(\int^{b}_{a}f(t)dX(t,\omega)\) in the sense of convergence in probability for any function \(f(t)\in L^{\alpha}(a,b)\). (The authors assume f(t) any function of L p(a,b), \(p\geq 1\), but perhaps, \(p\geq \alpha.)\) Now consider the random Fourier-Stieltjes series (RFS) \(\sum a_ nA_ n(\omega)e^{2\pi iny}\), where \(A_ n(\omega)=\int^{1}_{0}e^{2\pi int} dX(t,\omega)\), X(t,\(\omega)\) being a 1-periodic symmetric stable process of index \(\alpha\) and \(a_ n\) is the Fourier coefficient of \(f(t)\in L^{\alpha}(0,1)\). Suppose \(1<\alpha <2.\) It is shown that the sum \(S_ n(y)=\sum^{n}_{-n}a_ kA_ k(\omega)e^{2\pi iky}\) converges in probability to \(S(y)=\int^{1}_{0}f(y-t)dX(t,\omega)\) and S(y) is continuous in probability. A stronger result under stronger conditions on \(\{a_ n\}\) is given. When \(\alpha =2\), it is shown that \(S_ n(y)\) converges to S(y) almost surely, X(t,\(\omega)\) being a periodic Wiener process, and S(y) is L 2- continuous; and furthermore, if \(\sum | a_ n| <\infty\), it is almost surely continuous. When \(\alpha =1\), RFS is shown to be (C,1) summable in probability. The case \(0<\alpha <1\) is also treated.