Laws of large numbers for periodically and almost periodically correlated processes
For a second-order continuous-time complex-valued stochastic process \(X = \{X(t), t \in R\}\), define the functions \(m(t) = E\{X(t)\}\) and \(R(s,t) = E \{X(s) \overline X(t)\}\), \(s,t \in R\), and the random variables \(J_{A,X} (\lambda)\), \(\lambda > 0\), \(A > 0\), by \(J_{A,X} (\lambda, \omega) = A^{-1} \int^ A_ 0 X(t, \omega) \exp( - i\lambda t) dt\). \(X\) is called periodically correlated (PC) if, for some \(T>0\), \(m(t+T) = m(t)\), \(R(s+T,t+T) = R(s,t)\) for all \(s,t \in R\). Assume that \(X\) is a measurable PC process with \(m(t) \equiv 0\) and \(R(t,t) \in L_ 1 [0,T]\), and that \(a_ 0 (\tau) \equiv T^{-1} \int^ T_ 0 R(t+ \tau,t) \exp \{-i2 \pi kt/T\}dt\) is continuous at \(\tau = 0\). Then it is shown that \(J_{A,X} (\lambda)\) converges in quadratic mean as \(A\to \infty\). Moreover, if \(\Theta\) is uniformly distributed on \([0,T]\) and independent of \(X\), then \(J_{A,X} (\lambda)\) and \(J_{A,Y} (\lambda)\) both converge almost surely if either of them does, where \(Y(t)=X(t+\Theta)\). Combined with a strong law due to \textit{V. F. Gaposhkin} [Theory Probab. Appl. 22(1977), 286-310 (1978); translation from Teor. Veroyatn. Primen. 22, 295-319 (1977; Zbl 0377.60033)] for weakly stationary processes, this result leads to sufficient conditions for \(J_{A,X} (\lambda) \to 0\) a.s. as \(A \to \infty\). Analogous results are also presented for almost PC processes.
scientific article; zbMATH DE number 691016