Modified Sign Method for Testing the Fractality of Gaussian Noise

Fractal Gaussian noise is a stationary Gaussian sequence of zero-mean random variables whose sums possess the stochastic self-similarity property. If the random variables are independent, the self-similarity coefficient equals 1/2. The sign criterion for testing the hypothesis that the parameter equals 1/2 against the alternative $H\neq 1/2$ is based on counting the sign change rate for elements of the sequence. We propose a modification of the criterion: we count sign change indicators not only for the original random variables but also for random variables formed as sums of consecutive elements. The proof of the asymptotic normality of our statistics under the alternative hypothesis is based on the theorem on the asymptotics of the covariance of sign change indicators for a zero-mean stationary Gaussian sequence with a slowly decaying correlation function.