Asymptotic normality of a Hurst parameter estimator based on the modified Allan Variance

It has been observed that in many situations the network traffic is characterized by self-similarity and long-range correlations on various time-scales. The memory parameter of a related time series is thus a key quantity in order to predict and control the traffic flow. In the present paper we analyze the performance of a memory parameter estimator, d, defined by the log-regression on the so-called modified Allan variance. Under the assumption that the signal process is a fractional Brownian motion, with Hurst parameter H, we study the rate of convergence of the empirical modified Allan variance, and then prove that the log-regression estimator d converges to the memory parameter d_0= 2H - 2 of the process. In particular, we show that the deviation d-d_0, when suitably normalized, converges in distribution to a normal random variable, and we compute explicitly its asymptotic variance