The asymptotic codifference and covariation of log-fractional stable noise

Many econometric quantities such as long-term risk can be modeled by Pareto-like distributions and may also display long-range dependence. If Pareto is replaced by Gaussian, then one can consider fractional Brownian motion whose increments, called fractional Gaussian noise, exhibit long-range dependence. There are many extensions of that process in the infinite variance stable case. Log-fractional stable noise (log-FSN) is a particularly interesting one. It is a stationary mean-zero stable process with infinite variance, parametrized by a tail index α between 1 and 2, and hence with heavy tails. The lower the value of α, the heavier the tail of the marginal distributions. The fact that α is less than 2 renders the variance infinite. Thus dependence between past and future cannot be measured using the correlation. There are other dependence measures that one can use, for instance the “codifference” or the “covariation”. Since log-FSN is a moving average and hence “mixing”, these dependence measures converge to zero as the lags between past and future become very large. The codifference, in particular, decreases to zero like a power function as the lag goes to infinity. Two parameters play an important role: (a) the value of the exponent, which depends on α and measures the speed of the decay; (b) a multiplicative constant of asymptoticity c which depends also on α. In this paper, it is shown that for symmetric α-stable log-FSN, the constant c is positive and that the rate of decay of the codifference is such that one has long-range dependence. It is also proved that the same conclusion holds for the second measure of dependence, the covariation, which converges to zero with the same intensity and with a constant of asymptoticity which is positive as well.