|Stochastic Systems Group|
Dependence structures of some infinite variance stochastic processes
Murad S. Taqqu
Fractional Gaussian noise is a Gaussian process whose increments 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 number alpha between 1 and 2. The lower the value of alpha, the heavier the tail of the marginal distributions. The fact that alpha is less than 2 renders the variance infinite. Therefore 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. We show that the codifference decreases to zero like a power function as the lag goes to infinity. The value of the exponent, which depends on alpha, measures the speed of the decay. There is also a multiplicative constant of asymptoticity c which depends also on alpha and plays an important role. This constant c turns out to be positive for symmetric alpha-stable log-FSN, and the rate of decay of the codifference is such that one has long-range dependence. We also show that a second measure of dependence, the "covariation", converges to zero with the same intensity and that its constant of asymptoticity is positive as well.
This is joint work with Joshua B. Levy.
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