Stochastic Systems Group  

Dependence structures of some infinite variance stochastic processes
Murad S. Taqqu
Boston University
Fractional Gaussian noise is a Gaussian process whose increments exhibit longrange dependence. There are many extensions of that process in the infinite variance stable case. Logfractional stable noise (logFSN) is a particularly interesting one. It is a stationary meanzero 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 logFSN 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 alphastable logFSN, and the rate of decay of the codifference is such that one has longrange 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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