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Chúng ta nói rằng phân phối ổn định nếu nó là ổn định với các thành phần đặc trưng. Trường hợp = 2 tương ứng với sự phân bố bình thường = 1 được phân phối . nghịch đảo phân phối tương ứng với . Đối với | 38 MODELING LONG-TERM STOCK RETURNS It is possible to summarize the family in terms of the characteristic function Ộ X E eiXt exp iyt c t 1 ip sign t z t a 2.33 where c 0 a G 0 2 p G 1 1 and z t a tan - êlog ltl if a 1 if a 1 2.34 The y parameter is a location parameter the a component is called the characteristic exponent and is used to classify distributions within the stable family. We say that a distribution is a-stable if it is stable with characteristic component a. The case a 2 corresponds to the normal distribution and a 1 is the Cauchy distribution. The inverse Gaussian distribution corresponds to a 1 2 p 1. For a 2 the distribution is fat-tailed with infinite variance. If p 0 then the distribution is symmetric. As with the normal distribution stable distributions can be used to describe stochastic processes. Let Yt be a stochastic process such as the log-return process. If Yt has independent and stationary increments for any time unit then Yt is a stable or Levy process and Yt has an a-stable distribution. Stable processes have been popular for modeling financial processes because they can be very fat-tailed and because of the obvious attraction of being able to convolute the distribution. However they are not easy to use estimation requires advanced techniques and it is not easy to simulate a stable process although a method is given in Chambers et al. 1976 and software using that method is available from Nolan 2000 . The model specifically does not incorporate autocorrelations arising from volatility bunching and therefore does not in fact fit the data sets in the section on data particularly well. An excellent source of explanatory and technical information on the use of stable distributions is given in Nolan 1998 also on his Web site 2000 Nolan provides software for analyzing stable distributions. GENERAL STOCHASTIC VOLATILITY MODELS We can allow volatility to vary stochastically without the regime constraints of the RSLN model. For example let yt ỊI