Theoretical properties of the sample generalized codifference function of stable moving average process

The generalized codifference function as a dependence measure for stationary processes with infinite variance has been proposed as a generalization of the autocorrelation function. In this paper we investigate the theoretical properties of estimator of generalized codifference function of stable moving average process. Some theoretical properties of the sample codifference function of moving average process for small order are discussed.


Introduction
Various empirical studies typically show that most of the financial data are leptokurtic (i.e. heavy on the tail and peaked around the center). In other words, empirically derived fact that the chances of occurrence of extreme events and the variations that occur in the data is larger than can be modeled by a normal distribution. Thus, if a practitioner uses a financial model based on the normal distribution, there is a great risk of financial loss due to the emergence of extreme events and the variations in the data that cannot be modeled by this distribution. One of the powerful distributions that can represent the characteristics of such data is the stable distribution (see e.g., Rachev and Mittnik [1] and Rachev et al. [2]). Stable distribution with the normal distribution as a special case of this distribution class, is one of a relatively popular distribution for modeling leptokurtic data [3,4]. In this context, various empirical studies especially in the economics and finance field showed that the stable non-normal distribution is more suitable for modeling various financial data such as asset returns [3][4][5][6][7][8].
Most statistical models require the existence of second order moments or distribution with finite variance. Based on the second order moments, the dependencies structure of the model can be assessed. It notice that conditions of the second moments or variances of stable distribution depends on characteristic exponent parameters α. The second moments of stable random variable exist only for = 2 (Gaussian case) and for 0 < < 2, one can not use the covariance functions to describe the dependence structure. It is known that the covariance function can completely describe the dependence structure of Gaussian distributed random vectors. Some generalization of the autocovariance function as dependence measure of process with infinite variance have been proposed in the literature, e.g., the  [9]) and the codifference function (see e.g., [7], Kokoszka and Taqqu [10], Wyłomańska et al. [11], Rosadi [12], and Rosadi and Deistler [6]).

Generalized Condifference Function
The generalized codifference function (GCF) of the stationary processes at lag k as proposed in [10]

Estimation of the GCF
As the generalized codifference function is defined via characteristic function, it can be estimated by empirical characteristic function. Given a sample +, #, … , -, a consistent estimator for the generalized codifference function at lag ℤ can be defined for ℝ, as in [13] ̂ The asymptotic properties of the sample generalized codifference function for a class of linear time series models was investigated in [13]. It can be shown that for a stationary linear process where the coefficients @ A 's are real and majored by geometrically bounded weights and is i.i.d C C, by applying Theorem 1 in [13], we obtain ̂ , − ; ⟶ , − ; where the consistency is in the probability sense.
We summarize the following results from [13] regarding the asymptotic behavior of the sample generalized codifference function for a class of linear processes. Let Here r qs •• and r qs ‚‚ and r qs ‚• denote the partitions of r qs which correspond to the real and the imaginary components, respectively. The components r qs ‚‚ and r qs ‚• are defined similary as r qs •• . In (10),

Asymptotic Property of the Sample GCF
Let us consider the univariate discrete time process , Ÿ ℤ which is a moving average process of order 1 MA 1 with symmetric −stable innovation, i.e., = + @ )+ 12 Here is i.i.d. symmetric −stable (C C) distributed, i.e., it has a characteristic function of the form exp = exp −¢ £ | | £ where denotes the index of stability 0 < ≤ 2 and ¢ ≥ 0 denotes the scale parameter. In the following theorem, we derive the limiting distribution property of the sample normalized codifference function for moving average (MA) process of order 1 with symmetric α stable (SαS) innovation [see 13].
In order to find the component of r "" , we write  By using (16) we have completes the proof.