Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process X_t with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights b_j decaying as j^−β, 1<β<2/. We show that the empirical process (normalized by N^1/β) weakly converges, as the sample size N increases, to the process c_x⁺L⁺+c_x⁻L⁻, where L⁺,L⁻ are independent totally skewed β-stable random variables, and c_x⁺,c_x⁻ are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(X_s) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.     

(1/α)-Self similar α-stable processes with stationary increments

In this note we settle a question posed by Kasahara, Maejima, and Vervaat. We show that the α-stable Lévy motion is the only (1/α)-self-similar α-stable process with stationary increments if 0 < α < 1. We also introduce new classes of (1/α)-self-similar α-stable processes with stationary increments for 1 < α < 2.     

Dilated Fractional Stable Motions

Dilated fractional stable motions are stable, self-similar, stationary increments random processes which are associated with dissipative flows. Self-similarity implies that their finite-dimensional distributions are invariant under scaling. In the Gaussian case, when the stability exponent equals 2, dilated fractional stable motions reduce to fractional Brownian motion. We suppose here that the stability exponent is less than 2. This implies that the dilated fractional stable motions have infinite variance and hence they cannot be characterised by a covariance function. These dilated fractional stable motions are defined through an integral representation involving a nonrandom kernel. This kernel plays a fundamental role. In this work, we study the space of kernels for which the dilated processes are well-defined, indicate connections to Sobolev spaces, discuss uniqueness questions and relate dilated fractional stable motions to other self-similar processes. We show that a number of processes that have been obtained in the literature, are in fact dilated fractional stable motions, for example, the telecom process obtained as limit of renewal reward processes, the Takenaka processes and the so-called random wavelet expansion processes.     

More limit theory for the sample correlation function of moving averages

Let X_t = Σ^∞_j=-∞ c_jZ_{t - j} be a moving average process where {Z_t} is iid with common distribution in the domain of attraction of a stable law with index , 0 < < 2. If 0 < < 2, E|Z₁| < ∞ and the distribution of |Z₁|and |Z₁Z₂| are tail equivalent then the sample correlation function of {X₁} suitably normalized converges in distribution to the ratio of two dependent stable random variables with indices and /2. This is in sharp contrast to the case E|Z₁| ^{= ∞} where the limit distribution is that of the ratio of two independent stable variables. Proofs rely heavily on point process techniques. We also consider the case when the sample correlations are asymptotically normal and extend slightly the classical result.     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.     

Properties of spectral covariance for linear processes with infinite variance

We consider a measure of dependence for symmetric α-stable random vectors, which was introduced by the second author in 1976. We demonstrate that this measure of dependence, which we suggest to call the spectral covariance, can be extended to random vectors in the domain of normal attraction of general stable vectors. We investigate the asymptotic of the spectral covariance function for linear stable (Ornstein–Uhlenbeck, log-fractional, linear-fractional) processes with infinite variance and show that, in comparison with the results on the properties of codifference of these processes, obtained two decades ago, the results for the spectral variance are obtained under more general conditions and calculations are simpler.     

α-Stable characterization of Banach spaces (1 < α < 2)

This work characterizes some subclasses of α-stable (0 < α < 1) Banach spaces in terms of the extendibility to Radon laws of certain α-stable cylinder measures. These result extend the work of S. Chobanian and V. Tarieladze (J. Multivar. Anal.7, 183–203 (1977)). For these spaces it is shown that every Radon stable measure is the continuous image of a stable measure on a suitable L_β space with β = α(1 − α)⁻¹. The latter result extends some work of Garling (Ann. Probab.4, 600–611 (1976)) and Jain (Proceedings, Symposia in Pure Math. XXXI, p. 55–65, Amer. Math. Soc., Providence, R.I.).     

A note on convex ordering for stable stochastic integrals

We establish a convex ordering between stochastic integrals driven by strictly α-stable processes with index α ∈ (1,2). Our approach is based on the forward–backward stochastic calculus for martingales together with a suitable decomposition of stable stochastic integrals.     

Gaugeability for Feynman-Kac functionals with applications to symmetric -stable processes

For symmetric -stable processes, an analytic criterion for a measure being gaugeable was obtained by Z.-Q. Chen (2002), M. Takeda (2002) and M. Takeda and T. Uemura (2004). Applying it, we consider the ultracontractivity of Feynman-Kac semigroups and expectations of the number of branches hitting closed sets in branching symmetric -stable processes.

     

Test of association between multivariate stable vectors

Based on observations of d-dimensional random vectors in the domain of attraction of a stable distribution with (multi-)index α = (α₁, …, α_d), an estimator for the dependence function of the α_i-stable variables is constructed. The estimator utilizes the α-tail-estimator and an estimator of the spectral measure of the α-stable law. This estimator gives rise to a test of association of the stable components and various quantitative measures of association.