Symmetric Stable Law Research Articles

Estimating the Logarithm of Characteristic Function and Stability Parameter for Symmetric Stable Laws

Let \(X_1,\ldots ,X_n\) be an i.i.d. sample from symmetric stable distribution with stability parameter \(\alpha\) and scale parameter \(\gamma\). Let \(\varphi _n\) be the empirical characteristic function. We prove a uniform large deviation inequality: given preciseness \(\epsilon >0\) and probability \(p\in (0,1)\), there exists universal (depending on \(\epsilon\) and p but not depending on \(\alpha\) and \(\gamma\)) constant \(\bar{r}>0\) so that $$P\big (\sup _{u>0:r(u)\le \bar{r}}|r(u)-\hat{r}(u)|\ge \epsilon \big )\le p,$$where \(r(u)=(u\gamma )^{\alpha }\) and \(\hat{r}(u)=-\ln |\varphi _n(u)|\). As an applications of the result, we show how it can be used in estimation the unknown stability parameter \(\alpha\).

$$L^{\alpha -1}$$ distance between two one-dimensional stochastic differential equations driven by a symmetric $$\alpha$$-stable process

In this article, we consider a coefficient stability problem for one-dimensional stochastic differential equations driven by an $$\alpha$$ -stable process with $$\alpha \in (1,2)$$ . More precisely, we find an upper bound for the $$L^{\alpha -1}(\varOmega ,{\mathbb {P}})$$ distance between two solutions in terms of the $$L^{\alpha }\left( {\mathbb {R}},\mu^{\alpha }_{x_0}\right)$$ distance of the coefficients for an appropriate measure $$\mu^{\alpha} _{x_0}$$ which characterizes symmetric stable laws and depends on the initial value of the stochastic differential equation. We obtain this result using the method introduced by Komatsu (Proc Jpn Acad Ser A Math Sci 58(8):353–356, 1982) which is used in the proof of uniqueness of solutions together with an upper bound for the transition density function of the solution of the stochastic differential equation obtained by Kulik (The parametrix method and the weak solution to an SDE driven by an $$\alpha$$ -stable noise. arXiv:1412.8732 , 2014).

On the rates of convergence to symmetric stable laws for distributions of normalized geometric random sums

Let X1,X2,... be a sequence of independent, identically distributed random variables. Let ?p be a geometric random variable with parameter p?(0,1), independent of all Xj, j ? 1: Assume that ? : N ? R+ is a positive normalized function such that ?(n) = o(1) when n ? +?. The paper deals with the rate of convergence for distributions of randomly normalized geometric random sums ?(?p) ??p,j=1 Xj to symmetric stable laws in term of Zolotarev?s probability metric.

Asymptotic theory for M-estimates in unstable AR([formula omitted]) processes with infinite variance innovations

In this paper, we present the asymptotic distribution of M-estimates for parameters in unstable AR(p) processes. The innovations are assumed to be in the domain of attraction of a symmetric stable law with index 0<α≤2. In particular, in the case of repeated unit roots or conjugate complex unit roots, M-estimates have a higher asymptotic rate of convergence compared to the least square estimates and the asymptotic results can be written as Itô stochastic integrals.

Asymptotic Multivariate Dominance: A Financial Application

We propose a multivariate stochastic dominance relation aimed at ranking different financial markets/sectors from the point of view of a non-satiable risk averse investor. In particular, we assume that the vector of returns of a given market is in the domain of attraction of a symmetric stable Paretian law in order to take into account the asymptotic behaviour of the financial returns. We determine the stochastic dominance rule for stable symmetric distributions, where the stability parameter plays a crucial role. Consequently, the multivariate rule for ordering markets is based on a comparison between i) location parameters, ii) dispersion parameters, and iii) stability indices. Finally, we apply the method to the equity markets of the four countries with the highest gross domestic product in 2013, namely, the US, China, Japan and Germany. In this empirical comparison we examine the ex ante and ex post dominance between stock markets, either assuming that the returns are jointly (or conditionally, for a robust approach) Gaussian distributed, or in the domain of attraction of a stable sub-Gaussian law.

On the confidence intervals of parametric functions for Distributions Generated by Symmetric Stable Laws

In this paper we consider “Discrete Distributions Generated by Standard Symmetric Stable Densities” (DSSD in short) arising in Bioinformatics (Astola and Danielian, 2007). Using well-known asymptotic properties of the maximum likelihood (ML) estimators we obtain the respective asymptotic confidence intervals for some useful parametric functions of the DSSD.

Banks’ criterion and symmetric stable laws with index of stability between one-half and one

In this paper we compare the sample mean and the sample median of independent and identically distributed data that come from a symmetric stable distribution using Banks’ (1997) criterion, which is an alternative to Pitman’s (1937) criterion for comparing estimators of a parameter of interest. For data from a symmetric stable distribution with index of stability between one half and one and odd sample size, we show that Banks’ criterion choses the sample median as an estimator of the location parameter of the distribution. For data from a normal distribution with even sample size, we show that Banks’ criterion favours the sample mean (thus “complementing” a similar result about odd sample sizes in Karunaratne and Hadjicostas (2009)). In the process, we prove some trigonometric inequalities, which are interesting in their own right.

Record statistics and persistence for a random walk with a drift

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in the presence of a constant bias c. At each time step, the walker makes a random jump of length η drawn from a continuous distribution f(η), which is symmetric around a constant drift c. We focus in particular on the case where f(η) is a symmetric stable law with a Lévy index 0 < μ ⩽ 2. The record statistics crucially depends on the persistence probability, which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter μ. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and μ) of the jump distribution. We compute the asymptotic mean record number 〈Rn〉 after n steps as well as its full distribution P(R, n). We also compute the statistics of the ages of the longest and the shortest lasting record. Our exact computations show the existence of five distinct regions in the (c, 0 < μ ⩽ 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.

Complete Characterization of Convergence to Equilibrium for an Inelastic Kac Model

Pulvirenti and Toscani introduced an equation which extends the Kac caricature of a Maxwellian gas to inelastic particles. We show that the probability distribution, solution of the relative Cauchy problem, converges weakly to a probability distribution if and only if the symmetrized initial distribution belongs to the standard domain of attraction of a symmetric stable law, whose index $\alpha$ is determined by the so-called degree of inelasticity, $p>0$, of the particles: $\alpha=\frac{2}{1+p}$. This result is then used: (1) To state that the class of all stationary solutions coincides with that of all symmetric stable laws with index $\alpha$. (2) To determine the solution of a well-known stochastic functional equation in the absence of extra-conditions usually adopted.

Fractional multiplicative processes

Les mesures sur [0, 1] auto-similaires en loi sont limites de processus multiplicatifs construits a partir de poids aleatoires distribues sur les sous-intervalles b-adiques de [0, 1]. Ces poids sont i.i.d., positifs et d'esperance 1/b. Il est naturel d'etendre la construction a des poids prenant des valeurs negatives. On obtient alors des martingales a valeurs dans les fonctions continues sur [0, 1]. Nous nous interessons, pour H ∈ (0, 1), a la martingale (B n ) n≥1 de ce type construite en prenant des poids a valeurs dans {―b ―H , b ―H }, afin que B n converge presque surement uniformement vers une fonction B auto-similaire en loi dont la regularite Holderienne et les proprietes fractales soient comparables a celles du mouvement brownien fractionnaire d'exposant H. C'est bien le cas lorsque H ∈ (1 /2, 1), et la construction fournit alors un nouvel exemple de loi invariante par moyenne ponderee aleatoire. Cette loi satisfait la meme equation fonctionnelle qu'une loi stable symetrique usuelle d'indice 1/H. Si H ∈ (0, 1/2], B n diverge presque surement, mais il existe une normalisation naturelle par une suite (a n ) n≥1 telle que la marche aleatoire correlee normalisee B n /a n converge en loi vers la restriction a [0, 1] du mouvement brownien standard. Des theoremes limites sont egalement associes au cas H > 1/2.

Unit root tests and dramatic shifts with infinite variance processes

A model which explains data that is subject to sudden structural changes of unspecified nature is presented. The structural shifts are generated by a random walk component whose innovations belong to the normal domain of attraction of a symmetric stable law. To test the model against the stationarity case, several non-parametric, and regression-based statistics are studied. The non-parametric tests are a generalization of the variance ratio test to innovations with heavy-tailed distributions. The tests are consistent and shown to have good finite sample size and power properties and are applied to a set of economic variables.

Convex and star-shaped sets associated with multivariate stable distributions, I: Moments and densities

It is known that each symmetric stable distribution in R d is related to a norm on R d that makes R d embeddable in L p ( [ 0 , 1 ] ) . In the case of a multivariate Cauchy distribution the unit ball in this norm is the polar set to a convex set in R d called a zonoid. This work interprets symmetric stable laws using convex or star-shaped sets and exploits recent advances in convex geometry in order to come up with new probabilistic results for multivariate symmetric stable distributions. In particular, it provides expressions for moments of the Euclidean norm of a stable vector, mixed moments and various integrals of the density function. It is shown how to use geometric inequalities in order to bound important parameters of stable laws. Furthermore, covariation, regression and orthogonality concepts for stable laws acquire geometric interpretations.

Identifiability of Finite Mixtures of Elliptical Distributions

Abstract. We present general results on the identifiability of finite mixtures of elliptical distributions under conditions on the characteristic generators or density generators. Examples include the multivariate t‐distribution, symmetric stable laws, exponential power and Kotz distributions. In each case, the shape parameter is allowed to vary in the mixture, in addition to the location vector and the scatter matrix. Furthermore, we discuss the identifiability of finite mixtures of elliptical densities with generators that correspond to scale mixtures of normal distributions.

Nonparametric regression under dependent errors with infinite variance

We consider local least absolute deviation (LLAD) estimation for trend functions of time series with heavy tails which are characterised via a symmetric stable law distribution. The setting includes both causal stable ARMA model and fractional stable ARIMA model as special cases. The asymptotic limit of the estimator is established under the assumption that the process has either short or long memory autocorrelation. For a short memory process, the estimator admits the same convergence rate as if the process has the finite variance. The optimal rate of convergencen −2/5 is obtainable by using appropriate bandwidths. This is distinctly different from local least squares estimation, of which the convergence is slowed down due to the existence of heavy tails. On the other hand, the rate of convergence of the LLAD estimator for a long memory process is always slower thann −2/5 and the limit is no longer normal.

Estimating the index of a stable law via the pot-method

Abstract Consider an i.i.d. sample X1,…,Xn of random variables which are equal in distribution to |Y|, where Y follows a symmetric stable law with index β, 0

Tests for cointegration with infinite variance errors

This paper develops the asymptotic theory for residual-based tests and quasi-likelihood ratio tests for cointegration under the assumption of infinite variance errors. This article extends the results of Phillips and Ouliaris (1990)and Johansen (1988, 1991)which are derived under the assumption of square-integrable errors. Here the limit laws are expressed in terms of functionals of symmetric stable laws rather than Brownian motion. Critical values of the residual-based tests of Phillips and Ouliaris (1990)and likelihood-ratio-based tests of Johansen (1991)are calculated and tabulated. We also investigate whether these tests are robust to infinite variance errors. We found that regardless of the index of stability α, the residual-based tests are more robust to infinite variance errors than the likelihood-ratio-based tests.

Logarithmic averages of stable random variables are asymptotically normal

We show that most random walks in the domain of attraction of a symmetric stable law have a non-trivial almost sure central limit theorem with the normal law as the limit.

Robustness of the autoregressive spectral estimate for linear processes with infinite variance

Consider a discrete‐time linear process {xt}, a one‐sided moving average of independent identically distributed random variables {εt}, with the common distribution in the domain of attraction of a symmetric stable law of index δ∈ (0, 2) and the moving‐average coefficients b(j) such that εt is invertible in terms of the present and possibly infinite past values of {xt}. By treating {xt} as if it is second‐order stationary, a normalized spectral density function f(μ) is defined in terms of the b(j) and, having observed x1, ..., xT, an autoregression of order k is fitted by the well‐known Yule–Walker and least squares methods and the normalized autoregressive spectral estimators are constructed. On letting k←∞ as T←∞, but sufficiently slowly, these estimators are shown to be uniformly consistent for f(μ), the convergence rate being T−1/φ, φ &gt; δ. The finite sample behaviour is investigated by a simulation study which also examines possible effects of considering ‘non‐invertible’ models.

Mathematical modeling of the distribution of galaxies in the universe

On the base of a random walk model with the transition probability given as a three-dimensional spherically symmetric stable law, a structural function is derived for the distribution of galaxies in the Universe which satisfactorily describes the scaling properties (fractality) of the distribution at small distances as well as the violation of scaling at large distances, in accordance with the data in the CfA catalogue.

On Representation of Densities of Stable Laws by Special Functions

A representation of the density of a one-dimensional strongly stable law via Meijer’s G-functions is obtained. The paper also considers the expressions of integral transforms with Euler kernels of these densities in terms of G-functions which make it possible to obtain representations of distribution functions of strongly stable laws with rational parameters. Some examples are given of representations of densities of multivariate spherically symmetric stable laws in terms of Meijer functions.