A rate of convergence in the Poissonian representation of stable distributions
Let \( X, X_1,\ldots \) be i.i.d. real symmetric random variables (r.v.'s) with a distribution function (d.f.) F such that \( E(| X| ^\alpha) < \infty \) for some \( \alpha \in (0,2) \). Let \(\Gamma_j = \gamma_1 + \ldots + \gamma_j\), \(j\geq 1 \), where \( \gamma_1,\gamma_2,\ldots \) are i.i.d. r.v.'s such that \( {\mathbf P}(\gamma_1 > x) = e^{-x}, x \geq 0 \), and \( \{X_j\} \) and \( \{\Gamma_j \} \) are independent. Let, moreover, \( {\mathbf E}(| X| ^\alpha) = 2c_{\alpha}\) where \( c_{\alpha} = \lim\limits_{x \to \infty} x^{\alpha}(1 - {\mathcal G}_{\alpha}(x)) > 0, {\mathcal G}_{\alpha}(x) \) being the d.f. of a symmetric stable law with index \( \alpha \in (0,2)\). The authors study asymptotical behaviour as \( n \to \infty \) of \[ \Delta_n (\alpha,F) = \sup\limits_{x} \left| {\mathbf P}\left( \sum_{j = 1}^{n} \Gamma_j^{1/\alpha} X_j < x\right) - {\mathcal G}_{\alpha}(x)\right| \] which is the rate of Poissonian representation \( \sum_{j = 1}^{\infty} \Gamma_j^{1/\alpha} X_j \) for a stable distribution with index \(\alpha\). For instance, if \( {\mathbf E}(| X| ^r) \leq M \) for some \( r \in (\alpha,2] \) and the Cramer condition holds for the characteristic function \(\varphi \) of \(F\), namely, \( | \varphi(t)| < \gamma \) for all \( | t| > \beta \), then \(\Delta_n(\alpha,F) \leq c_1 n^{-(r-\alpha)/\alpha} (\log(n + 1))^{\kappa}, n \geq 1\), where positive \( c_1 \) and \( \kappa \) depend on \(M,\gamma, \beta, \alpha \). Slower rates are obtained without the Cramer condition.
Poissonian representation for a stable distribution
scientific article; zbMATH DE number 1135559
scientific article