On approximation of sums of random variables in the domain of attraction of a stable law with parameter \(\alpha=1\). (English. Russian original) Zbl 0840.60041

Zinchenko, N. M.

Theory Probab. Math. Stat. 47, 29-33 (1993); translation from Teor. Jmovirn. Mat. Stat. 47, 29-34 (1992).

Let \(S_n = \sum^n_{i = 1} \xi_i\), where \(\xi_i\), \(i \geq 1\), are i.i.d. random variables with characteristic function \(f(u)\) in the domain of normal attraction of a stable law with parameters \(\alpha \in (0,2)\) and \(|\beta |\leq 1\). The possibility of a.s. approximation of \(S_n\) by sums of stable random variables \(\eta_i\) has been studied recently (an exhaustive reference in the paper). More precisely, conditions on \(f(u)\) have been found under which it is possible to define, on some probability space, sequences \((\xi_i)\) and \((\eta_i)\) so that \(|S_n - \sum_{i \leq n} \eta_i |= o(\varphi (n))\) a.s. for some special error term \(\varphi (n)\). A similar problem has been posed concerning the approximation of \(S_n\) by a stable process \(Y_{\alpha, \beta} (t)\). Nevertheless attraction by a stable law with parameter \(\alpha = 1\) has escaped attention. It is precisely this case that is studied in the paper. Two methods are known for proving such approximation assertions. The first one is the so-called quantile method, the second one leans general approximation theorems. The author uses the second method. Denote by \(g_1 (u)\) the characteristic function of a stable law with parameter \(\alpha = 1\). Let there exist \(a_1\), \(a_2 > 0\) and \(l > 1\) such that for \(|u |< a_1 : |f(u) - g_1 (u) |\leq a_2 |u |^l\), then \(\varphi (n) = n^{1 - \rho}\) with \(0 < \rho < 1/(A + 1)\) where \(A = \max \{10/(l - 1) + 1;3\}\).

Reviewer: A.I.Volodin (Kazan’)

MSC:

60G50	Sums of independent random variables; random walks
60F15	Strong limit theorems
60E07	Infinitely divisible distributions; stable distributions

Keywords:

stable law with parameters; domain of normal attraction of stable law; almost sure approximation