Summary
Let (X n ) be i.i.d. random variables belonging to the domain of normal attraction of a symmetric stable law with parameter 0<p<2. We study the a.s. and weak approximation of the partial sum process\(S(t) - \sum\limits_{n \leqq t} {X_n (t \geqq } 0)\) by a symmetric stable processG p(t). Stout proved an upper bound for the optimal remainder term in this approximation; we prove here a lower bound, leaving only a small gap between the upper and lower estimates. We also give a new method to obtain upper bounds. Finally, we prove analogues of these results in the case when a.s. approximation is replaced by approximation in probability.
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deAcosta, A., Giné, E.: Convergence of moments and related functionals in the general central limit theorem in Banach spaces. Z. Wahrscheinlichkeitstheor. Verw. Geb.48, 213–231 (1979)
Berkes, I., Dabrowski, A., Dehling, H., Philipp, W.: A strong approximation theorem for sums of random vectors in the domain of attraction to a stable law. Acta. Math. Hungar.48, 161–172 (1986)
Cramér, H.: On the approximation to a stable probability distribution. In: Szegö, G. (ed.) Studies in mathematical analysis and related topics (Essays in honor of G. Pólya), pp. 70–76. Stanford, Calif.: Stanford University Press 1962
Feller, W.: An introduction to probability theory and its applications, vol. II. New York: Wiley 1966
Fisher, E.: An almost sure invariance principle for random variables in the domain of attraction of a stable law. Z. Wahrscheinlichkeitstheor. Verw. Geb.67, 461–471 (1984)
Heyde, C.C.: On large deviation problems for sums of random variables which are not attracted by the normal law. Ann. Math. Statist.38, 1575–1578 (1967)
Lévy, P.: Théorie de l'addition des variables aléatoires. Paris: Gauthiers-Villars 1937
Loève, M.: Probability theory. 2nd ed. New York: Van Nostrand 1963
Major, P.: An improvement of Strassen's invariance principle. Ann. Probab.7, 55–61 (1979)
Mijnheer, J.: Strong approximations of partial sums of i.i.d. random variables in the domain of attraction of a symmetric stable distribution. Trans. of the 9th Prague conference on information theory, stat. decision functions, random processes. Prague: Academia 1983
Seneta, E.: Regularly varying functions. (Lect. Notes. Math., vol. 508) Berlin Heidelberg New York: Springer 1976
Simons, G., Stout, W.F.: A weak invariance principle with applications to domains of attraction. Ann. Probab.6, 294–315 (1978)
Stout, W.F.: Almost sure invariance principles whenEX 21 =∞. Z. Wahrscheinlichkeitstheor. Verw. Geb.49, 23–32 (1979)
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Research supported by Hungarian National Foundation for Scientific Research, grant no. 18081 f(x)~g(x),f(x)\(<< \) g(x),f(x)≈g(x)(x→∞) mean, as usual,\(\mathop {\lim }\limits_{x \to \infty } f(x)/g(x) = 1\),\(\mathop {\overline {\lim } }\limits_{x \to \infty } |f(x)/g(x)|\infty \),\(0< \mathop {\lim }\limits_{x \to \infty } f(x)/g(x)\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \mathop {\overline {\lim } }\limits_{x \to \infty } f(x)/g(x)< + \infty \), respectively
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Berkes, I., Dehling, H. Almost sure and weak invariance principles for random variables attracted by a stable law. Probab. Th. Rel. Fields 83, 331–353 (1989). https://doi.org/10.1007/BF00964369
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DOI: https://doi.org/10.1007/BF00964369