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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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
Keywords
- Stochastic Process
- Probability Theory
- Mathematical Biology
- Lower Estimate
- Invariance Principle