Abstract
Let X i i =1,2,3,... be a sequence of independent and identically distributed random variable which belong to the domain of attraction of a stable law of index α. Wrie \(S_0=0,\quad S_n\sum\nolimits^{n}_{i}=1 \,{\mathbf{X}}_i, n\geqq 1,\) and \(M_n={\rm max}_{0\leqq k \leqq n} S_k.\) In the case where the X i are such that \(\sum\nolimits^{\infty}_{1}\, n^{-1}\mathbf{P}{\rm r}\left(S_n > 0\right) < \infty, \) we have limn→∞M n=M which is finite with probability one, while in the case where \(\sum\nolimits^{\infty}_{1}\, n^{-1}\mathbf{P}{\rm r}\left(S_n < 0\right) < \infty,\) a limit theorem for M n has been obtained by Heyde [9]. The techniques used in [9], however, break down in this case \(\sum\nolimits^{\infty}_{1}{{n}^{-1}}\mathbf{P}{\rm r}\left(S_n < 0\right)= \infty, \sum\nolimits^{\infty}_{1}{{n}^{-1}}\mathbf{P}{\rm r}\left(S_n > 0\right)=\infty\) (the case of oscillation of the random walk generated bu the S n) and the only result available with case α=2 (Erdös and Kac [5]) and the case where the X i themselves have a symmetric stable distribution (Darling [4]). In this paper we obtain a general limit therome for M n in the case of oscillation. Specifically, if {B n,n=1,2,3,...} is a monotone sequence of constants such that B -1 n S n converges in distribution to the stable with charactistic function
In connection with the parameter restrictions, we note that the stable law with characteristic function (1) is one-sided if \(\alpha < 1, | \beta | = 1\) (e.g., Lukacs [12], page 106) so that the random walk generated by the S n does not oscillate ([9], Lemma). The case \(\alpha = 1, \beta \neq 0\) introduces a normalization complication and is not amenable to tratment by the methods of this paper.
Received in revised forms 26 August 1968.
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Heyde, C.C. (2010). On the Maximum of Sums of Random Variables and the Supremum Functional for Stable Processes. In: Maller, R., Basawa, I., Hall, P., Seneta, E. (eds) Selected Works of C.C. Heyde. Selected Works in Probability and Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-5823-5_17
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