Summary
This paper gives extensions of Mori's strong law for (r) S n =S n −X (1)}n ...−X (r) n , where S n =X1+X2+...+X n ,X i are iidrv's and (X ni ()) is (X i ) arranged in decreasing order of absolute magnitude. The methods differ from Mori's. Continuity of the distribution of the X i is assumed throughout. Necessary and sufficient conditions for relative stability ((r) S n /B n →±1 a.s. for some B n ), including a generalised condition of Spitzer's and a dominated ergodic theorem, are proved. A one-sided version of the relative stability results is also given. A theorem of Kesten's is generalised to show that if ((r) S n −A n )/B n is bounded almost surely for constants A n ,B n ↑ +∞ then \((^{(r)} s_n - \alpha _n )/B_n \xrightarrow{P}0\) for some α n . A corollary to this is that if ¦ (r) S n ¦/B n is bounded away from 0 and +∞ a.s. then (r) S n is relatively stable. This generalises a result of Chow and Robbins, apart from the continuity assumption.
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Maller, R.A. Relative stability of trimmed sums. Z. Wahrscheinlichkeitstheorie verw Gebiete 66, 61–80 (1984). https://doi.org/10.1007/BF00532796
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DOI: https://doi.org/10.1007/BF00532796