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| ==== Another approach ==== | ==== Another approach ==== | ||
| - | Another approach, due to Etamadi ((An Elementary Proof of the Strong Law of Large Numbers, 1981, Etamadi)), shows the result with identically distributed and pairwise independent random variables. He starts with nonnegative random variables. He first shows that for $Y_i=X_i{\mathsf 1}_{\{X_i<i\}}$, the normalized sum $S_{k_n}/k_n$ converges almost surely using that $k_n=\lfloor \alpha^n\rfloor$. The second step shows that $Y_i\neq X_i$ only a finite number of times ans the last step is to let $\alpha$ goes to 1. | + | Another approach, due to Etamadi ((An Elementary Proof of the Strong Law of Large Numbers, 1981, Etamadi)), shows the result with identically distributed and pairwise independent random variables. He starts with nonnegative random variables. He first shows that for $Y_i=X_i{\mathsf 1}_{\{X_i<i\}}$, the normalized sum $S_{k_n}/k_n$ converges almost surely using that $k_n=\lfloor \alpha^n\rfloor$. The second step shows that $Y_i\neq X_i$ only a finite number of times and the last step is to let $\alpha$ goes to 1. |
| [[?do=backlink|Linked pages]] | [[?do=backlink|Linked pages]] | ||
world/lln.1770449980.txt.gz · Last modified: 2026/02/07 08:39 by rdouc
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