User Tools
world:lln
Differences
This shows you the differences between two versions of the page.
| Both sides previous revision Previous revision Next revision | Previous revision | ||
|
world:lln [2026/02/07 08:39] rdouc |
world:lln [2026/02/23 19:10] (current) rdouc |
||
|---|---|---|---|
| Line 12: | Line 12: | ||
| We follow the approach of **Bernard Delyon** in his unpublished lecture notes on dynamical systems. We just made minor adaptations to fit with the particular case of iid random variables. The beginning of the proof is close to **Neveu**'s approach but then it differs substantially. The proof is based on the following elementary lemma: | We follow the approach of **Bernard Delyon** in his unpublished lecture notes on dynamical systems. We just made minor adaptations to fit with the particular case of iid random variables. The beginning of the proof is close to **Neveu**'s approach but then it differs substantially. The proof is based on the following elementary lemma: | ||
| - | <WRAP center round box 70%> | + | <WRAP center round todo 70%> |
| __Lemma__: Let $(Y_i)$ be iid random variables such that $\PE[|Y_1|]<\infty$ and $\PE[Y_1]>0$, then a.s., | __Lemma__: Let $(Y_i)$ be iid random variables such that $\PE[|Y_1|]<\infty$ and $\PE[Y_1]>0$, then a.s., | ||
| Line 31: | Line 31: | ||
| ==== 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.1770449946.txt.gz · Last modified: 2026/02/07 08:39 by rdouc
Page Tools
