{"id":5261,"date":"2012-11-05T20:38:08","date_gmt":"2012-11-05T19:38:08","guid":{"rendered":"http:\/\/djalil.chafai.net\/blog\/?p=5261"},"modified":"2014-06-17T21:04:14","modified_gmt":"2014-06-17T19:04:14","slug":"about-the-first-borel-cantelli-lemma","status":"publish","type":"post","link":"https:\/\/djalil.chafai.net\/blog\/2012\/11\/05\/about-the-first-borel-cantelli-lemma\/","title":{"rendered":"About the first Borel-Cantelli lemma"},"content":{"rendered":"

\"Dice\"<\/a><\/p>\n

The first Borel<\/a>-Cantelli<\/a> lemma is one of rare elementary providers of almost sure convergence in probability theory. It states that if \\( {(A_n)} \\) is a sequence of events in a probability space \\( {(\\Omega,\\mathcal{F},\\mathbb{P})} \\) such that \\( {\\sum_n\\mathbb{P}(A_n)<\\infty} \\) then \\( {\\mathbb{P}(\\varlimsup_nA_n)=0} \\). The classical proof consists in noticing that \\( {B_n=\\cup_{m\\geq n}A_m} \\) is non increasing in \\( {n} \\), and thus, since \\( {\\sum_n\\mathbb{P}(A_n)<\\infty} \\),<\/p>\n

\\[ \\mathbb{P}(\\varlimsup A_n) =\\mathbb{P}(\\cap_nB_n) =\\lim_n\\mathbb{P}(B_n) \\leq \\lim_{n}\\sum_{m\\geq n}\\mathbb{P}(A_m)=0. \\]<\/p>\n

There is another proof of this lemma that I prefer, and which goes as follows: by the Fubini-Tonelli theorem or by the monotone convergence theorem,<\/p>\n

\\[ \\mathbb{E}(\\sum_n\\mathbf{1}_{A_n}) =\\sum_n\\mathbb{E}(\\mathbf{1}_{A_n}) =\\sum_n\\mathbb{P}(A_n) <\\infty \\]<\/p>\n

therefore \\( {\\mathbb{P}(\\sum_n\\mathbf{1}_{A_n}=\\infty)=0} \\). Now \\( {\\{\\sum_n\\mathbf{1}_{A_n}=\\infty\\}=\\varlimsup A_n} \\). I like very much this proof because it concentrates useful ingredients for students:<\/p>\n