Timeline for Dependent Bernoulli sequence for which the strong law fails to hold
Current License: CC BY-SA 3.0
15 events
| when toggle format | what | by | license | comment | |
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| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Mar 8, 2016 at 2:24 | vote | accept | Chee | ||
| Mar 5, 2016 at 16:01 | answer | added | Serguei Popov | timeline score: 4 | |
| Mar 5, 2016 at 14:39 | history | edited | Chee | CC BY-SA 3.0 |
added a related proble on finite-range dependence
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| Mar 5, 2016 at 14:27 | comment | added | Chee | @Anthony: I have clarified $m^2$ in condition 4; Anthony and Fedor: I have summarized your examples into an abstract formulation. Maybe the techniques you guys have presented is a general way to construct dependent Bernoulli's for which SLLN fails to hold. Thank you! | |
| Mar 5, 2016 at 14:16 | history | edited | Chee | CC BY-SA 3.0 |
added folow up question and thinking
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| Mar 5, 2016 at 14:02 | history | edited | Chee | CC BY-SA 3.0 |
modified the condition 4
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| Mar 5, 2016 at 13:02 | comment | added | Fedor Petrov | Example may be modified to satisfy condition 4. For example, choose independent $Y_i$'s and set $X_n=[(Y_1+Y_2+Y_n)/3]$. | |
| Mar 5, 2016 at 7:04 | comment | added | Anthony Quas | What is $m^2$ in your 4th condition? I'm wondering if the example can be weakened to have "fairly large" blocks of totally dependent random variables that still satisfies your condition. | |
| Mar 5, 2016 at 1:24 | history | edited | Chee | CC BY-SA 3.0 |
added a constraint to make examples nontrivial
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| Mar 5, 2016 at 1:21 | comment | added | Chee | @AnthonyQuas: Thanks! Your example has a special feature, i.e., the sequence of random variables are totally linearly dependent. (It can be classified as a "trivial" example). I have edited the question. | |
| Mar 5, 2016 at 1:05 | comment | added | Anthony Quas | Here's an example with $p_n=0.5$ for each $n$. It is possible, but annoying to modify it to make all the $p_n$'s distinct. Let $X_1$ be a random variable taking the values $\pm 1$ with equal probability. Then set $X_n=X_1$ for each $n$. | |
| Mar 5, 2016 at 0:53 | history | edited | Chee | CC BY-SA 3.0 |
added "fails to hold"
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| Mar 5, 2016 at 0:47 | history | edited | user9072 | CC BY-SA 3.0 |
deleted 12 characters in body; edited tags; edited title
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| Mar 5, 2016 at 0:42 | history | asked | Chee | CC BY-SA 3.0 |