Assume $\{B_t:t\ge0\}$ be a brownian motion process. Is $B_s-\frac{s}{t}B_t$ and $B_t$ independent given that ($s\le t$)
Are these 2 random variable independent???
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$\begingroup$
probability
brownian-motion
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6Why, Oh why does this question have 4 upvotes? Have you made attempts to solve this yourself? Where do you get stuck? What level is this? Masters? The more context you provide the better answer you will get from others. – 2012-12-03
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3@SimonHayward On MSE, **ANYTHING** can garner upvotes (and I agree with everything you wrote in your comment). – 2012-12-03
1 Answers
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Hint: The random vector $(B_s-(s/t)B_t,B_t)$ is centered normal. There is a simple way to check that some entries of a normal vector are independent, you could try it.
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0@saz. I think the argument should go like this. The finite dimensional distributions of Brownian motion are all joint normal. So $(B_s,B_t)$ is bivariate normal. A linear transformation of multivarite normal is normal, and $(B_s−B_t s/t ,B_t)$ is one such transformation. – 2012-12-03
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1@learner Yes, thanks... (Sorry, I solved it some seconds ago on my own, that's why I deleted my comment. The question was how one can prove that $(B_s-s/t \cdot B_t,B_t)$ is normal.) – 2012-12-03
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1@Mathematics Well, one of the first things one should learn about **gaussian families** is that any collection of linear combination of random variables from a gaussian family is a gaussian family. Here the vector $(B_s,B_t)$ is gaussian hence the vector $(B_s-(s/t)B_t,B_t)$ is. (Say, would you be the one who downvoted my answer, by any chance?) – 2012-12-04
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0@did But definitely it is not obvious to show the normal vector independent. I down vote it because this answer is not a good one. I think it is a comment instead of an answer, and it didn't mention the most important thing, how to show it is independent. – 2012-12-04
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0@Mathematics Downvoting is for everybody to decide, naturally. But downvoting "silently" (as you did) is objectionable, and, more importantly, the reasons you present now for this particular downvote are... peculiar, to say the least: (1.) my answer begins by **Hint** and it provides exactly this, a hint; (2.) the second most basic thing one learns about gaussian families (for the first one, see my first comment) is that it suffices to compute scalar products to check independence. This fact being a mystery to you is more revealing about your background than about the pertinence of my answer. – 2012-12-04
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0@Mathematics Why did you erase your previous comment? – 2012-12-04
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0@did I found i misunderstood your hints so the comment erased after i realised what you mean. Well about downvote without announcement,i think it is nothing special. I don't quite see many post people would announce that they have downvoted. I don't mean none of the post or people would do it, but definitely not majority of the member. Also, i think if you want give hint, you may comment it especially your short non informative hint, better in comment rahter than answer. Also, i think what you mean by most basic thing depends on what courses you take, it is just too subjective. – 2012-12-04
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0@Mathematics Indeed you misunderstand a lot, so much that you might as well keep your very personal conceptions to yourself. To criticize a hint, announced as such, because it is a hint is simply beyond my understanding. All in all, I see nothing positive in your contributions to this page (a nonsensic answer, now erased, a false comment, now erased, and the unwarranted slander of someone else's answer). You do not deserve the user's name you chose. – 2012-12-04
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0@did well, erase a false comment is reasonable, if you don't think so, then that's your problem. Also, here is a forum to everyone, not just you, people can just downvote without announcement if they disagree, need not to reveal who they are. I announce it just for giving advices to this non informative, non contributing answer, at least for me, is an upvote really that important to you? If you do, i may give you back, but this is still a downvote answer for me. – 2012-12-05
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0@did If you really think people erasing answer or comment is a big deal, then you can just complain, but this is your own business, and don't tell me such thing. You just like a gang making up your own rules from your own experience and your own style here – 2012-12-05
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0@Mathematics Hello?? Please try to come back to Earth some day. – 2012-12-05
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1@did: I know very little about brownian-motion, but perhaps it might help to indicate how one might show independence. That might satisfy Mathematics and cool the flames a bit. – 2012-12-05
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0@robjohn Not sure I understand what you mean. Note that to *satisfy* misnamed user Mathematics is the least of my concerns. Note also that I already indicated in a comment *how one might show independence* and that I do not intend to modify my answer: believe it or not, some people here *think* about their answers before posting them. (To tell you the truth, the more I ponder your comment, the less I like it.) – 2012-12-05