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Math Help - Strong Law of Large Numbers Application

  1. #1
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    Strong Law of Large Numbers Application

    This was a test question from last semester that I stared at blankly for a long time and couldn't figure out where to begin. Obviously since I need to show convergence almost surely, its the Strong Law of Large Numbers, but beyond that I got really stuck. I'd love some input on at least how to get started. Thanks!

    Problem:
    Define the sequence X_{n} inductively by setting X_{0} = 1, and selecting X_{n+1} randomly and uniformly from the interval [0, X_{n}]. Prove that n^{-1} \log X_{n} converges almost surely to a constant, and evaluate the limit.

    Hint given (I'm having trouble with the /sum so I wrote out the sum expanded):
    let \log X_{n}= \log X_{n} - \log X_{n-1} + \log X_{n-1} + ... + \log X_{1} + \log X_{0}
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  2. #2
    MHF Contributor matheagle's Avatar
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    Re: Strong Law of Large Numbers Application

    I don't see a SLLN here.
    Why don't you obtain the distribution of X_n and then take the log of that rv?
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  3. #3
    MHF Contributor chisigma's Avatar
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    Re: Strong Law of Large Numbers Application

    Quote Originally Posted by puggles View Post
    This was a test question from last semester that I stared at blankly for a long time and couldn't figure out where to begin. Obviously since I need to show convergence almost surely, its the Strong Law of Large Numbers, but beyond that I got really stuck. I'd love some input on at least how to get started. Thanks!

    Problem:
    Define the sequence X_{n} inductively by setting X_{0} = 1, and selecting X_{n+1} randomly and uniformly from the interval [0, X_{n}]. Prove that n^{-1} \log X_{n} converges almost surely to a constant, and evaluate the limit.

    Hint given (I'm having trouble with the /sum so I wrote out the sum expanded):
    let \log X_{n}= \log X_{n} - \log X_{n-1} + \log X_{n-1} + ... + \log X_{1} + \log X_{0}
    With \ln X_{n} You mean E \{\ln X_{n}\}... don't You?...

    Kind regards

    \chi \sigma
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  4. #4
    MHF Contributor chisigma's Avatar
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    Re: Strong Law of Large Numbers Application

    Quote Originally Posted by chisigma View Post
    With \ln X_{n} You mean E \{\ln X_{n}\}... don't You?...
    If the answer is 'yes', then the quantity \mu_{n}= E \{X_{n}\} is the solution of the difference equation...

    \mu_{n+1}= \frac{\mu_{n}}{2}\ ,\ \mu_{0}=1 (1)

    ... so that is...

    \mu_{n}= \frac{1}{2^{n}} (2)

    Kind regards

    \chi \sigma
    Last edited by chisigma; October 19th 2011 at 07:00 AM.
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  5. #5
    MHF Contributor chisigma's Avatar
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    Re: Strong Law of Large Numbers Application

    Quote Originally Posted by chisigma View Post
    If the answer is 'yes', then the quantity \mu_{n}= E \{X_{n}\} is the solution of the difference equation...

    \mu_{n+1}= \frac{\mu_{n}}{2}\ ,\ \mu_{0}=1 (1)

    ... so that is...

    \mu_{n}= \frac{1}{2^{n}} (2)
    ... and setting \lambda_{n}= E \{\ln X_{n}\} is...

    \lambda_{n+1}= \frac{1}{X_{n}}\ \int_{0}^{X_{n}} \ln x\ dx = \lambda_{n}-1\ ,\ \lambda_{0}=0 (1)

    ... so that is...

    \lambda_{n}= -n (2)

    Kind regards

    \chi \sigma
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  6. #6
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    Re: Strong Law of Large Numbers Application

    Thanks everyone for your help. Sorry I took so long to respond... the assignment is already turned in, but your comments have helped me understand this much better! Thank you
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