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Homework Help: Conditions on random variable to satisfy limit property

  1. Sep 4, 2011 #1
    1. The problem statement, all variables and given/known data
    The problem is to find sufficient and preferably also necessary conditions on random variable X such that its characteristic function g(x) satisfies the limit property:
    [itex]\lim_{t\to0}\frac{1-g(\lambda t)}{1-g(t)}=\lambda^2[/itex]
    I may assume X is symmetric around 0, so the characteristic function is real and even.

    2. Relevant equations
    [itex]g(t)=\int_{-\infty}^{\infty} e^{itx}f_{X}(x) dx[/itex]

    3. The attempt at a solution

    I'm stuck immediately after trying to apply l'Hopital's rule. Any suggestions would be helpful.
    Thank you.
  2. jcsd
  3. Sep 4, 2011 #2
    What is the value of [itex]g(0)[/itex]?
  4. Sep 4, 2011 #3
    [itex]g(0)=1[/itex] since it is a characteristic function.
  5. Sep 4, 2011 #4
    So, what can you say about your limit? Can you evaluate it?
  6. Sep 4, 2011 #5
    Using l'Hopital's rule and chain rule, the equivalent condition I need will be
    [itex]\lim_{t\to0}\frac{g'(\lambda t)}{g'(t)} = \lambda [/itex].
  7. Sep 4, 2011 #6
    Not really, evaluate these derivatives explicitly:

    \frac{d}{d t} \left(1 - g(\lambda t)\right)

    \frac{d}{d t} \left(1 - g(t)\right)
  8. Sep 4, 2011 #7
    I must be confused. If I'm not mistaken, the first evaluates to [itex]-\lambda g'(\lambda t)[/itex] and the second [itex]-g'(t)[/itex].
  9. Sep 4, 2011 #8
    True, so your limit becomes:

    \lim_{t \rightarrow 0}{\frac{-\lambda g'(\lambda t)}{-g'(t)}} = \lambda \, \lim_{t \rightarrow}{\frac{g'(\lambda t)}{g'(t)}}

    Before you run in evaluating this limit, you need to consider two cases:

    1) [itex]g'(0) \neq 0[/itex]. Then the limit is simply 1 and you get the result you posted in post #5. However, this is not what you have in the condition of the problem.

    2) [itex]g'(0) = 0[/itex] What is the limit in this case?

    BTW, what does [itex]g'(0)[/itex] mean?
  10. Sep 4, 2011 #9
    The condition I posted on #5 is what I desire to have, rather than what will follow from the assumption. Sorry for the confusion.
    By the assumption that X is symmetric around 0 and thus its expected value is 0, it follows g'(0)=0, so it is necessary to use l'Hopital's rule again.
    Thus it seems a sufficient condition is that 0 < var(X) < infinity. Please let me know if I am wrong, but otherwise thank you for your help!
  11. Sep 4, 2011 #10
    Yes, you are correct.
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