+ Reply to Thread
Results 1 to 3 of 3

Thread: Convergence question

  1. #1
    Points: 5,567, Level: 48
    Level completed: 9%, Points required for next Level: 183

    Location
    Cincinnati, OHIO
    Posts
    48
    Thanks
    0
    Thanked 0 Times in 0 Posts

    Convergence question




    X1, X2,.......Xn is a sequence of random variables. Show that Xn converges to b in quadratic mean if and only if

    The limit (n approaching infinity) of E(Xn) = b and
    The limit (n approaching infinity) of V(Xn) = 0.

    We know that Xn converges to X in quadratic mean if E(Xn - X)^2 -> 0 as n approaches infinity and Xn converges to c in quadratic mean if E(Xn - c)^2 -> 0 as n approaches infinity.

    What's throwing me is the presence of V in terms of solving the problem. Any assistance would be greatly appreciated. Thanks.

  2. #2
    Points: 1,916, Level: 26
    Level completed: 16%, Points required for next Level: 84
    Buckeye's Avatar
    Location
    Ohio
    Posts
    104
    Thanks
    32
    Thanked 4 Times in 4 Posts

    Re: Convergence question

    I'm assuming V(Xn) is variance. Is there not a way to write variance in terms of Expectation? This is beyond my scope, but that's one thing I'm thinking about.
    "I have discovered a truly remarkable proof of this theorem which this margin is too narrow to contain." Pierre de Fermat

  3. #3
    Super Moderator
    Points: 13,151, Level: 74
    Level completed: 76%, Points required for next Level: 99
    Dragan's Avatar
    Location
    Illinois, US
    Posts
    2,014
    Thanks
    0
    Thanked 223 Times in 192 Posts

    Re: Convergence question


    Quote Originally Posted by greg6363 View Post
    X1, X2,.......Xn is a sequence of random variables. Show that Xn converges to b in quadratic mean if and only if

    The limit (n approaching infinity) of E(Xn) = b and
    The limit (n approaching infinity) of V(Xn) = 0.

    We know that Xn converges to X in quadratic mean if E(Xn - X)^2 -> 0 as n approaches infinity and Xn converges to c in quadratic mean if E(Xn - c)^2 -> 0 as n approaches infinity.

    What's throwing me is the presence of V in terms of solving the problem. Any assistance would be greatly appreciated. Thanks.
    Would it not be better to think of this in the following manner:

    Suppose that \left \{X_{n}  \right \}_{n=1}^{\infty } is a sequence of independent random variables from a common distribution,
    that has a mean \mu and variance \sigma ^{2}<\infty.

    Let \bar{X_{n}} be the sample mean. Thus,

    \lim_{n\rightarrow \infty }E\left (\left | \bar{X_{n}}-\mu  \right | ^{2} \right )

    =\lim_{n\rightarrow \infty }V\left ( \bar{X_{n}} \right )
=\lim_{n\rightarrow \infty }n^{-1}\sigma ^{2}=0.

    As such, this implies that \bar{X_{n}}\rightarrow \mu, i.e., convergence in quadratic mean, as n\rightarrow \infty.

+ Reply to Thread

           




Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts






Advertise on Talk Stats