1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Convergence in probability of the sum of two random variables

  1. Oct 25, 2011 #1
    1. The problem statement, all variables and given/known data

    [itex] X, Y, (X_n)_{n>0} \text{ and } (Y_n)_{n>0} [/itex] are random variables.

    Show that if

    [itex] X_n \xrightarrow{\text{P}} X [/itex] and [tex] Y_n \xrightarrow{\text{P}} Y [/tex] then [itex] X_n + Y_n \xrightarrow{\text{P}} X + Y [/itex]

    2. Relevant equations

    If [itex] X_n \xrightarrow{\text{P}} X [/itex] then [itex] \text{Pr}(|X_n-X|>\epsilon)=0 \text{ } \forall \epsilon > 0 \text{ as } n \to \infty[/itex]

    3. The attempt at a solution

    First, let the sets [itex] A_n(\epsilon) = \{|X_n - X|<\epsilon\} [/itex] and [itex] B_n(\epsilon) = \{|Y_n - Y|<\epsilon\} [/itex]

    The sum of the two moduli will always be less than [itex]2\epsilon[/itex] if both of the moduli are less than [itex]\epsilon[/itex] but the converse is not generally true.

    [itex] C_n(\epsilon)=\{|X_n-X|+|Y_n-Y|<2\epsilon\}\supset{A_n(\epsilon)\cap B_n(\epsilon)}[/itex]


    Using the triangle inequality:
    [itex] |X_n + Y_n - X - Y | \le |X_n-X|+|Y_n-Y| [/itex]


    [itex]D_n(\epsilon) =\{|X_n-X+Y_n-Y|<2\epsilon\} \supset C_n(\epsilon) [/itex]


    I think this has gone wrong in several places but from here I hope to say that

    [itex] \text{Pr}(D_n) \ge \text{Pr}(C_n) \ge \text{Pr}(A_n\cap B_n ) \ge \text{Pr}(A_n) \to 1 \text{ as } n\to\infty [/itex]

    [tex]\text{Pr}(D_n^c) \to 0 \text{ as } n\to \infty[/tex]
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?



Similar Discussions: Convergence in probability of the sum of two random variables
Loading...