Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

If $$X_1, X_2,...$$ is a sequence of pairwise independent, identically distributed random variables then $$|X_1|\ I_{|X_1|>b}, |X_2|\ I_{|X_2|>b},... $$ is also pairwise independent, identically distributed?

($I_{|X_i|>b}$ means $I_{|X_i|>b} = 1$ if $|X_i|>b$ and $I_{|X_i|>b} = 0$ otherwise). Thanks.

share|improve this question

1 Answer 1

If you are talking about the pairwise independence, you always only compare pairs hence it is sufficient to consider a single pair. You have $X_1 1_{(X_1>b)} = f_b(x_1)$ where $$ f_b(x) = x\cdot 1_{(x>b)}. $$ Since measurable functions of independent randon variables are independent random variables, you have the pairwise independence. Identical distribution comes from the fact that whenever $X_i\sim \mu_i$ then $f(X_i)\sim \mu_i\circ f^{-1}$ which is an image probability measures. Since $\mu_i = \mu_j$ you have that image measures (distribution of $f_b(X_i)$ and of $f_b(X_j)$) are equal as well.

So, the answer is: yes. The answer will still hold to be the same if you ask whether $f_b(X_i)$ are mutually independent provided the mutual independence of $X_i$.

share|improve this answer
    
Good answer, but I think he might mean independence as a whole not pairwise. –  mez Jan 28 '13 at 12:11
    
@mezhang: thanks, added as a comment –  Ilya Jan 28 '13 at 12:17

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.