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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.

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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 $X_i$.

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  • $\begingroup$ Good answer, but I think he might mean independence as a whole not pairwise. $\endgroup$
    – mez
    Jan 28, 2013 at 12:11
  • $\begingroup$ @mezhang: thanks, added as a comment $\endgroup$
    – SBF
    Jan 28, 2013 at 12:17

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