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.
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 $X_i$.
