For each natural number $n \geq 2$, define the set $A_n$ to be the set of points $p/n$ with $0 < p < n, \gcd(p,n) = 1$. Now define a sequence of independent random variables $X_1, X_2, \cdots$, where $X_n \in [0, f(n)]$ for some non-negative function $f$ satisfying $0 \leq f \leq 1/2$ and $f > 0$ infinitely often. Now consider the random sets $B_n$ which are unions of the form $\bigcup_{a \in A_n} [a - X_n/n, a+ X_n/n]$. Now, in this setting, does the independence of the sequence of random variables $X_n$ imply any notion of independence or 'almost' independence for the new random sets defined? Here the probability being considered is the Lebesgue measure of the unit interval $[0,1]$, though if another natural measure $\mu \ll m$ would be fine as well.

Edit: Another attempt at rescuing this question.

Edit: I have redefined my original question; which is now listed below. I figured out what I am trying to ask more specifically.

Suppose that I am given a sequence of events $A_1, A_2, \cdots \subset [0,1]$, where $A_n = \displaystyle \bigcup_{j=1}^{k_n} [u_j, v_j]$, a finite union of closed intervals. and we don't know whether they are independent. Suppose we give a sequence of random variables $X_1, X_2, \cdots$, where for each $n$ we have $X_n$ takes on values in $[0, f(n)]$ with respect to some probability distribution, for some non-negative function $f$ with $0 \leq f(n) \leq 1/2$ for all $n$ and $f(n) > 0$ infinitely often. For my purposes, it suffices to assume that $f(n)$ is small; say $f(n) = o(1)$. Now suppose that $X_1, X_2, \cdots$ is independent. Now consider the (random) sets $B_n = \displaystyle \bigcup_{j=1}^{k_n} [u_j - X_n, v_j + X_n]$. What can we say (if anything) about the independence of the sequence of events $B_1, B_2, \cdots$?

  • $\begingroup$ Independent with respect to which measure on $[0,1]$? $\endgroup$ – Did May 29 '11 at 20:55

Let $f=0$. Then we do not know whether the $A_n=B_n$ are independent.


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