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Suppose that $\mu$ and $\lambda$ are two equivalent measures in a probability space $(\Omega, \mathscr F)$. Suppose that random variables $\xi_1,..,\xi_n$ in $(\Omega, \mathscr F)$ are independent according to $\mu$, does this imply that they are independent also according to $\lambda$ ?

BCLC
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AlmostSureUser
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1 Answers1

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No, let $\xi_1,\xi_2$ uniformly distributed on the unit square i.e. $\xi_1,\xi_2$ are random variables on $(\Omega, \mathscr F) = ([0,1] \times [0,1], \mathscr F)$. So they are independent with respect to the Lebesgue measure.

Now, take the following common density instead

$$f_{\xi_1,\xi_2}(x,y)=x+y, \,\,\, 0\le x\le 1, 0\le y\le 1.\tag 1$$

With respect to this common distribution the two random variables are not independent.

$$f_{\xi_1}(x)=x+\frac12,\,\,0\le x \le 1$$ and

$$f_{\xi_2}(y)=y+\frac12,\,\,0\le y \le 1.$$

So, $$f_{\xi_1,\xi_2}(x,y)\not=f_{\xi_1}(x)f_{\xi_2}(y).$$

At the same time the two measures (the Lebesgue measure and the one defined by $(1)$) are equivalent. They are absolute continuous with respect to each other.

BCLC
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zoli
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  • Suppose then that, under $\mathbb{P}$ the random variables are independent and identically distributed with $\mathbb{P}\left[\xi_t=a\right]=p$ and $\mathbb{P}\left[\xi_t=b\right]=1-p$. I am wondering whether (and for which conditions) independence is preserved by any new measure such that $\mathbb{Q}\left[\xi_t=a\right]=p^{\prime}$ and $\mathbb{Q}\left[\xi_t=b\right]=1-p^{\prime}$. – AlmostSureUser Nov 17 '15 at 11:15
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    What exactly are the $\lambda$ and $\mu$ here? – BCLC Nov 17 '15 at 21:10
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    @BCLC: $\lambda$ is the Lebesgue measure and $\mu$ is the measure defined by the density $x+y$ over the unit square. – zoli Nov 17 '15 at 21:13
  • @zoli So $$\mu(E) = \int\int_{E} \ x+y \ dA$$ for $E \in \mathscr{F}$ where $E \subseteq [0,1] \times [0,1] = \Omega$? – BCLC Nov 17 '15 at 21:17
  • @DP1981 $\mu = \mathbb P, \lambda = \mathbb Q$ ? – BCLC Nov 17 '15 at 21:18
  • @zoli Nice. So what do you call the 'natural' $\mathscr F$ here? $\mathscr{B}(\Omega)$ ? Not familiar w/ multivariate prob/measure theory – BCLC Nov 17 '15 at 21:22
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    @DP1981: yes. $\mathbb P$ and $\mathbb Q$ are the two distributions. – zoli Nov 17 '15 at 21:23