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For sochastic processes $X_n$ and $Y_n$ it holds that if they are indenpendant for all $n$ then $$E[X_nY_n] = E[X_n]E[Y_n]$$ But can you go the other way. I mean if $E[X_nY_n] = E[X_n]E[Y_n]$, does that mean that $X_n \text{ and }Y_n$ HAVE to be indenpendant. Basically what I am asking is which of these following statements is true:

Statement 1: $X_n$ and $Y_n$ are indenpendant $\Rightarrow $ $E[X_nY_n] = E[X_n]E[Y_n]$

Statement 2: $X_n$ and $Y_n$ are indenpendant $\Leftrightarrow $ $E[X_nY_n] = E[X_n]E[Y_n]$

KinkyLaura
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    Many counterexamples are discussed on site (and elsewhere). See also https://stats.stackexchange.com/questions/12842/covariance-and-independence , https://stats.stackexchange.com/questions/301180/difference-between-dependent-and-correlated-variables , https://stats.stackexchange.com/questions/179511/why-zero-correlation-does-not-necessarily-imply-independence , ... and also https://math.stackexchange.com/questions/444408/zero-correlation-does-not-imply-independence – Glen_b Jan 28 '18 at 05:56

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Statement 1 is true if the expectations exist; statement 2 is not necessarily true.

Assuming $X_n$ and $Y_n$ have finite mean and variance then $E[X_nY_n] = E[X_n]E[Y_n]$ is equivalent to saying they have zero covariance and correlation. But this does not mean they are independent.

As a counterexample, consider $X_n$ having a standard normal distribution $N(0,1)$, and independently $Z_n$ taking the values $-1$ and $1$ with equal probability, and $Y_n = X_n Z_n$.

Then $E[X_n]=0$, $E[Y_n] = E[X_n]E[Z_n]=0$, and $E[X_nY_n] = E[X_n^2]E[Z_n]=0$ so $E[X_nY_n] = E[X_n]E[Y_n]$

but (apart from sign) $X_n$ determines $Y_n$ and they are not independent

Henry
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