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I have a Bernoulli process $\Phi(t)$ with a symmetric distribution $p=1/2$. The random variable can take values $a,b$. My question is what is the covariance of this process \langle\Phi(t)\Phi(t')\rangle?

Thanks.

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    @Jon Please see updates to both this and the previous question of yours.2012-01-18

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Per your comment, let $\Phi(t) = \operatorname{sign}(W(t))$. Using results from your previous question it would be $\begin{eqnarray} \mathbb{E}(\Phi(t) \Phi(t^\prime)) &=& \mathbb{P}(W(t) \geqslant 0, W(t^\prime)\geqslant 0) + \mathbb{P}(W(t) < 0, W(t^\prime) < 0) \\ &\phantom{=} &- \mathbb{P}(W(t) < 0, W(t^\prime)\geqslant 0) - \mathbb{P}(W(t) \geqslant 0, W(t^\prime) < 0) \\ &=& \frac{2}{\pi} \arcsin\left( \sqrt{\frac{\min(t,t^\prime)}{\max(t,t^\prime)}} \right) \end{eqnarray} $

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    @Sasha: I posted my paper http://arxiv.org/abs/1201.5091. Thank you again.2012-01-25