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Let $(Ω,\mathcal F,P)$ be a probability space. Assume that the two sequences $(X_n)_{n\in\Bbb N}$ and $(Y_n)_{n\in\Bbb N}$ are independent. For all $n\in\Bbb{N}$, define $Z_n=X_nY_n$ and $$\mathcal{G}_n=\sigma(X_1,\ldots,X_n)\qquad\mathcal{H}_n=\sigma(Y_1,\ldots,Y_n)\qquad\mathcal{F}_n=\sigma(Z_1,\ldots,Z_n)$$

If $(X_n,\mathcal{G}_n)_{n\in\Bbb N}$ and $(Y_n,\mathcal{H}_n)_{n\in\Bbb N}$ are martingales, is $(Z_n,\mathcal{F}_n)_{n\in\Bbb N}$ still a martingale?

JACK
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    Any thoughts on the problem? What have you tried? Do you have an example where the statement holds or fails to hold? – saz Nov 13 '17 at 14:11
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    To write indices in math mode, please use _{...}. – Did Nov 13 '17 at 14:18
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    That's too difficult for me, the only intuition that i had, is to say that it's still a martingale by the fact that the two squences are independent and so the result remains in L^1 – JACK Nov 13 '17 at 14:21
  • Interesting, why did you see fit to delete the mark signaling where the question is in your post? – Did Nov 13 '17 at 14:34
  • sorry, i didn't know that! I thought was a mistake. – JACK Nov 13 '17 at 14:35
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    What else does a Martingale need to satisfy? It's not enought to be in $L^{1}$. – madprob Nov 13 '17 at 14:43
  • See here: https://math.stackexchange.com/questions/22367/sum-and-product-of-martingale-processes – Math1000 Nov 29 '17 at 02:57

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