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Let $X_0,X_1,X_2,...$ be independent random variables with $$P(X_n =1)=P(X_n =-1)={1\over 2},\forall n$$ Let $$Z_n = \prod_{i=0}^n X_i .$$

Show that $Z_1,Z_2,...$ are independent.


An intuition is to prove it by induction. By definition a sequence of r.v.s $(X_n)_{n\ge1}$ are independent if the sigma algebra generated by them are independent. Here, I try to prove $P(Z_1=1,Z_2=-1)=P(Z_1 =1)P(Z_2 =-1)$ and this is easy to do, but when includes like n R.V.s should I write all the possible combination of events? I am not so sure how to write this proof.

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This will help you , I think math.stackexchange.com/questions/215687/… –  user5656 Jan 11 at 12:30

1 Answer 1

For any finite $S\subset\left\{ 0,1,2,\ldots\right\} $ define $Z_{S}:=\prod_{s\in S}X_{s}$. Note that here $Z_{n}$ and $Z_{\left\{ 0,\ldots,n\right\} }$ denote the same random variable.

By induction it can be shown that $Z_{n}$ has exactly the same distribution as - let's pick one out - $X_{0}$ . It uses the apparant independency of $Z_{n-1}$ and $X_{n}$. Note that $Z_{n}$ can only take values in $\left\{ -1,1\right\} $ so that it is enough to prove that $P\left\{ Z_{n}=1\right\} =\dfrac{1}{2}$.

$P\left\{ Z_{n}=1\right\} =P\left\{ Z_{n-1}=1\wedge X_{n}=1\right\} +P\left\{ Z_{n-1}=-1\wedge X_{n}=-1\right\} =P\left\{ Z_{n-1}=1\right\} P\left\{ X_{n}=1\right\} +P\left\{ Z_{n-1}=-1\right\} P\left\{ X_{n}=-1\right\} =\dfrac{1}{2}\times\dfrac{1}{2}+\dfrac{1}{2}\times\dfrac{1}{2}=\dfrac{1}{2}$.

In fact this means that $Z_{S}$ has the same distribution as $X_{0}$ for any finite $S\subset\left\{ 0,1,2,\ldots\right\} $.

Let $n,k$ be integers with $k>0$ and let $S:=\left\{ n+1,\cdots,n+k\right\} $. Then $Z_{n}$ and $Z_{S}$ are independent and for $\varepsilon,\delta\in\left\{ -1,1\right\} $ we find:

$P\left\{ Z_{n}=\varepsilon\wedge Z_{n+k}=\delta\right\} =P\left\{ Z_{n}=\varepsilon\wedge Z_{S}=\delta\varepsilon^{-1}\right\} =P\left\{ Z_{n}=\varepsilon\right\} P\left\{ Z_{S}=\delta\varepsilon^{-1}\right\} =\dfrac{1}{2}\times\dfrac{1}{2}=P\left\{ Z_{n}=\varepsilon\right\} P\left\{ Z_{n+k}=\delta\right\} $.

This proves that $Z_n$ and $Z_{n+k}$ are independent.

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