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Problem: Let $U_{n}$ be the discrete-time iid Gaussian random process with mean 0 and variance 1. Two filtered version of $U_{n}$ is defined by $X_{n}=U_{n}+U_{n-1 }$ and $Y_{n} = U_{n} - U_{n-1}$. Find the joint characteristic function when $n\neq m$ and find joint pdf of $X_{n}$ and $Y_{m}$ ($n$ and $m$ may or may not be equal)

My approach:

First of all $X_{n}$ and $Y_{n}$ are gaussian with mean 0 and variance 2.

And i thinks $X_{n}$ and $Y_{n}$ are independent.

Because they consist of iid gaussian random process.

So to find joint characteristic function just find chf of one gaussian random variable and multiply two gaussian chf.

My answer is $exp(-t^2)$

Finding joint pdf of $X_{n}$ and $Y_{m}$ is just multiplying pdf of $X_{n}$ and pdf of $Y_{m}$

If there is any mistake please let me know thank you!

Kim
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  • By "the discrete-time iid Gaussian random process" you mean that $U_n\stackrel{\mathsf{iid}}\sim\mathcal N(0,1)$, correct? – Math1000 Mar 03 '16 at 13:14

1 Answers1

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The joint characteristic function of $(X_n,Y_m)$ is defined as $$\varphi_{X_n,Y_m}(s,t) = \mathbb E\left[e^{i(sX_n + tY_m)} \right]. $$ Since $$ sX_n+tY_m = \begin{cases} sU_n+(s+t)U_{n-1}-tU_{n-2},& m=n-1\\ (s+t)U_n+(s-t)U_{n-1},& m=n\\ tU_{n+1}+(s-t)U_n+sU_{n-1},& m=n+1\\ s(U_n + U_{n-1})+ t(U_m - U_{m-1}),& |m-n|>1, \end{cases} $$ it follows that $$\varphi_{X_n,Y_m}(s,t) =\begin{cases} e^{-(s^2+st+t^2)},& m=n-1\\ e^{-2 (s^2+t^2)} ,& m=n\\ e^{-(s^2-st+t^2)} ,&m=n+1\\ e^{-2(s^2+t^2)} ,& |m-n|>1, \end{cases} $$ and hence $$ f_{X_n,Y_m}(x,y) = \begin{cases} \frac1{2\sqrt 3\pi}e^{-\frac13(x^2-xy+y^2)} ,& m=n-1\\ \frac1{8\pi}e^{-\frac18(x^2+y^2)} ,& m=n\\ \frac1{2\sqrt 3\pi}e^{-\frac13(x^2+xy+y^2)} ,& m=n+1\\ \frac1{8\pi}e^{-\frac18(x^2+y^2)} ,& |m-n|>1. \end{cases} $$

Math1000
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