Given a set of independent and identically distributed random variables $(\xi_i)_{i\in I}$ i know that $(X_i)_{i\in I}$ and $(Y_i)_{i\in I}$ ar conditionally independent and identically distributed. And $(X_i)_{i\in I}$ is a family of identically distributed random variables.

So I want to show, that $(X_i, Y_i)_{i\in I}$ are identically distributed. So let $\mathcal{F}:=\sigma( \xi_i: i\in I)$ then we have for $\Delta \in I$ and $f,h$ bounded and continous:

$$ E\left[ f(X_\mathbf{i}) \cdot h(Y_\mathbf{i}) \right]\\ =E\left[ E\left[ f(X_\mathbf{i}) \cdot h(Y_\mathbf{i}) | \mathcal{F} \right] \right]\\ =E\left[ E\left[ f(X_\mathbf{i}) | \mathcal{F} \right] E\left[ h(Y_\mathbf{i}) | \mathcal{F}_{\mathbf{i} } \right] \right]\\ =E\left[ E\left[ f(X_\mathbf{i}) | \mathcal{F} \right] E\left[ h(X_\mathbf{i}) | \mathcal{F} \right] \right]\\ \overset{ X_\mathbf{i}\overset{(d)}{=}X_\Delta }{=}E\left[ E\left[ f(X_\Delta) | \mathcal{F} \right] E\left[ h(X_\Delta) | \mathcal{F} \right] \right]\\ =E\left[ E\left[ f(X_\Delta) | \mathcal{F} \right] E\left[ h(Y_\Delta) | \mathcal{F} \right] \right]\\ =E\left[ E\left[ f(X_\mathbf{i}) \cdot h(Y_\Delta ) | \mathcal{F} \right] \right]\\ = E\left[ f(X_\Delta) \cdot h(Y_\Delta) \right]$$

So that is true for every $f,h$ bounded and continous. That means $(X_i, Y_i)\overset{(d)}{=} (X_j, Y_j)$ for $i,j\in i$.

Is t really so easy or am I missing something important here?



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