Assume we have a sequence $\xi=(\xi_1,\xi_2,\dots)$ of random variables such that $$\eta=\left(\frac{\sum_{i=1}^n \delta_{\xi_i}}{n}\right)_{n\geq 1}$$ is a reverse-martingale with respect to its own tail $\mathcal{T}_n=\sigma(\eta_n,\eta_{n+1},\dots)$. Does it necessarily hold that $\xi$ is stationary? A counterexample would also be of great help.
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$\begingroup$ What is $\delta_{\xi_i}$? $\endgroup$– MichaelCommented Mar 14, 2017 at 23:36
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$\begingroup$ It is the Dirac measure: $\delta_x(A)=1_{A}(x)$. $\endgroup$– mbeCommented Mar 14, 2017 at 23:38
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$\begingroup$ In this case, what is $A$? $\endgroup$– MichaelCommented Mar 14, 2017 at 23:42
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$\begingroup$ Any measurable set on the space of definition of $x$. For simplicity assume that this space is the real numbers with the borel field. $\endgroup$– mbeCommented Mar 14, 2017 at 23:46
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