In the above, we are told that both |Xt$\preceq$| $\preceq$ |Xt| and |Xt$\succ$| $\preceq$ |Xt|.
Would appreciate clarification on 1) why this is the case, and 2) what the superscripts for |Xt$\preceq$| and |Xt$\succ$| are actually representing.
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The definition of $X_t^{\leq}$ and $X_t^{>}$ is given in the second and fourth line, respectively. The claimed inequalities follow directly from this definition. – saz Nov 10 '15 at 19:21
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This follows the fact that by definition $$\left|X_t^{\leq} \right|=\left|X_t\mathbf 1_{\{ t\leqslant 100\} } \right|=\left|X_t \right|\underbrace{\mathbf 1_{\{ t\leqslant 100\} }}_{\leqslant 1}\leqslant \left|X_t \right|.$$ The other inequality can be proved in a completely similar way.
Davide Giraudo
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