Confused regarding this notation for scalar Xt?

Question

In the above, we are told that both |X_t^$\preceq$| $\preceq$ |X_t| and |X_t^$\succ$| $\preceq$ |X_t|.

Would appreciate clarification on 1) why this is the case, and 2) what the superscripts for |X_t^$\preceq$| and |X_t^$\succ$| are actually representing.

Answer 1

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.