# Levy process: reversed $\overset{d}{=}$ dual : $X_{t-s}-X_t \overset{d}{=} - X_s$

Let $X$ be a Levy process, there is a theorem which says $$X_{(t-s)^-} - X_t \overset{d}{=} - X_s,$$ for $0 \le s \le t$, $t$ fixed.
How do we proove this without a graphical argument?

In Kyprianou, it goes like this \begin{align} X_{(t-s)^-}-X_t &\overset{a.s.}{=} X_{t-s} - X_t && \text{(quasi left-continuity)} \tag{1} \\ &\overset{a.s}{=} - \left( X_t - X_{t-s} \right) \tag{2} \\ &\overset{d}{=} - X_s \tag{3} \end{align}

Why does (3) follows?

*Kyprianou, Fluctuation of Levy process,lemma 3.4

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Isn't that the stationary increments? – Stefan Hansen Feb 12 '13 at 16:28
@StefanHansen You are right, I understand now. Thanks. – Nicolas Essis-Breton Feb 12 '13 at 16:30