Let $F$ be a CDF that is symmetric about $t^* \in T$.

Consider the following integral:


Using change of variable and the symmetry of $F$ we can re-write this integral as

$$ \int_{t^*-\overline{\delta}}^{t^*+\overline{\delta}}cos(t)f(t)dt =\int_{-\overline{\delta}}^{\overline{\delta}} cos(t^*+\delta)f(t^*+\delta)d\delta = \int_{0}^{\overline{\delta}} \left(cos(t^*+\delta) + cos(t^*-\delta)\right)f(t^*+\delta)d\delta $$

How is Symmetry being used in the above re-writing of the integral?

I believe its in the second equality (that is changing the bottom limit of integration from $-\overline{\delta}$ to $0$), but it's not clear to me exactly how.

Is the integral just being split and then "flipped" (non-technical term, sorry) using symmetry? i.e. something like $$ \int_{-\overline{\delta}}^{\overline{\delta}} (stuff) = \int_{-\overline{\delta}}^0 (stuff) + \int_0^{\overline{\delta}} $$ and then use symmetry to rewrite $\int_{-\overline{\delta}}^0 (stuff)$ as $\int_0^{\overline{\delta}} ( other stuff)$?


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