Theorem for independent and symmetrically distributed random variables.

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I trying to understand the following proof. My main difficulty is understanding the transformations from Eq. 14 to Eq. 15, specially the step after the integration.

Can someone help me?

This theorem is from Bertsimas, D., & Sim, M. (2004). The price of robustness. Operations research, 52(1), 35-53.

https://pubsonline.informs.org/doi/abs/10.1287/opre.1030.0065?journalCode=opre

Thank you in advance.

Proof - Theorem 2