A process of the invention allows a cycloalkane to react with oxygen in the presence of a catalytic imide compound having an N-hydroxy (or N-oxo) cyclic imide skeleton and thereby yields a corresponding bis(1-hydroxycycloalkyl) peroxide. The catalytic imide compound includes, for example, a compound represented by following Formula (1): wherein R1 and R2 are each, for example, a hydrogen atom, a halogen atom, an alkyl group, an aryl group or a cycloalkyl group, where R1 and R2 may be combined to form a double bond, an aromatic or non-aromatic ring; and X is an oxygen atom or a hydroxyl group. The cycloalkane includes, for example, a cycloalkane having from 5 to 15 members. The invention can easily produce bis(1-hydroxycycloalkyl) peroxide from an inexpensive material.