To obtain the subject compound corresponding to the original compound in high selectivity and concentration by using an oxo metal compound as catalyst.

Using (A) an oxo metal compound as catalyst, (B) an arylalkyl hydrocarbon of formula (P and Q are each H or an alkyl; (n) is 1-3; Ar is an n-valent aromatic hydrocarbon)(e.g. cumene) is oxidized with an oxygen-contg. gas into the corresponding arylalkyl hydroperoxide. The component A is a compound of formula II (M is a metal; L is a ligand; (n) is 1-8), pref. an oxyvanadium compound, oxytitanium compound or oxyzirconium compound's salt or complex, for example, titanium (II) dioxide acetylacetonate.