A catalyst is used which comprises a cocatalyst component wherein a non-coordinating ion-containing compound is chemically bonded to a fine particulate carrier, together with a metallocene compound and a specific hydrocarbon. There are provided an olefin polymerization catalyst which produces olefin polymers containing few solvent-soluble components without a wider molecular weight distribution, and which exhibits no significant reduction in activity even after storage, as well as olefin polymerization catalyst components and a method for their storage, and a process for production of propylene polymers using them.