To polymerize a cycloolefin at a high conversion in the presence of a very small amt. of a catalyst to produce an amorphous olefin polymer by polymerizing a cycloolefin in the presence of a catalyst which comprises a transition metal compd. and at least either an aluminum compd. selected from among organoaluminum compds., cyclic aluminoxanes, and linear aluminoxanes or a boron compd.

A cycloolefin represented by formula II (wherein R9 to R20 are each H, an org. group, etc., is polymerized in the presence of a catalyst which comprises a compd. represented by formula I [wherein M is an element of the group 4 or lanthanide; A and A' are each a hydrocarbon group, etc.; Y is a (halogenated) hydrocarbon group, etc.; L is a Lewis base; etc.], and at least either a compd. selected from among compds. represented by the formulas: E1aAlZ3-a, {-Al(E2)-O-}b, and E3{-Al(E3)-O-}cAlE32 (wherein E1 to E3 are each a hydrocarbon group; Z is H or halogen; etc.), or a compd. selected from among compds. represented by the formulas: BQ1Q2Q3, G+(BQ1Q2Q3Q4)-, and (L-H)+(BQ1Q2Q3Q4)- [wherein Q1 to Q4 are each halogen, a (halogenated) hydrocarbon group, etc.].