NOVELTY - A method (P1) for preparing cyclic olefin addition polymer involves addition polymerizing cyclic olefin-functional siloxane or mixture of cyclic olefin-functional siloxane and cyclic olefin compound in presence of multi-component catalyst containing a zero-valent palladium compound, an ionic boron compound, and a phosphine compound having substituent group selected from 3-6C alkyl, cycloalkyl and aryl, thus obtaining cyclic olefin addition polymer, where a ratio of structural units derived from cyclic olefin-functional siloxane is 10-100 mole% of the addition polymer.    USE - For preparing cyclic olefin addition polymer used in film, sheet or coating (claimed); as oxygen-enriching film for air conditioners and fuel cells and also as contact lens.    ADVANTAGE - The cyclic olefin addition polymer has excellent gas permeability, particularly, excellent oxygen permeability, along with high thermal stability (heat resistance) and film strength (mechanical strength), excellent solubility in organic solvents and polysiloxane solvents and hence can be applied to filming agents for medicines, foods and cosmetics. The polymer exhibits excellent film-forming property. The cyclic olefin addition polymer of high gas permeability is easily prepared by use of multi-component catalyst of specific type of structure having excellent polymerization activity, selectivity and molecular weight controllability.    DETAILED DESCRIPTION - A method (P1) for preparing a cyclic olefin addition polymer having gas permeability and a number average molecular weight of 100000-2000000 as determined by gel permeation chromatography relative to polystyrene standards, involves: addition polymerizing cyclic olefin-functional siloxane of formula (I), or mixture of cyclic olefin-functional siloxane (I) and cyclic olefin compound of formula (II), in presence of multi-component catalyst containing a zero-valent palladium compound, an ionic boron compound, and a phosphine compound having substituent group selected from 3-6C alkyl, cycloalkyl and aryl, thus obtaining a cyclic olefin addition polymer, where a ratio of structural units derived from cyclic olefin-functional siloxane (I) is at 10-100 mole% of addition polymer.    R1=aliphatic unsaturated bond-free monovalent organic group;    s=0-2;    j=0 or 1;    A1-A4=H, halo, 1-10C alkyl, alkenyl, cycloalkyl, aryl, alkoxy, aryloxy or halogenated hydrocarbyl, or a polar group selected from oxetanyl and alkoxycarbonyl;or    A1 and A2 or A1 and A3+the C atoms bonded to them=alicyclic structure, aromatic ring structure, carbonimide or acid anhydride;    i=0 or 1.