An object of the invention is to provide a solid polyaluminoxane composition suitably used as a cocatalyst and a catalyst carrier in combination with an olefin oligomerization or polymerization catalyst, without the use of solid inorganic carriers such as silica. The solid polyaluminoxane composition of the invention includes a polyalkylaluminoxane and a trialkylaluminum, and has a solubility in n-hexane at 25° C. of less than 0.50 mol % as measured by a specific method (i), a solubility in toluene at 25° C. of less than 1.0 mol % as measured by a specific method (ii), and a 13 mol % or more molar fraction of alkyl groups derived from the trialkylaluminum moieties relative to the total number of moles of alkyl groups derived from the polyalkylaluminoxane moieties and the alkyl groups derived from the trialkylaluminum moieties as measured with respect to tetrahydrofuran-d8 soluble components by a specific method (iii).