We consider the problem of the convergence of the so-called LePage series in the Skorokhod space D{double-struck}^d=D{double-struck}([0,1],R{double-struck}^d) and provide a simple criterion based on the moments of the increments of the random process involved in the series. This provides a simple sufficient condition for the existence of an α-stable distribution on D{double-struck}^d with given spectral measure.