A cross-linked polymer composition includes a cross-linked polyolefin. Peroxide in an amount of less than 35 mmol —O—O—/kg polymer composition may be included in the polymer composition prior to crosslinking the polyolefin. Suitable polyolefins that can be cross-linked in the polymer composition include an unsaturated LDPE homopolymer and an unsaturated LDPE copolymer of ethylene with one or more comonomers. Such unsaturated LDPE polymer can contain vinyl groups, and the total amount of vinyl groups present in the unsaturated LDPE can be higher than 0.05/1000 carbon atoms. The cross-linked polymer composition can be used, for example, in producing a cross-linkable and cross-linked article such as a cross-linkable and cross-linked direct current (DC) or other type of power cable.