Show that if both transition kernels K and K2 have p() as stationary density, so do...

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Show that if both transition kernels K and K2 have p() as stationary density, so do KK2 and AK+ (1X)K2 for any = [0, 1]. In practice, the former corresponds to sampling from K and K2 cyclically and the latter draws either K with probability A or K2 otherwise. Although it is not required to show, extension to more than 2 kernels should be straightforward. Show that if both transition kernels K and K2 have p() as stationary density, so do KK2 and AK+ (1X)K2 for any = [0, 1]. In practice, the former corresponds to sampling from K and K2 cyclically and the latter draws either K with probability A or K2 otherwise. Although it is not required to show, extension to more than 2 kernels should be straightforward.