when i run a gibbs sampler (lets say for $n=200$ Iterations) with two full condiontoals, i get the output $\mathbf{x} = (x_1^{(n)},x_2^{(n)})_{n \in [1,...,200]}$.

So $\mathbf{x}$ are the realazations of a gibbs markov chain, the so called gibbs sequenz. but are $(x_1^{(n)})_{n \in [1,...,200]}, (x_2^{(n)})_{n \in [1,...,100]}$ both realizations of a markov chain too ?