The invariance principle type theorems for recurrent semi-Markov processes with a general state space

The invariance principle type theorems for processes of sums of random variables defined on a recurrent semi-Markov process with a general state space are proved. The results are obtained in assumption that for a fixed subset of state space the recurrent times have a stable limit distribution $L_\alpha$, $\alpha\in(0,1]$ (the case $\alpha=1$ corresponds to the ergodic process).