Convergence of Limiting Matrix

Question

If the matrix $A_n$ is a markov transition matrix. Literature define de limiting matrix as $$\lim_{N \rightarrow \infty} \frac{1}{N} \sum_{n=0}^{N-1} A_n $$ where $\{A_n, n= 0,1,2,... \}$ is a sequence if matrices. My question is if $$ \lim_{N \rightarrow \infty} A_n$$ exist, the limit above also exist? I think the answer is yes but I am not sure. The both limits are equals? Thanks