Irreducibility of markov chains

Question

Let $A=(a(x,y))_{x,y\in X}$ be a finite irreduzibel nonnegative matrix.

Let $b,c >0$, and $\alpha=a^{(b+c)}(x,x)>0$.

So $a^{(n(b+c))}(x,x)\ge \alpha^n$. And therefore $\lim \sup_{n\rightarrow\infty}{a^{(n)}(x,y)^{1/n}}\ge\alpha^{1/(b+c)}>0$

I don't understand several things here:

1. Why ist this wrong: $a^{(n)}(x,y)^{1/n}=a(x,y)$

2. I don't understand how to follow the whole 3th line.

Thanks for any kind of help!

Answer 1

1. $a^{(n)}$ is the $n$th power of the matrix $a$. hence $a^{(n)}(x,y) \neq a(x,y)^n$.