# What does the arrows mean? (notation for markov chain topic)

I found this arrow notation while reading about markov chain and random walks and dont understand this notation. I can't find the answer anywhere. Help?

{T0 < Tc} (slanted arrow up, from left bottom to right top) {T0 < T1}, T0 represents inf{n>=0: Xn=0}.

there's one with slanted arrow down too

As $$c \to \infty$$, the set $$\{\tau_0 < \tau_c\}$$ gets larger, presumably. So $$\{\tau_0 < \tau_c\} \nearrow \{\tau_0 < \infty\}$$ probably means $$\bigcup_{c > 0} \{\tau_0 < \tau_c\} = \{\tau_0 < \infty\}$$. For the other arrow, the sets get smaller, so it would be an intersection instead of a union.
$$A_n\nearrow A$$ means you have a nested sequence $$A_1\subseteq A_2\subseteq\dots$$ with $$\bigcup_{n\in\mathbb{N}} A_n=A$$. Similarly, $$A_n\searrow A$$ means $$A_1\supseteq A_2\supseteq\dots$$ with $$\bigcap_{n\in\mathbb{N}} A_n=A$$.
Of course you could change the indexing set from $$\mathbb{N}$$ to say $$\mathbb{R}$$.