1
$\begingroup$

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

attached image

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

$A_n\nearrow A$ 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 $\mathbb{R}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.