What does $\lim_{\theta \uparrow 1}$ mean in criticality of Branching process?
Assume that $\lim_{\theta \uparrow 1} F'(\theta)=F'(1) < \infty$.
Where $F$ is is prob. generating function:
$$F(\theta):= \sum_{l=0}^{\infty} \theta^l p_l$$
Asked
Active
Viewed 18 times
1 Answers
1
It is simply the left hand limit of $F'(\theta)$ as $\theta \to 1$. You take limit through values of $\theta <1$.
Kavi Rama Murthy
- 311,013
-
You mean "approaches from 'negative' side"? – mavavilj May 07 '19 at 10:07
-
Yes, $F$ and its derivative may not exist for $\theta >1$ so we can only take limit from the left side. – Kavi Rama Murthy May 07 '19 at 10:08