0

This is a rather basic question, is just that I don't know a thing about this subject. Let's say $x$ is an integer. At $t=0$, the value of $x$ is 1. Then, at each time step, one of the following happens:

  • $x$ goes to $x+1$, with probability $p_1 < \frac{1}{2}$
  • $x$ remains with the same value with probability $ 1 - 2 p_1 $
  • $x$ goes to $x-1$, with probability $p_1 < \frac{1}{2}$

I'm interested in knowing the distribution of the time $t^*$ when $x$ reaches the value zero for first time.

dsign
  • 111
  • 2
    Are you familiar with the answer if p_1 = 1/2? – John Jun 26 '13 at 16:25
  • @John That would be this, but even in this article I don't see anything about $t^*$. How can I connect both however? – dsign Jun 26 '13 at 16:26
  • How accurate an answer do you need? – John Jun 26 '13 at 16:27
  • @John I would rather like some pointers. For example, what results are out there if we say $p_1 = 1/2 $ ? – dsign Jun 26 '13 at 16:34
  • 1
    This is covered by my very first answer in MSE. Equation ($\ast$) gives you the distribution for the hitting time of state "1" starting at "0", but this is the same as hitting time of state "0" starting at "1". In the line following ($\ast$) I assume that $p_1=1/2$, but not before.
    http://math.stackexchange.com/questions/4234/random-walk-0/4235#4235     –  Jun 26 '13 at 16:37
  • See also http://www.win.tue.nl/~rhofstad/monthly753-756-hofstad.pdf –  Jun 26 '13 at 16:43
  • Thanks @ByronSchmuland, this will get me started. – dsign Jun 26 '13 at 16:45
  • @dsign Glad to help! –  Jun 26 '13 at 16:46

0 Answers0