A random walker starts at the origin, and experiences unbiased diffusion along a continuous line in 1d. What is the probability for this walker to return to the origin for the first time as a function of time? In other words, I am looking for the probability of first return to the starting point. I would greatly appreciate to learn the functional form of this probability, or to learn of references where it is discussed or stated.
Some other links to direct solutions found by MrGoogle under key words
random walk first return
http://www.stat.berkeley.edu/~arturof/Teaching/STAT150/Notes/I_First_Return_Probabilities.pdf
http://www.stat.cmu.edu/~genovese/class/iprob-S06/notes/lec5-calcs.pdf
and references therein. For systematic study, however, some book(s) on probability with fully explained basics on generating functions and their advanced application will be needed. It is a good start with Chapter III of William Feller's bible:
An Introduction to Probability Theory and Its Application
Regards
Hi,
See the link below.
Good Luck.
http://mathworld.wolfram.com/RandomWalk1-Dimensional.html
Some other links to direct solutions found by MrGoogle under key words
random walk first return
http://www.stat.berkeley.edu/~arturof/Teaching/STAT150/Notes/I_First_Return_Probabilities.pdf
http://www.stat.cmu.edu/~genovese/class/iprob-S06/notes/lec5-calcs.pdf
and references therein. For systematic study, however, some book(s) on probability with fully explained basics on generating functions and their advanced application will be needed. It is a good start with Chapter III of William Feller's bible:
An Introduction to Probability Theory and Its Application
Regards
Quite correct. A beautiful related result concerns the same problem for random walks with arbitrary independent identically distributed symmetrical step lengths: see Theorem of Sparre-Anderssen. The return probability does not depend on the distribution of step lengths: it need not have either a variance or a finite expectation value.
Dear F. Leyvraz,
It's OK with the existing extensions which currently develope within domain called cumulative processes driven by a renewal processes; in particular they are intensively used in modelling the ruin processes. The same correctness concerns in some cases the probability of the ruin, which, however, depends on the parameters of the components like distance between renewals, size of the increments in much more weak degree. Contrarily, the dependence of the time to the ruin (or, the instant of the ruin) is much stronger, of course (eg. if one extends the time between jumps twice, the same would concern the hitting time of whatever:).
Regards
Yes, see Feller vol. 1, chapter III. In particular, Lemma 2 says that the probability that the first return to the origin occurs at the nth step is (1/(2n-1))*binomial(2n,n)*2^(-2n). The probability of returning to the origin is 1, but the expected time to do so is infinite.
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