Hello,
Let us consider the classic random walk:
x0 = 0, xn+1 = xn + rn,
where rn are equally distributed zero-mean Gaussian random variables. It is well-known that the probability distribution of xn is a zero-mean Gaussian with standard deviation proportional to sqrt(n).
Now, I will say that the instant k is a zero-crossing for a particular realization of the random walk iff xk has opposite sign than xk+1, and I will consider the random variable T defined as the temporal distance between two consecutive zero-crossing.
What can we say about the probability distribution of T?
My intuition would suggest such probability distribution is time dependent and its expected value would also grow as sqrt(n). Am I correct or wrong?
I thank you all, Giuseppe Papari
Please have a look at the arcsine-laws for random walks, Wiener processes etc. For a start
http://en.wikipedia.org/wiki/Arcsine_laws_(Wiener_process)
There is a vast literature about this topic...
If I understand your question correctly then you are sampling consecutive pairs which are drifting away with a distance ~sqrt(n). You are going to get less and less 'zero crossing' pairs. A quick simulation suggests that your sampling rate is ~ 1000/1600000 while for 2000 such pairs you already need over 10 million steps. It appears that the variance of the distributed zero crossing distance is ~2/3 of the Variance[rn] and what I understand as T is proportional to the population size of which you sample from.
.
there is no reason for the distribution of the duration beween successive zero-crossings of a simple random walk to be time-dependent : every time the random walker crosses the zero line, you can imagine it is "reset" : as there is no memory, crossing the zero line at time 100 or crossing the zero line at time 100000 leaves the random walker in the same - physical time-independent - situation (see the notion of "random walk local time" for technical details)
in addition to Markus references above, you might be interested in
https://www.researchgate.net/publication/221423038_Lengths_and_heights_of_random_walk_excursions
and references therein
.
Conference Paper Lengths and heights of random walk excursions
Thinking about your problem a little more I believe that this is a version of the 'ruin problem' where by 'ruin' you can define a zero crossing event. For a 1D Brownian motion it can be shown that the probability for such events diminishes as time passes by roughly as t-2/3.
Indeed, I get a good fit of my simulated data of T to an Inverse Gaussian (a.k.a Wald -- special case Lévy) distribution as in the enclosed link. Please see attached file.
http://en.wikipedia.org/wiki/Inverse_Gaussian_distribution
You are interested at the length of the excursion from 0, in a random walk.
If r_n is a bernoulli distribution, the study is done in http://www.dmtcs.org/pdfpapers/dmAC0105.pdf
x_n, correctly renormalised converges to a Brownian Motion. The length of the excursion of a Browninan motion is studied by Pitman and Yor in https://projecteuclid.org/euclid.aop/1024404422
Cheers
Marie
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