Can someone help me find the moment generating function of a random walk?

[humenghu]

Hi, can someone who is familiar with the analysis of random walks (statistical mechanics, condensed matter physics etc.) help me on solving a particular problem?

We define the following random walk, the random variable w(t) is evolved as

w(t+1)=w(t), with probability of p;

w(t+1)=w(t)+C*(1-w(t))*f, with probability 1-p.

C is a constant, f is a random variable with known distribution, and f is not correlated with w.

Following the way of analyzing a Brownian motion, what I have tried is to write down
E(w(t)|w(t-1),p), then show the E(w(t)|p) using the induction, for a given w(0).

Next, at least I need to find the Var(w(t)) , or equivalently E(w(t)^2) since I have known (E(w(t)))^2 for a given w(0).

I hope that someone can help me find the moment generating function of this random walk

(E(w(t)^n)), or alternatively just the mean and the variance of w(t). Any hint on this is badly needed.

 

[haruspex]

You can obtain E(w(t)) as something of the form A+Bλ^t, right?
You should also get a recurrence relation on E(w(t)²) which involves E(w(t)). This should have a solution like C+Dλ^t+Eμ^t.
Plugging in the coefficients from the recurrence relations and the initial conditions should allow you to determine the 7 constants.

 

[humenghu]

You can obtain E(w(t)) as something of the form A+Bλ^t, right?
You should also get a recurrence relation on E(w(t)²) which involves E(w(t)). This should have a solution like C+Dλ^t+Eμ^t.
Plugging in the coefficients from the recurrence relations and the initial conditions should allow you to determine the 7 constants.

Thanks for the hint.

Yeah the E(W_{n})=K^{n}W_{0}+K^{n-1}L+K^{n-2}L+...+KL+L

Where K=p-C(1-p)E(f), L=C(1-p)E(f).

However I found that the E(Wt^{2}) is a little bit hard to show since the terms grow somehow fast with n. I could write down:

E(W_{n}^{2})=A^{n-1}M+ A^{n-2}[BE(W_{1})+C]+ A^{n-3}[BE(W_{2})+C]+...A[BE(W_{n-2})+C]+[BE(W_{n-1})+C]

where A,B,C are terms consisting of p, C and random variable f.M=AE(W_{0}^{2})+BE(W_{0})+C

so do you think this is what you meant? or?

Addtionally I am still curious about the moments of the Wt, in a general sense. Is it possible to show them in closed form here?

 

[haruspex]

Yeah the E(W_{n})=K^{n}W_{0}+K^{n-1}L+K^{n-2}L+...+KL+L

Where K=p-C(1-p)E(f), L=C(1-p)E(f).

I don't think it should be that bad.
K^{n}W_{0}+K^{n-1}L+K^{n-2}L+...+KL+L
= K^{n}W_{0}+L(K^n-1)/(K-1)
= K^{n}(W_{0}+L/(K-1))-L/(K-1)
So as I said, it is possible to represent the first moment as A+BKⁿ, and the second moment will have only two additional constants.

 

[humenghu]

Now I see what you mean. Thanks a lot!

Did you magically predict the first two moments just by looking at the evolution rules? or did you actually write down the terms like I did, any additional tips on this ?

 

[haruspex]

I wrote out the recurrence relation for w(t) and saw it took the form
w(t+1) = Aw(t) + B.
Clearly this is basically multiplying by A at each step, so the answer will be something with A^t. To cover the inhomogeneous term (the B), look for a stationary value, i.e. w(t+1) = w(t) = w:
w = Aw+B
w = B/(1-A)
So now we have a general solution of w(t+1) = Aw(t), and a specific solution of w(t+1) = Aw(t) + B. Since the relationship is linear, we can add any multiple of the homogeneous solution to the inhomogeneous one:
w(t) = cA^t + B/(1-A)

The recurrence relation for w² is of the form
w²(t+1) = Dw²(t) + Ew(t) + F
= Dw²(t) + E(cA^t + B/(1-A)) + F
Collecting up constants, this reduces to
w²(t+1) = Dw²(t) + E'A^t + F'
Again, the homogeneous part gives w²(t) = gD^t. What to do with E'A^t? Iterating the recurrence will result in adding terms like that for successive values of t, i.e. the sum of a geometric series. So we know this will add a term like A^t to the answer. So just assume an answer of the form gD^t + hA^t + k, plug it into the recurrence relation and initial conditions, and see what you get for g, h and k.

 

[humenghu]

I see. Thanks!