# Modeling a 1D random walk with nonconstant probabilities

I have (what I will term, for lack of a better word) a random walk that has a particular property: it tends to be right of the origin some fraction k of the time and left 1-k of the time (and on the origin a negligible fraction). So although the probabilities of going up or down are close to 1/2, there is a central tendency as a result of this propensity. (Note: $k\neq1/2$ else I would use a Poisson process.)

It would be best if the process was memoryless in the sense that while the probability can vary (slightly) from 1/2 based on the current position, the count so far is ignored. Thus, it would perform the same at 0 whether it had just spent the last million above 0 as if below. Is such a model possible within my constraints?

I want to analyze the probability that the process is, say, left of the origin over an interval of varying lengths, given that the process starts at the origin.

I'm looking for a model as natural (parsimonious) as possible in hopes that I can compare my measurements to the model to see if the event is as rare as it (naively) appears to be. The combination of it being a walk (so that it can 'dig itself into a hole' as it were) and its unusual central tendency have confused my intuition as to what is normal...

Edit: For example, one possible model that seems to fit my requirements: some (N, p) where for position n > N go left with probability p and right with probability 1-p; for position n < N go left with probability 1-p and right with probability p; for n = N use 1/2 either way. (Think of p as just slightly more/less than 50%.) The process doesn't know its score but by choosing N sufficiently far from the origin and p close enough to 50% I get my requirements. It's not quite parsimonious, having one too many degrees of freedom (multiple choices for a given k) and it really doesn't fit my intuition as to how the process works, but maybe it inspires some better ideas.

The underlying problem, for those interested or looking for inspiration on the problem: this models the crossing point where $\pi(n)-\operatorname{li}(n)=0$ after Rubinstein & Sarnak 1994, where k is about $2.6\times10^{-7}$.

• Why would a Poisson process (which is defined on $\mathbb{Z}^+$ and nondecreasing in time) ever make sense for your problem? – JMS Jul 14 '11 at 17:22
• This reminds me of <a href="en.wikipedia.org/wiki/Sequential_analysis">sequential analysis</a> and <a href="springerlink.com/content/w3144338312n2743/">sequential designs of clinical trials</a>. Traditionally, randomization of patients into different treatments has been completely blind. However, more recently, in order to benefit patients while still learning something about performance of the drug, one can randomize allocation to the treament arm so that the better performing arm receives more patients. So instead of say 33%:33%:33% allocation to the control and two treatme – StasK Aug 12 '11 at 22:05
• ...treatment groups, one can entertain increasing the probability of the better performing arm to 50%:25%:25%. [This continuation of @StasK's comment was cut off in the conversion from reply format.] – whuber Aug 12 '11 at 22:14
• @JMS: Call it a difference of Poisson distributions with equal lambdas if that bothers you. But in any case that's irrelevant since $k\neq1/2$ in my case. – Charles Aug 15 '11 at 13:35

If at $0$, go right with probability $k$ and left with probability $1-k$. If right of $0$, go right with probability $p$ and left with probability $1-p$; if left of $0$, go right with probability $1-p$ and left with probability $p$.
Providing $p \lt \frac{1}{2}$, the expected time that the walk is right of $0$ as a proportion of the time it is not at $0$ is $k$ (which I think is really what you are asking for) and the closer $p$ is to $\frac{1}{2}$ the smaller the proportion of the total time the walk will be at $0$.
For $p \gt \frac{1}{2}$, there is a positive probability the walk will never return to $0$ while for $p = \frac{1}{2}$ the expected time for each return is infinite.
How about something like $$dx = a*dt + b*d W^1_t - c*\text{sgn}(x)dW^2_t$$ where $W^1_t$ and $W_t^2$ are two distinct Wiener processes? It has a central tendency as long as $c>a$, and $a$ should determine the fraction $k$ it spends over the origin.