I'm not sure that I completely understand your question, but this sounds like effectively a one dimensional random walk where you're flipping a coin to decide whether you move closer to the starting point or further from it every step, so the distribution of distances your particle has from the starting point at large \$t\$ ought to be normal. %PDF-1.5

A random walk is the process by which randomly-moving objects wander away from where they started.

By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. Journal of Applied Probability fields are published. /Subtype /Form The simplest random walk to understand is a 1-dimensional walk.

The question is: calling \$P(t)\$ the probability to find the particle at the point \$(x(t),y(t))\$ at time \$t\$, after how many time steps the probability to find the particle outside the circle \$(x(t)^2+y(t)^2)\gt R\$ is: \$P(t)\ge P_0?\$

/Filter /FlateDecode Particles chasing one another around a circle, limiting distribution of the random walk from irrational rotation, The probability that a 2d continuous time random walk avoids the origin, First collision time of \$n\$ random walkers on a cycle, Non-erasure probability in a loop-erased random walk in three dimensions.

single random walk on the continuous circle S', induced by the same function f. For large n, the random walk on Z/(n) will be 'similar' to the random walk on S', and thus the rates of convergence will be related to the single rate of convergence for the random walk on S'.