# Random walk on $\mathbb{R}$ with “sticky” origin

Let $$P_i$$, $$N_i$$, and $$Z_i$$, $$i\in\mathbb{N}$$ be r.v.'s with the $$P_i$$, $$N_i$$, and $$Z_i$$ being identically distributed with known pdf's $$f_P$$, $$f_N$$, and $$f_Z$$, respectively; and with no dependence between any $$P_i$$, $$N_i$$, or $$Z_i$$. Define the following recurrent process:

$$T_0:=0$$

$$T_i:=\left\{ \begin{array}{ll} Z_i & \quad T_{i-1} = 0 \\ min(T_{i-1}+N_i, 0) & \quad T_{i-1} < 0 \\ max(T_{i-1}+P_i, 0) & \quad T_{i-1} > 0 \end{array} \right.$$

In English: a particle walks randomly on the real number line (starting from the origin) in discrete time steps, where its position cannot change sign from positive to negative or vice-versa in a single step - were it to do so, it is set back at the origin before the next step - and where its evolution depends on the sign of its current position.

What can be said about the pdf's of the $$T_i$$ in relation to $$f_P$$, $$f_N$$, and $$f_Z$$?

• If $P_i, N_i, Z_i$ are all standard normals, then $T_2$ has a pdf of $$\frac{1+\text{erf}(|x|/2)}{4\sqrt{\pi}}e^{-x^2/4}$$ plus a $1/4$ chance of being exactly 0, for a total variance of $3/2+1/\pi$. That complexity does not suggest simple asymptotics. – Matt F. Oct 21 '18 at 19:55
• Also: $T_2$ has $1/4$ chance of being exactly 0 whenever $P, N, Z$ have identical continuous distributions with symmetry about the origin. – Matt F. Oct 22 '18 at 0:31