Considering a random walk where, at each step, a person can move either left or right with a step size of 1 and an equal probability of 0.5 for each direction. However, there's a unique feature: a wall is positioned at $x = 0$. If the person reaches this wall, they are compelled to move right on the subsequent step. The walk begins at $x = 0$, so the first step is always to the right. This scenario can be modeled as a Markov chain.
For each step, let's denote the random variable as $S_i $, where $i = 1, 2, ..., n$, and define $S = S_1 + S_2 + ... + S_n$.
Q1: Will the distribution of this random walk converge to a steady distribution as the number of steps $n$ approaches infinity?
Q2: If it does converge, what will the steady distribution be? Specifically, will the normalized sum $\frac{S}{\sqrt n}$ converge in distribution to $exp(-x) $ for $ 0 < x < + \infty$ ?
One way of solving this problem is to note that the reflecting wall can be handled by taking the absolute value. For example, one potential random walk without the reflecting wall might be $$ +1 \to 0 \to -1 \to -2 \to -1 \to 0 \to \dots $$ After applying the absolute value, we then get $$ +1 \to 0 \to +1 \to +2 \to +1 \to 0 \to \dots $$ which can be seen as the reflected version.
Then, the central limit theorem then shows that, for the unreflected $S/\sqrt{n}$, it converges in distribution to the normal distribution. After taking absolute values, we thus get the Folded Normal Distribution.