Let
$R_n$ be the set of simple random walk paths such that $S_n=0.$
$P_n$ be the set of simple random walk paths such that $\forall i \in \{1,2,...,n\},$ $S_i > 0$.
$N_n$ be the set of paths such that $\forall i \in \{1,2,...,n\}, S_i \geq 0$.
Assume that all random walk paths start at the origin.
How can I show that there is a bijection between $P_{2n}$ and $N_{2n-1}$ and that there is a bijection between $R_{2n}$ and $N_{2n}$. Basically I want to show that a path in $R_{2n}$ with minimum value $k$ corresponds to a path in $N_{2n}$ with terminal value $2k$.
For this I'm thinking about cutting or shifting or reflecting paths. I don't think probability matters here. But I'm stuck on formulating the proofs.
If we have sequence $S_0,S_1,...,S_n$ which is represented by a polygonal line with segments $(k-1,S_{k-1}) \rightarrow (k,S_k)$ a path is a polygonal line that is a possible outcome of simple random walk.