 Uniform depletion and refill Markov chain

Refill the stock to a fixed positive size $N$ immediately whenever a step empties it. The resulting Markov chain has state space $\{1,\dots ,N\}$ and transitions $P(i,j)=1/i$ for $j<i$ and $P(i,N)=1/i$. It is irreducible and aperiodic, since every state reaches $N$, $N$ reaches every state, and $P(N,N)>0$. Its unique stationary distribution is ${\pi }_N=1/H_N$ and ${\pi }_k=1/((k+1)H_N)$ for $k<N$. Normalization is the harmonic number identity. For $k<N$, stationarity follows from ${\sum }_{j=k+1}^{N-1}1/(j(j+1))+1/N=1/(k+1)$; the same telescoping sum starting at $1$ gives stationarity at $N$. The finite-state convergence theorem therefore makes this stationary distribution the limiting distribution from every initial state.