This is an exercise 1.47 from Richard Durrett's Essentials of Stochastic Processes p.85 (doi: 10.1007/978-1-4614-3615-7_1 or Google Books).
On each request the ith of the $n$ possible books is the one chosen with probability $p_i$. To make it quicker to find the book next time, the librarian moves the book to the left end of the shelf. Define the state at any time to be the sequence of the books we see as we examine the shelf from left to right. Since all the books are distinct, this list is a permutation of the set $\{1,2,…,n\}$, i.e each number is listed exactly once. How to show that
$$\pi (j_1,…,j_n)=p_{j_1}\frac{p_{j_2}}{1-p_{j_1}}…\frac{p_{j_n}}{1-p_{j_1}-…-p_{j_{n-1}}}$$ is a stationary distribution?
Thank you for your time and help.
Best Answer
I got it. There is actually a paper by W. Hendricks (published in journal of applied probability, 1972) on this particular problem where the author explicitly computed the stationary distribution. Thanks anyway.