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[1] Hint: First show that if $ P $ and $ Q $ are stochastic matrices then so is their product — to check the row sums, try postmultiplying by a column vector of ones. Finally, argue that $ P^n $ is a stochastic matrix using induction." ] } ], "metadata": { "filename": "finite_markov.rst", "kernelspec": { "display_name": "Julia 1.2", "language": "julia", "name": "julia-1.2" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.2.0" }, "title": "Finite Markov Chains" }, "nbformat": 4, "nbformat_minor": 2 }