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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": { "date": 1591310627.4928222, "download_nb": 1, "download_nb_path": "https://julia.quantecon.org/", "filename": "finite_markov.rst", "filename_with_path": "tools_and_techniques/finite_markov", "kernelspec": { "display_name": "Julia 1.4.2", "language": "julia", "name": "julia-1.4" }, "language_info": { "file_extension": ".jl", "mimetype": "application/julia", "name": "julia", "version": "1.4.2" }, "title": "Finite Markov Chains" }, "nbformat": 4, "nbformat_minor": 2 }