Optimal prediction of Markov chains with and without spectral gap

We study the following learning problem with dependent data: Given a trajectory of length $n$ from a stationary Markov chain with $k$ states, the goal is to predict the distribution of the next state. For $3 \leq k \leq O(\sqrt{n})$, the optimal prediction risk in the Kullback-Leibler divergence is shown to be $\Theta(\frac{k^2}{n}\log \frac{n}{k^2})$, in contrast to the optimal rate of $\Theta(\frac{\log \log n}{n})$ for $k=2$ previously shown in Falahatgar et al in 2016. These nonparametric rates can be attributed to the memory in the data, as the spectral gap of the Markov chain can be arbitrarily small. To quantify the memory effect, we study irreducible reversible chains with a prescribed spectral gap. In addition to characterizing the optimal prediction risk for two states, we show that, as long as the spectral gap is not excessively small, the prediction risk in the Markov model is $O(\frac{k^2}{n})$, which coincides with that of an iid model with the same number of parameters.