Minimax and adaptive inference for branching ratios under long memory

Abstract

The branching ratio $\rho$ of a self-exciting point process determines the branching margin $1-\rho$ to the critical boundary. We study inference on $\rho$ from a finite record for stationary Hawkes processes with completely monotone kernels whose slow relaxation mass obeys an envelope near zero rate. Slow excitation can then be exchanged for exogenous immigration; a spectral bound on the Kullback--Leibler rate and a coupling argument for the unobserved past give a minimax lower bound of order $T^{-\gamma /(2\gamma +1)}$ for a power-law envelope with tail exponent $\gamma$. A block-dispersion estimator whose block length grows as $T^{1/(2\gamma +1)}$ attains this rate. A quantitative central limit theorem, obtained from the cluster representation and a second-order Poincaré inequality on Poisson space, yields honest confidence intervals of nearly minimax expected length. For unknown $\gamma$, a Lepski-type rule adapts in probability at a logarithmic cost; free adaptation between memory classes is impossible, and honest intervals cannot adapt to lighter memory. Simulations are consistent with the rates.