Phase transition in a long-memory log-Gaussian Cox process

We study a stochastic point process with power-law temporal correlations driven by hidden variables. We show that a generalized Merton-type model under an exponential-tail asset assumption, obtained by replacing the Gaussian cumulative distribution function with a logistic CDF, together with an appropriate double-scaling limit, converges to a log-Gaussian Cox process (LGCP) with log-normal intensity. The resulting LGCP exhibits a phase transition at the critical power index $\gamma=1$. This transition separates regimes of short-memory dynamics from long-memory behavior characterized by anomalous diffusion. We further demonstrate that temporal correlations persist even in the Poisson limit when the scaling is properly defined, in contrast to conventional Poisson convergence where memory effects vanish. We also compare this LGCP with self-exciting processes such as Hawkes processes, highlighting fundamental differences in correlation structure and extreme-event behavior. The theoretical results are illustrated using credit risk time series, and empirical estimation of the temporal correlation parameter from historical default data provides evidence for long-memory behavior in pre-1980 credit portfolios.