Exponential decay of spatial pair correlations in 1D point processes follows from geometric ergodicity and exponential moments of Markovian spacings via an exponential Markov renewal theorem.
Br´ emaud,Point process calculus in time and space—an introduction with applications, Probability Theory and Stochastic Modelling, vol
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Markov Renewal Theory for Transfer Operators and Point Processes on the Line
Exponential decay of spatial pair correlations in 1D point processes follows from geometric ergodicity and exponential moments of Markovian spacings via an exponential Markov renewal theorem.