Shot-noise processes with logarithmic response functions converge weakly to Hadamard fractional Brownian motion, a process with Brownian marginals and fractional long memory of reduced intensity.
Iksanov, Functional limit theorems for renewal shot noise processes with increasing response functions,Stochastic Processes and their Applications, 123(6), 2013, 1987-2010
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Shot-noise processes with logarithmic response function and their scaling limits
Shot-noise processes with logarithmic response functions converge weakly to Hadamard fractional Brownian motion, a process with Brownian marginals and fractional long memory of reduced intensity.