Analytical ACF solution for the α=2, β>2 regime

Derive an explicit analytical expression for the autocorrelation function A(t_d) of the model event sequence x(t) defined in Section 2, under the power-law case where reduced interevent times follow ψ(τ̄)=C_ψ τ̄^{-α} on [1, τ̄_c] with α=2 and burst sizes follow Q(b)=C_Q b^{-β} on [1, b_c] with β>2. Specifically, obtain closed-form or asymptotically exact formulas for Z(t_d) and A(t_d) in this parameter regime, for which the authors report no available analytical solution.

Background

The paper proposes a generative model for bursty event sequences with independently specified reduced interevent-time and burst-size distributions, and derives analytically the autocorrelation function via Laplace-transform techniques. Assuming power-law tails for both distributions, the authors obtain scaling relations for the ACF decay exponent γ as a function of the interevent-time exponent α and the burst-size exponent β.

While many parameter regimes admit analytical solutions, the authors explicitly note that for α=2 and β>2 there are no analytical solutions available, referencing the Laplace-transform difficulties encountered. Although symmetry arguments allow them to conclude γ=0 when α=2 or β=2, the explicit ACF solution for α=2, β>2 remains unresolved, motivating a concrete analytical derivation in this case.