Functional Limit Theorem Without Centering for General Shot-Noise Processes

A general shot-noise process is defined as the convolution of a deterministic càdlàg function with a locally finite counting process concentrated on the nonnegative semiaxis. We establish sufficient conditions guaranteeing that a general shot-noise process properly normalized without centering weakly converges in the Skorokhod space. We present several examples of specific counting processes satisfying sufficient conditions and formulate the corresponding limit theorems. The present work continues our investigation originated in [Iksanov and Rashytov (2020)], where a functional limit theorem with centering was proved under the condition that the limit process is a Riemann–Liouville-type (Gaussian) process.