Shot noise, weak convergence and diffusion approximations

Shot noise processes have been extensively studied due to their mathematical properties and relevance in several applications. Here, we consider nonnegative shot prove weak convergence Lévy-driven Ornstein–Uhlenbeck (OU) process, whose features depend on the underlying jump distributions. Among others, obtain OU-Gamma OU-Inverse Gaussian processes, having gamma inverse gaussian as background Lévy respectively. Then, derive necessary conditions guaranteeing diffusion limit a OU show that they are not met unless allowing for negative jumps happening with probability going zero, quantify error occurred when replacing process non-Gaussian processes. The results offer new class of models be used instead commonly applied approximate synaptic input currents, membrane voltages or conductances modelled by single neuron modelling.