Local limit theorems for complex valued sequences: Old & New

In probability theory, local limit theorems provide an asymptotic expansion of the convolution powers of a probability distribution supported on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℤ</mml:mi> </mml:math> with uniform bounds on the remainders. In this review, we present some recent results for the iterated convolution of complex valued integrable sequences in one space dimension. In the so-called parabolic case, we give a complete expansion, at any accuracy order, for these convolution powers and we provide sharp, pointwise, generalized Gaussian bounds for the remainders. We also present an extension of our main result to the semi-discrete setting (time-continuous convolution problems), and discuss several natural perspectives.