International Statistical Review (2011), 79, 2, 255–271 doi:10.1111/j.1751-5823.2011.00148.x Autocorrelation Functions Richard Finlay 1 , Thomas Fung 2, * and Eugene Seneta 1 1 School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia E-mails: richardf@maths.usyd.edu.au, eseneta@maths.usyd.edu.au 2 Department of Statistics, Macquarie University, NSW 2109, Australia ∗ Honorary Associate, University of Sydney E-mail: thomas.fung@mq.edu.au Summary We explain the connection between autocorrelation functions of stationary continuous time processes and real characteristic functions, and review sufficient conditions for a function to be an autocorrelation function. We also give probabilistic constructions for time series reformulations of P´ olya’s (1949) Theorem on characteristic functions, and Young’s (1913) classical theorem in Fourier analysis. Our constructions allow the marginal distribution of the process to be any infinitely divisible distribution with finite variance. Key words: Autocorrelation function; real characteristic function; stationary process; infinitely divisible distribution; normal variance-mixing. 1 Introduction An important problem in time series analysis in both discrete and continuous time is to see whether a particular real-valued function can be used as an autocorrelation function of a stationary process. In general, for a real continuous function ρ (s), s ∈ R with ρ (0) = 1, to be a permissible autocorrelation function, it is necessary and sufficient for ρ (s) to satisfy the non-negative definiteness condition (for instance see Feller, 1966, section XIX.3): for every choice of finitely many real numbers t 1 , ... , t n and complex numbers z 1 , ... , z n ,  j ,k ρ (t j − t k )z j ¯ z k ≥ 0. (1) However, this condition is difficult to check in practice. The aim of this note is to provide a brief review of the more practical sufficient conditions used in the literature in both discrete and continuous time. We begin by explaining the connection between autocorrelation functions of second-order stationary continuous time processes and real characteristic functions. (The interrelationship between complex-valued characteristic functions and autocorrelations of complex-valued random processes is implicit in Feller, 1966, sections XIX.2 and XIX.3.) We then examine some known sufficient conditions for a real function to be an autocorrelation function, and discuss little-known links of such functions with the probabilistic concepts of characteristic functions, distribution theory, and time series analysis, as well as with Fourier analysis. C  2011 The Authors. International Statistical Review C  2011 International Statistical Institute. Published by Blackwell Publishing Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.