Constructing processes with prescribed mixing coefficients

The rate at which dependencies between future and past observations decay in a random process may be quantified in terms of mixing coefficients. The latter in turn appear in strong laws of large numbers and concentration of measure inequalities for dependent random variables. Questions regarding what rates are possible for various notions of mixing have been posed since the 1960's, and have important implications for some open problems in the theory of strong mixing conditions. This paper deals with [eta]-mixing, a notion defined in [Kontorovich, Leonid, Ramanan, Kavita, 2008. Concentration inequalities for dependent random variables via the martingale method. Ann. Probab. (in press)], which is closely related to [phi]-mixing. We show that there exist measures on finite sequences with essentially arbitrary [eta]-mixing coefficients, as well as processes with arbitrarily slow mixing rates.