Summary: We construct a process with gamma increments, which has a given convex autocorrelation function and asymptotically a self-similar limit. This construction validates the use of long-range dependent \(t\) and variance-gamma subordinator models for actual financial data as advocated in C. C. Heyde and N. N. Leonenko [Adv. Appl. Probab. 37, No. 2, 342–365 (2005; Zbl 1081.60035)] and R. Finlay and E. Seneta [J. Appl. Probab. 43, No. 2, 441–453 (2006; Zbl 1103.62103)], in that it allows for noninteger-valued model parameters to occur as found empirically by data fitting.