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 Heyde and Leonenko (2005) and Finlay and Seneta (2006), in that it allows for noninteger-valued model parameters to occur as found empirically by data fitting.