Empirical studies of the high frequency data in stock markets show that the time series of trade signs or signed volumes has a long memory property.
In this paper, we present a discrete time stochastic process for polymer model which describes trader's trading strategy, and show that a scale limit of the process converges to superposition of fractional Brownian motions with Hurst exponents and Brownian motion, provided that the index gamma of the time scale about the trader's investment strategy coincides with the index delta of the interaction range in the discrete time process. The main tool for the investigation is the method of cluster expansion developed in the mathematical study of statistical mechanics.