  We propose a stochastic process driven by memory effect with novel
distributions including both exponential and leptokurtic heavy-tailed
distributions. A class of distribution is analytically derived from the
continuum limit of the discrete binary process with the renormalized
auto-correlation and the closed form moment generating function is obtained,
thus the cumulants are calculated and shown to be convergent. The other class
of distributions are numerically investigated. The concoction of the two
stochastic processes of the different signs of memory under regime switching
mechanism does incarnate power-law decay behavior, which strongly implies that
memory is the alternative origin of heavy-tail.
