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