On the principle of minimum growth rate in multiplicatively interacting stochastic processes

A method of moment inequalities is used to derive the principle of minimum growth rate in multiplicatively interacting stochastic processes(MISPs). When a value of a power-law exponent at the tail of probability distribution function exists in a range $0 < s \le 1$, a first-order moment diverges and an equality for a growth rate of systems breaks down. From the estimate of inequalities, we newly find a conditional inequality which determines the growth rate, and then the exponent in $0 < s \le 1$.