Moderate Laws of Large Numbers via Weak Laws

By a $moderate$ $law$ $of$ $large$ $numbers$ we mean any theorem whose conclusion includes the $L^{p}$-vanishment of the sequence of the sample means of some centered random variables with $1 \leq p < +\infty$ given.Given any $1 \leq p < +\infty$ and any $\eps > 0$,we prove a moderate law of large numbers for $L^{p+\eps}$-bounded random variables that obey a weak law.Thus our moderate laws in particular complement those obtained from the martingale theory,and establish the counterintuitive fact that (for$L^{p+\eps}$-bounded random variables) where there is a weak law there is a moderate law.