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Let X be an observable random variable with unknown distribution function F(x)=P(X≤x), −∞, and let θ=sup{r≥0:E|X|r<∞}. We call θ the power of moments of the random variable X. Let X1,X2,…,Xn be a random sample of size n drawn from F(⋅). In this paper we propose the following simple point estimator of θ and investigate its asymptotic properties: θˆn=lognlogmax1≤k≤n|Xk|, where logx=ln(e∨x), −∞. In particular, we show that θˆn→Pθif and only iflimx→∞xrP(|X|>x)=∞∀r>θ. This means that, under very reasonable conditions on F(⋅), θˆn is actually a consistent estimator of θ. [ABSTRACT FROM AUTHOR]