An Expectile Strong Law of Large Numbers

We show that Kolmogorov's classical strong law of large numbers applies to all expectiles uniformly. The expectiles of a random sample converge almost surely (uniformly) to the true expectiles if and only if the true data generating process has a finite first moment. The result holds for expectile functions of scalar and vector-valued random variables and can be reformulated to state that the mean (or any expectile) of a random sample converges almost surely to the true mean (or expectile) if and only if any arbitrary expectile exists and is finite