Y is a non-negative random variable, not necessary integrable. How to show
$$\lim_{n\rightarrow \infty }\frac{1}{n}\int_{Y
My idea is to find a good upper bound that converges to zero. However, I didn't find a good one. My upper bound is
$$\frac{1}{n}\int_{Y
Answer
Note that the sequence of random variables Xn=1nI{Y<n}Y is bounded by 1 and converges pointwise to 0. The assertion now follows from the dominated convergence theorem.
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