Let \(\{X_i:i\geq 1\}\) be i.i.d. integrable random variables. Prove that \[ \lim _{n\to \infty } \frac {1}{\log n}\sum _{i=1}^{n}\frac {X_i}{i} = \mathbb {E}[X_1]. \]
Question
Math 6010 • Linear Models
Fall 2018 • Question 4
Question
Fall 2018 • Question 4
Let \(\{X_i:i\geq 1\}\) be i.i.d. integrable random variables. Prove that \[ \lim _{n\to \infty } \frac {1}{\log n}\sum _{i=1}^{n}\frac {X_i}{i} = \mathbb {E}[X_1]. \]