Let $X_1, X_2, \ldots$ be iid random variables with bounded second moment. The question is to calculate the exact value of $$\mathbb{E} \max_{1 \le j < \infty} \frac{X_1 + \cdots + X_j}{j}.$$

Is there any principled approach to calculate or approximate this expectation in the literature? If not, can we still do something for some special distiributions of $X_1$?


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