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Let $X, Y$ be two non-negative random variables without first moment, i.e. $\mathbb{E}[X] = \infty = \mathbb{E}[Y]$. Suppose for all $t > 0$ sufficiently large we have the following inequality for truncated first moments:

$$ \mathbb{E}\left[X 1_{\{X \le t \}}\right] \le \mathbb{E}\left[Y 1_{\{Y \le t \}}\right].$$

My question is: is it true that there exists $C \ge 1$ such that $$ \mathbb{P}(X > t) \le C \mathbb{P}(Y > t) $$ for all $t$ sufficiently large?

(I would expect this to hold for $C = 1$ but I do not have a clue.)

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