$Existence of a Non-negative Integrable Random Variable with Supremum-Constrained Survival Function$

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$Existence of a Non-negative Integrable Random Variable with Supremum-Constrained Survival Function$

Let $( X_1,\dots,X_n,\dots )$ be a sequence of non-negative integrable random variables, Denote by $S_i(t) = \mathbb{P}(X_i \geq t)$, the survival function of $X_i, i \ge 1$

Does there exist a non-negative and integrable random variable X with survival function S, such that $ \sup_nS_n(t) \leq S(t), \, \forall t \ge 0$?

Probability Measure Theory

Tpolo

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Maybe I'm not understanding the question, but...

Let $X_n = n$, that is, the function that is constantly equal to $n$ on the whole domain. Since the domain (say, $\Omega$) is a probability space, it has measure one, so $\int_\Omega X_n d \mu = n$ so $X_n$ is integrable.

With this choice, $S_n(t) = 1$ if $t \leq n$, and $S_n(t) = 0$ otherwise. In particular, $\sup_n S_n(t) = 1$ for each $t$, so $S(t) = 1$ everywhere (as any survival function $S$ must satisfy $S(t) \leq 1$ everywhere).

So if $S$ is the survival function of some random variable $X$, then $\mathbb{P}(X \geq t) = 1$ for all $t$, which is impossible. It follows that no such random variable can exist.

Erdos

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$Existence of a Non-negative Integrable Random Variable with Supremum-Constrained Survival Function$

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