Is monotonic and convergent series bounded by the limit?

Asked Oct 19 '16 at 18:19

Active Oct 19 '16 at 18:39

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Problem: Suppose that $X_1\leq X_2\leq\cdots $ and $X_n \stackrel{P}{\to} X$, is the series bounded such that $X_n \leq X$ for all $n$?

Similar arguments are made in the first answer by Saz at Monotone increasing sequence of random variable that converge in probability implies convergence almost surely, but I cannot convince myself that monotonic and convergent series bounded by the limit. I know that there's a theorem stating that bounded and monotonic would lead to convergence but there's no theorem claiming that monotonic and convergence would lead to bounded.

Or how to show that: $$\left\{\omega: |X_k(\omega)-X(\omega)|>\epsilon\right\} \subset \left\{\omega: |X_n(\omega)-X(\omega)|>\epsilon\right\},$$ for all $k\geq n$.

edited Apr 13 '17 at 12:19

Community

asked Oct 19 '16 at 18:19

Sheldon

By monotonicity, you have pointwise convergence to a function $Y$, which may attain $+\infty$ as a value. Can you show that pointwise convergence to $Y$ and convergence in probability to $X$ imply that $X(\omega) = Y(\omega)$ almost everywhere? – Daniel Fischer Oct 19 '16 at 18:28
Intuitively, it is bounded. But how to show it is bounded at $X$? I just cannot find a theorem to rely on. – Sheldon Oct 19 '16 at 18:32
You don't generally have $X_n \leqslant X$ everywhere, only almost everywhere. – Daniel Fischer Oct 19 '16 at 18:36
Please refer to my new edit for a mathematical statement. – Sheldon Oct 19 '16 at 18:39
That need only hold modulo null sets. – Daniel Fischer Oct 19 '16 at 18:43

Is monotonic and convergent series bounded by the limit?

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