Bounded in Probability

Definition

Let

• $\{X_n\}$ : Sequence of Random variables

If

$$\forall ϵ>0,\exists B_{ϵ}>0,N_{ϵ}\in \mathbb{N}\ni n\ge N_{\epsilon }⟹P[|X_n|\le B_{\epsilon }]\ge 1-ϵ$$

Then we say $\{X_n\}$ is bounded in probability

Remark

Notice the resemblence with definition of bounded sequence.

Related theorems

• Theorem 5.2.6 : If $X_n\stackrel{D}{\to }X$