# $X_n\leq Y_n+o_p(1)$ or $X_n\geq Y_n+o_p(1)$ imply $X_n$ converges to $Y_n$

Consider the sequences of real-valued random variables $\{X_n\}_{\forall n \in \mathbb{N}}, \{Y_n\}_{\forall n \in \mathbb{N}}$. A book that I'm reading makes the following claims whose interpretation is not clear to me.

Case 1: $X_n\leq Y_n+o_p(1)$, i.e. $X_n$ converges to $Y_n$.

Case 2: $X_n\geq Y_n+o_p(1)$, i.e. $X_n$ converges to $Y_n$.

I guess that $o_p(1)$ is a sequence converging in probability to zero as $n\rightarrow \infty$.

I would like your help to clarify firstly what does it mean $X_n\leq Y_n+o_p(1)$ (and, symmetrically, $X_n\geq Y_n+o_p(1)$), why they imply convergence, and which type of convergence.

• $o_p(1)$ needs to be defined (by the author). – herb steinberg May 3 '18 at 0:30
• It is not defined unfortunately. I think it is a sequence converging in probability to zero. – TEX May 3 '18 at 7:09