Definition

Extension of Big O Notation for Random Variable

Let ${r_{n}} \in (0, \infty)$ be the rate of convergence, and ${X_{n}}$ be a Sequence of random variables, then $X_{n} = o_{P} (r_{n}) \Leftrightarrow n \to \infty lim \frac{X _{n}}{r _{n}} \to P 0 \Leftrightarrow \forall ϵ, M > 0, \exists N \in N, \forall n \geq N, P (\frac{X _{n}}{r _{n}} \geq M) \leq ϵ$

$X_{n} = O_{P} (r_{n}) \Leftrightarrow \forall ϵ > 0, \exists M \in R^{+}, \exists N \in N, \forall n \geq N, P (\frac{X _{n}}{r _{n}} \geq M) \leq ϵ$

Facts

If $X_{n} = O_{P} (1)$ , then ${X_{n}}$ is called Boundedness in Probability

$X_{n} \to D X \Rightarrow X_{n} = O_{P} (1)$

$(X_{n} = O_{P} (1)) \land (Y_{n} \to P 0) \Rightarrow X_{n} Y_{n} \to P 0$

$(Y_{n} = O_{P} (1)) \land (X_{n} = o_{P} (Y_{n})) \Rightarrow X_{n} \to P 0$

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Big O in Probability Notation

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