xbeta
Stochastic order

Absolute stochastic order

A sequence of random variables $\{X_n\}$ is bounded in probability if for any $\epsilon >0$ there exists a number $B_{\epsilon }<\mathrm{\infty }$ such that ${\sup }_n\mathrm{Pr}(\mid X_n\mid >B_{\epsilon })<\epsilon$. In this case we write $X_n=O_p(1)$.

A sequence of random variables $\{X_n\}$ converges to 0 in probability if for any $\epsilon >0$, ${\lim }_{n\to \mathrm{\infty }}\mathrm{Pr}(\mid X_n\mid >0)=0$. In this case we write $X_n=o_p(1)$.

Source: Davidson (1994, p. 187).

Relative stochastic order

Let $\{Y_n\}$ be another sequence, stochastic or deterministic. If $X_n/Y_n=O_p(1)$, we write $X_n=O_p(Y_n)$ and say that $X_n$ is at most of order $Y_n$ in probability. If $X_n/Y_n=o_p(1)$, we write $X_n=o_p(Y_n)$ and say that $X_n$ is of order less than $Y_n$ in probability.

Source: Davidson (1994, p. 187).

Notes

• The use of $O_p(1)$ and $o_p(1)$ for stochastic order is due to Mann and Wald (1943) (Source: Davidson (1994, p. 187)).

References

• Davidson, James (1994): Stochastic Limit Theory, Oxford University Press.

• Mann, H.B. and A. Wald (1943): ”On stochastic limit and order relationships,” Annals of Mathematical Statistics 14, 217-226.