Big O in probability notation

The order in probability notation is used in probability theory and statistical theory in direct parallel to the big-O notation which is standard in mathematics. Where the big-O notation deals with the convergence of sequences or sets of ordinary numbers, the order in probability notation deals with convergence of sets of random variables, where convergence is in the sense of convergence in probability.^[1]

For a set of random variables X_n and a corresponding set of constants a_n (both indexed by n which need not be discrete), the notation

$X_n=o_p(a_n) \,$

means that the set of values X_n/a_n converges to zero in probability as n approaches an appropriate limit. Equivalently, X_n = o_p(a_n) can be written as a_n o_p(1) where X_n = o_p(1) is defined as,

$\lim_{n \to \infty} P(|X_n| \geq \varepsilon) = 0,$

for every positive ε^[2].

The notation

$X_n=O_p(a_n) \,$

means that the set of values X_n/a_n is bounded in the limit in probability.

References

1. ^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN 0-19-920613-9
2. ^ Yvonne M. Bishop, Stephen E. Fienberg, Paul W. Holland. (1975,2007) Discrete multivariate analysis, Springer. ISBN 0387728058, ISBN 9780387728056