# 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 that 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]}

## Contents

## DefinitionsEdit

### Small O: convergence in probabilityEdit

For a set of random variables *X _{n}* and a corresponding set of constants

*a*(both indexed by

_{n}*n*, which need not be discrete), the notation

means that the set of values *X _{n}*/

*a*converges to zero in probability as

_{n}*n*approaches an appropriate limit. Equivalently,

*X*

_{n}= o

_{p}(

*a*

_{n}) can be written as

*X*

_{n}/

*a*

_{n}= o

_{p}(1), where

*X*

_{n}= o

_{p}(1) is defined as,

for every positive ε.^{[2]}

### Big O: stochastic boundednessEdit

The notation,

means that the set of values *X _{n}*/

*a*is stochastically bounded. That is, for any ε > 0, there exists a finite M > 0 and a finite N > 0 such that,

_{n}### Comparison of the two definitionsEdit

The difference between the definition is subtle. If one uses the definition of the limit, one gets:

- Big O
_{p}(1): - Small o
_{p}(1):

The difference lies in the δ: for stochastic boundedness, it suffices that there exists one (arbitrary large) δ to satisfy the inequality, and δ is allowed to be dependent on ε (hence the δ_{ε}). On the other side, for convergence, the statement has to hold not only for one, but for any (arbitrary small) δ. In a sense, this means that the sequence must be bounded, with a bound that gets smaller as the sample size increases.

This suggests that if a sequence is o_{p}(1), then it is O_{p}(1), i.e. convergence in probability implies stochastic boundedness. But the reverse does not hold.

## ExampleEdit

If is a stochastic sequence such that each element has finite variance, then

(see Theorem 14.4-1 in Bishop et al.)

If, moreover, is a null sequence for a sequence of real numbers, then converges to zero in probability by Chebyshev's inequality, so

- .

## ReferencesEdit

**^**Dodge, Y. (2003)*The Oxford Dictionary of Statistical Terms*, OUP. ISBN 0-19-920613-9**^**Yvonne M. Bishop, Stephen E.Fienberg, Paul W. Holland. (1975,2007)*Discrete multivariate analysis*, Springer. ISBN 0-387-72805-8, ISBN 978-0-387-72805-6