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]
For a set of random variables Xn and corresponding set of constants an (both indexed by n, which need not be discrete), the notation
Xn=op(an)
means that the set of values Xn/an converges to zero in probability as n approaches an appropriate limit.Equivalently, Xn = op(an) can be written as Xn/an = op(1),i.e.
\limnP\left[\left|
Xn | |
an |
\right|\geq\varepsilon\right]=0,
for every positive ε.[2]
The notation
Xn=Op(an)asn\toinfty
P\left(| | Xn |
an |
|>M\right)<\varepsilon, \forall n>N.
The difference between the definitions is subtle. If one uses the definition of the limit, one gets:
Op(1)
\forall\varepsilon \existsN\varepsilon,\delta\varepsilon suchthatP(|Xn|\geq\delta\varepsilon)\leq\varepsilon \foralln>N\varepsilon
op(1)
\forall\varepsilon,\delta \existsN\varepsilon,\delta suchthatP(|Xn|\geq\delta)\leq\varepsilon \foralln>N\varepsilon,
The difference lies in the
\delta
\delta
\delta
\varepsilon
\delta\varepsilon
\delta
This suggests that if a sequence is
op(1)
Op(1)
If
(Xn)
Xn-E(Xn)=Op\left(\sqrt{\operatorname{var}(Xn)}\right)
If, moreover,
-2 | |
a | |
n |
\operatorname{var}(Xn)=
-1 | |
\operatorname{var}(a | |
n |
Xn)
(an)
-1 | |
a | |
n |
(Xn-E(Xn))
Xn-E(Xn)=op(an).