Theory of Statistics - George Mason University

854 Appendix C. Notation and Definitions
O(f(n)) The order class big O with respect to f(n).
g(n) ∈ O(f(n))
means there exists some fixed c such that g(n) ≤
cf(n) ∀n. In particular, g(n) ∈ O(1) means g(n) is
bounded.
In one special case, we will use O(f(n)) to represent
some unspecified scalar or vector x ∈ O(f(n)). This is
the case of a convergent series. An example is
s = f1(n) + · · · + fk(n) + O(f(n)),
where f1(n), . . ., fk(n) are finite constants.
We may also express the order class defined by convergence
as x → a as O(f(x))x→a (where a may be infinite).
Hence, g ∈ O(f(x))x→a iff
lim sup g(n)/f(n) < ∞.
x→a
o(f(n)) Little o; g(n) ∈ o(f(n)) means for all c > 0 there exists
some fixed N such that 0 ≤ g(n) < cf(n) ∀n ≥ N.
(The functions f and g and the constant c could all also
be negative, with a reversal of the inequalities.) Hence,
g(n) = o(f(n)) means g(n)/f(n) → 0 as n → ∞.
In particular, g(n) ∈ o(1) means g(n) → 0.
We also use o(f(n)) to represent some unspecified scalar
or vector x ∈ o(f(n)) in special case of a convergent
series, as above:
Spaces of Functions
s = f1(n) + · · · + fk(n) + o(f(n)).
We may also express this kind of convergence in the form
g ∈ o(f(x))x→a as x → a (where a may be infinite).
C k For an integer k ≥ 0, the class of functions whose derivatives
up to the k th derivative exist and are continuous.
L p For a real number p ≥ 1, the class of functions f on
a measure space (Ω, F, ν) with a metric · such that
f p dν < ∞.
Theory of Statistics c○2000–2013 James E. Gentle