O, o Notation

O means “order of” and o means “of lower order than.” If {u _n } and {v _n } are two sequences such that |u _n /v _n | < K for sufficiently large n, where K is a constant independent of n, then u _n = O(v _n ); for example, (2n − 1)/(n² + 1) = O(1/ n). The symbol O (colloquially called “big O”) also extends to the case of functions of a continuous variable; for example, (x + 1) = O(x). O(1) denotes any function that is defined for all values of x sufficiently large, and which either has a finite limit as x tends to infinity, or at least for all sufficiently large values of x remains less in absolute value than some fixed bound; for example, sin x = O(1).

If lim_n−>∞u _n /v _n = 0, then u _n = o(v _n ) (colloquially called “little o”); for example, log n = o(n), where again the notation extends to functions of a continuous variable; for example, sin x = o(x). Furthermore, u _n = o(1) means that u _n tends to 0 as n tends to infinity; for example, (log n)/n = o(1). In probability modeling (e.g.,...