Logarithmically Concave Sequence on:
[Wikipedia]
[Google]
[Amazon]
In mathematics, a sequence = of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if holds for .
''Remark:'' some authors (explicitly or not) add two further conditions in the definition of log-concave sequences:
* is non-negative
* has no internal zeros; in other words, the Support (mathematics), support of is an interval of .
These conditions mirror the ones required for Logarithmically_concave_function, log-concave functions.
Sequences that fulfill the three conditions are also called Pólya Frequency sequences of order 2 (PF2 sequences). Refer to chapter 2 of for a discussion on the two notions. For instance, the sequence satisfies the concavity inequalities but not the internal zeros condition.
Examples of log-concave sequences are given by the binomial coefficients along any row of Pascal's triangle and the Newton's inequalities, elementary symmetric means of a finite sequence of real numbers.
In mathematics, a sequence = of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if holds for .
''Remark:'' some authors (explicitly or not) add two further conditions in the definition of log-concave sequences:
* is non-negative
* has no internal zeros; in other words, the Support (mathematics), support of is an interval of .
These conditions mirror the ones required for Logarithmically_concave_function, log-concave functions.
Sequences that fulfill the three conditions are also called Pólya Frequency sequences of order 2 (PF2 sequences). Refer to chapter 2 of for a discussion on the two notions. For instance, the sequence satisfies the concavity inequalities but not the internal zeros condition.
Examples of log-concave sequences are given by the binomial coefficients along any row of Pascal's triangle and the Newton's inequalities, elementary symmetric means of a finite sequence of real numbers.
References
*See also
*Unimodality *Logarithmically concave function *Logarithmically concave measure Sequences and series {{combin-stub