Properties
- The characteristic function of a real-valued random variable always exists, since it is an integral of a bounded continuous function over a space whose measure is finite.
- A characteristic function is uniformly continuous on the entire space
- It is non-vanishing in a region around zero: φ(0) = 1.
- It is bounded: |φ(t)| ≤ 1.
- It is Hermitian: φ(−t) = φ(t). In particular, the characteristic function of a symmetric (around the origin) random variable is real-valued and even.
- There is a bijection between distribution functions and characteristic functions. That is, for any two random variables X1, X2
- If a random variable X has moments up to k-th order, then the characteristic function φX is k times continuously differentiable on the entire real line. In this case
- If a characteristic function φX has a k-th derivative at zero, then the random variable X has all moments up to k if k is even, but only up to k – 1 if k is odd.
- If X1, …, Xn are independent random variables, and a1, …, an are some constants, then the characteristic function of the linear combination of Xi's is
One specific case would be the sum of two independent random variables X1 and X2 in which case one would have .
- The tail behavior of the characteristic function determines the smoothness of the corresponding density function.
Read more about this topic: Characteristic Function (probability Theory)
Famous quotes containing the word properties:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)