In article <1155970245.104912.16680@m79g2000cwm.googlegroups.com>,
"Luna Moon" <lunamoonmoon@gmail.com> wrote:

Hi guys,
I've just found that this "positive definite function" requirement for
a function to be a probability characteristic function is not a
constructive and operational one and it is hard to use it to construct
characteristic functions... Suppose I want to create some function that
is characteristic function, what guidelines should I follow to make the
function "postive definite"?

There is one simple sufficient condition ("polya's theorem") for a
function to be a characteristic function, which leads to many examples.
If f:[0,infinity) --> R is continuous and convex, with f(0) = 1, then
g(t) := f(|t|) is a characteristic function.

Luna Moon wrote:

Hi guys,

I've just found that this "positive definite function" requirement

for

a function to be a probability characteristic function is not a
constructive and operational one and it is hard to use it to

construct

characteristic functions... Suppose I want to create some function
that is characteristic function, what guidelines should I follow to
make the function "postive definite"?

Thanks again!

Luna

You could start from the methods commonly used for constructing new
distributions when working from the "probability distribution" side of
the question and see if any of these carry over in a useful way. Two
obvious possibilities are:

(i) construction by addition ... the distribution function of the sum
of two independent random variables is given by the convolution of the
pair of distribution functions. This is quite complicated from the
"probability distribution" side, but rather simpler for characteristic
functions, since you just need the product of the characteristic
functions. This can be extended to the product of more than two and,
for appropriate conditions (weightings of the underlying rv's) to
infinite products.

(ii) construction by compounding .. this is an extension of simple
finite mixtures, which I assume are an obvious possibility (giving a
weighted sum of the characteristic functions). Write the distribution
function of a rv in terms of a number of parameters and let some or
all of these parameters be selected at random from given probability
distributions. Then the distribution function or characteristic
function of the final random variable is the integral of the original
distribution function or characteristic function (as a function of the
parameters) over the distribution of the parameters.

For example, the above possiblities lead to different ways of
combining an "ordinary" distribution such as the Normal, with an
unusual one such as a stable distribution.

You could also try consulting a table or listing of known
characteristic functions to see if there is a family that suits your
needs. You can find some in books which are a collection of tables,
but one separate table is available as:
Oberhettinger F (1973) Fourier Transforms of Distributions and Their
Inverses: a collection of Tables. Academic Press, NY.

Before you go too far with this, you should consider whether you need
to consider including the multivariate case in what you are trying to
do, as this may infulence some of your decisions.