Integrating

The results of this section are presented as a series of theorems leading quickly to the estimators of the moments of the entropy, presented in the next section. Note that the conditions of existence of the results are given in full generality in the complex domain, although the actual event counts are non-negative integers.

Define the Laplace convolution operator  by its operation on two functions

. If  then

Proof: Define . Define . Define . The integral in the statement of the theorem, after substituting for  and , becomes

Integrating over  the integral takes the form

The inner integral may be written as a convolution, and noting that  must be positive we find

Clearly, this can be extended by induction and the associativity of the convolution operator to give the desired result. QED.

For the following, the Laplace transform operator L and its inverse are discussed in appendix 9.6.3.

. (Laplace convolution) If  exists for , then

Proof: The proof is lengthy, but may be found in [36]. The result is mentioned in [50, 54].

Define the Gamma function ,  by

. If , , then

Proof: By theorem 1

By theorem 2

Note that

Substitute from equation 9.27 in the right side of equation 9.26. Finally use equation 9.27 again to take the inverse. Setting  finishes the proof. QED.

. If , , then

Proof: Note that . The interchange of derivative and integral is given in appendix 9.7.

In using theorem 4 we will take the derivatives in the expression on the right side of equation 9.28 where N appears in  (see equation 9.24); thus note that since , we have .

Define the polygamma function

(see [3] for properties of the polygamma function), and define

Introduce

. For , ,

Proof: By theorems 3 and 4

Taking the derivative and substituting for the  function yields the result. QED.

. For , ,

(6.1)

(6.2)

Proof: Apply theorems 3 and 4 in a manner similar to the proof of theorem 5. Here two derivatives are needed.

Tue Mar 25 08:11:49 CST 1997