Expected Logarithm and Negative Integer Moments of a Noncentral χ²-Distributed Random Variable ^†

Abstract

Closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral ${\chi }^2$-distributed random variable are presented in the cases of both even and odd degrees of freedom. Moreover, some basic properties of these expectations are derived and tight upper and lower bounds on them are proposed.

1. Introduction

The noncentral ${\chi }^2$ distribution is a family of probability distributions of wide interest. They appear in situations where one or several independent Gaussian random variables (RVs) of equal variance (but potentially different means) are squared and summed together. The noncentral ${\chi }^2$ distribution contains as special cases among others the central ${\chi }^2$ distribution, the exponential distribution (which is equivalent to a squared Rayleigh distribution), and the squared Rice distribution.

In this paper, we present closed-form expressions for the expected logarithm and for arbitrary negative integer moments of a noncentral ${\chi }^2$-distributed RV with even or odd degrees of freedom. Note that while the probability density function (PDF), the moment-generating function (MGF), and the moments of a noncentral ${\chi }^2$-distributed RV are well-known, the expected logarithm and the negative integer moments have only been derived relatively recently for even degrees of freedom [1,2,3,4,5,6], but—to the best of our knowledge—for odd degrees of freedom they have been completely unknown so far. These expectations have many interesting applications. So, for example, in the field of information theory, there is a close relationship between the expected logarithm and entropy, and thus the expected logarithm of a noncentral ${\chi }^2$-distributed RV plays an important role, e.g., in the description of the capacity of multiple-input, multiple-output noncoherent fading channels [1,2]. Many more examples in the field of information theory can be found in [7].

We will see that the expected logarithm and the negative integer moments can be expressed using two families of functions $g_m(·)$ and $h_n(·)$ that will be defined in Section 3. Not unexpectedly, $g_m(·)$ and $h_n(·)$ are not elementary, but contain special functions like the exponential integral function ([8], Sec. 8.21), the imaginary error function [9], or a generalized hypergeometric function ([8], Sec. 9.14). While numerically this does not pose any problem as the required special functions are commonly implemented in many mathematical programming environments, working with them analytically can be cumbersome. We thus investigate $g_m(·)$ and $h_n(·)$ more in detail, present important properties, and derive tight elementary upper and lower bounds to them.

The structure of this paper is as follows. After a few comments about our notation, we will formally define the noncentral ${\chi }^2$ distribution in the following Section 2 and also state some fundamental properties of the expected logarithm and the negative integer moments. In Section 3 we present the two families of functions $g_m(·)$ and $h_n(·)$ that are needed for our main results in Section 4. Section 5 summarizes properties of $g_m(·)$ and $h_n(·)$, and Section 6 presents tight upper and lower bounds on them. Many proofs are deferred to the appendices.

We use upper-case letters to denote random quantities, e.g., U, and the corresponding lower-case letter for their realization, e.g., u. The expectation operator is denoted by $\mathsf{E}\left[·\right]$; $ln(·)$ is the natural logarithm; $\Gamma (·)$ denotes the Gamma function ([8], Sec. 8.31–8.33); and $\mathsf{i}$ is the imaginary number $\mathsf{i}\triangleq \sqrt{-1}$. We use ${\mathbb{N}}^{\mathrm{even}}$ to denote the set of all even natural numbers:

$$\begin{array}{c}\hfill {\mathbb{N}}^{\mathrm{even}}\triangleq \{2,4,6,8,\dots \}.\end{array}$$

(1)

Accordingly, ${\mathbb{N}}^{\mathrm{odd}}\triangleq \mathbb{N}\backslash {\mathbb{N}}^{\mathrm{even}}$ is the set of all odd natural numbers.

For a function $\xi \mapsto f(\xi )$, $f^{(\ell )}(\xi )$ denotes its ${\ell }^{th}$ derivative:

$$\begin{array}{c}\hfill f^{(\ell )}(\xi )\triangleq \frac{{\mathrm{d}}^{\ell }f(\xi )}{\mathrm{d}{\xi }^{\ell }}.\end{array}$$

(2)

The real Gaussian distribution of mean $\mu \in \mathbb{R}$ and variance ${\sigma }^2>0$ is denoted by ${\mathcal{N}}_{\mathbb{R}}\left(\mu ,{\sigma }^2\right)$, while ${\mathcal{N}}_{\mathbb{C}}\left(\eta ,{\sigma }^2\right)$ describes the complex Gaussian distribution of mean $\eta \in \mathbb{C}$ and variance ${\sigma }^2>0$. Thus, if $X_1$ and $X_2$ are independent standard Gaussian RVs, $X_1,X_2\sim {\mathcal{N}}_{\mathbb{R}}\left(0,1\right)$, $X_1\perp \perp X_2$, then

$$\begin{array}{c}\hfill Z\triangleq \frac{1}{\sqrt{2}}{X}_1+\mathsf{i}\frac{1}{\sqrt{2}}{X}_2\end{array}$$

(3)

is circularly symmetric complex Gaussian, $Z\sim {\mathcal{N}}_{\mathbb{C}}\left(0,1\right)$.

2. The Noncentral ${\chi }^2$ Distribution

Definition 1.

For some $n\in \mathbb{N}$, let ${\left\{X_k\right\}}_{k=1}^n$ be independent and identically distributed (IID) real, standard Gaussian RVs, $X_k\sim {\mathcal{N}}_{\mathbb{R}}\left(0,1\right)$, let ${\left\{{\mu }_k\right\}}_{k=1}^n\in \mathbb{R}$ be real constants, and define

$$\begin{array}{c}\hfill \tau \triangleq \sum _{k=1}^n{\mu }_k^2.\end{array}$$

(4)

Then the nonnegative RV

$$\begin{array}{c}\hfill U\triangleq \sum _{k=1}^n{(X_k+{\mu }_k)}^2\end{array}$$

(5)

is said to have a noncentral ${\chi }^2$ distribution with n degrees of freedom and noncentrality parameter $\tau$. Note that the distribution of U depends on the constants $\left\{{\mu }_k\right\}$ only via the sum of their squares (4). The corresponding PDF is ([10], Ch. 29)

$$\begin{array}{c}\hfill p_U(u)=\frac{1}{2}{\left(\frac{u}{\tau }\right)}^{\frac{n-2}{4}}{e}^{-\frac{\tau +u}{2}}{I}_{n/2-1}\left(\sqrt{\tau u}\right),u\ge 0,\end{array}$$

(6)

where $I_{\nu }(·)$ denotes the modified Bessel function of the first kind of order $\nu \in \mathbb{R}$ ([8], Eq. 8.445):

$$\begin{array}{c}\hfill I_{\nu }(x)\triangleq \sum _{k=0}^{\infty }\frac{1}{k!\Gamma (\nu +k+1)}{\left(\frac{x}{2}\right)}^{\nu +2k},x\ge 0.\end{array}$$

(7)

For $\tau =0$ we obtain the central ${\chi }^2$ distribution, for which the PDF (6) simplifies to

$$\begin{array}{c}\hfill p_U(u)=\frac{1}{2^{\frac{n}{2}}\Gamma \left(\frac{n}{2}\right)}{u}^{\frac{n}{2}-1}{e}^{-\frac{u}{2}},u\ge 0.\end{array}$$

(8)

Note that in this work any RV U will always be defined as given in (5). Sometimes we will write $U^{[n,\tau ]}$ to clarify the degrees of freedom n and the noncentrality parameter $\tau$ of U.

If the number of degrees of freedom n is even (i.e., if $n=2m$ for some natural number m), there exists a second, slightly different definition of the noncentral ${\chi }^2$ distribution that is based on complex Gaussian random variables.

Definition 2.

For some $m\in \mathbb{N}$, let ${\left\{Z_j\right\}}_{j=1}^m$ be IID $\sim {\mathcal{N}}_{\mathbb{C}}\left(0,1\right)$, let ${\left\{{\eta }_j\right\}}_{j=1}^m\in \mathbb{C}$ be complex constants, and define

$$\begin{array}{c}\hfill \lambda \triangleq \sum _{j=1}^m{\left|{\eta }_j\right|}^2.\end{array}$$

(9)

Then the nonnegative RV

$$\begin{array}{c}\hfill V\triangleq \sum _{j=1}^m{\left|Z_j+{\eta }_j\right|}^2\end{array}$$

(10)

is said to have a noncentral ${\chi }^2$ distribution with $2m$ degrees of freedom and noncentrality parameter $\lambda$. It has a PDF

$$\begin{array}{c}\hfill p_V(v)={\left(\frac{v}{\lambda }\right)}^{\frac{m-1}{2}}{e}^{-v-\lambda }{I}_{m-1}\left(2\sqrt{\lambda v}\right),v\ge 0,\end{array}$$

(11)

which in the central case of $\lambda =0$ simplifies to

$$\begin{array}{c}\hfill p_V(v)=\frac{1}{\Gamma (m)}{v}^{m-1}{e}^{-v},v\ge 0.\end{array}$$

(12)

Note that in this work any RV V will always be defined as given in (10). Sometimes we will write $V^{[m,\lambda ]}$ to clarify the degrees of freedom $2m$ and the noncentrality parameter $\lambda$ of V.

Lemma 1.

Let $n\in {\mathbb{N}}^{\mathrm{even}}$ be an even natural number and $\tau \ge 0$ a nonnegative constant. Then

$$\begin{array}{c}\hfill U^{[n,\tau ]}\stackrel{\mathcal{L}}{=}2V^{[n/2,\tau /2]},\end{array}$$

(13)

where “$\stackrel{\mathcal{L}}{=}$” denotes equality in probability law.

Proof. 

Let ${\left\{X_k\right\}}_{k=1}^n$ be IID $\sim {\mathcal{N}}_{\mathbb{R}}\left(0,1\right)$ and ${\left\{{\mu }_k\right\}}_{k=1}^n\in \mathbb{R}$ as given in Definition 1. Define $m\triangleq n/2\in \mathbb{N}$ and for all $j\in \{1,\dots ,m\}$,

$$\begin{array}{ccc}\hfill Z_j& \triangleq & \frac{1}{\sqrt{2}}{X}_{2j-1}+\mathsf{i}\frac{1}{\sqrt{2}}{X}_{2j},\hfill \end{array}$$

(14)

$$\begin{array}{ccc}\hfill {\eta }_j& \triangleq & \frac{1}{\sqrt{2}}{\mu }_{2j-1}+\mathsf{i}\frac{1}{\sqrt{2}}{\mu }_{2j}.\hfill \end{array}$$

(15)

Then

$$\begin{array}{ccc}\hfill \lambda & \triangleq & \sum _{j=1}^m{\left|{\eta }_j\right|}^2\hfill \end{array}$$

(16)

$$\begin{array}{ccc}& =& \sum _{j=1}^m\left(\frac{1}{2}{\mu }_{2j-1}^2+\frac{1}{2}{\mu }_{2j}^2\right)\hfill \end{array}$$

(17)

$$\begin{array}{ccc}& =& \frac{1}{2}\sum _{k=1}^n{\mu }_k^2=\frac{1}{2}\tau \hfill \end{array}$$

(18)

and

$$\begin{array}{ccc}\hfill V^{[m,\lambda ]}& =& \sum _{j=1}^m{\left|Z_j+{\eta }_j\right|}^2\hfill \end{array}$$

(19)

$$\begin{array}{ccc}& =& \sum _{j=1}^m{\left|\frac{1}{\sqrt{2}}{X}_{2j-1}+\mathsf{i}\frac{1}{\sqrt{2}}{X}_{2j}+\frac{1}{\sqrt{2}}{\mu }_{2j-1}+\mathsf{i}\frac{1}{\sqrt{2}}{\mu }_{2j}\right|}^2\hfill \end{array}$$

(20)

$$\begin{array}{ccc}& =& \frac{1}{2}\sum _{j=1}^m\left({\left(X_{2j-1}+{\mu }_{2j-1}\right)}^2+{\left(X_{2j}+{\mu }_{2j}\right)}^2\right)\hfill \end{array}$$

(21)

$$\begin{array}{ccc}& =& \frac{1}{2}\sum _{k=1}^n{\left(X_k+{\mu }_k\right)}^2\hfill \end{array}$$

(22)

$$\begin{array}{ccc}& =& \frac{1}{2}{U}^{[n,\tau ]}.\hfill \\ \hfill & & \end{array}$$

(23)

 □

Proposition 1

(Existence of Negative Integer Moments). For $\ell \in \mathbb{N}$, the negative ${\ell }^{th}$ moment of a noncentral ${\chi }^2$-distributed RV of $n\in \mathbb{N}$ degrees of freedom and noncentrality parameter $\tau \ge 0$,

$$\begin{array}{c}\hfill \mathsf{E}\left[{\left(U^{[n,\tau ]}\right)}^{-\ell }\right],\end{array}$$

(24)

is finite if, and only if,

$$\begin{array}{c}\hfill \ell <\frac{n}{2}.\end{array}$$

(25)

Proof. 

See Appendix A.1. □

Proposition 2

(Monotonicity in Degrees of Freedom). The expected logarithm of a noncentral ${\chi }^2$-distributed RV of $n\in \mathbb{N}$ degrees of freedom and noncentrality parameter $\tau \ge 0$,

$$\begin{array}{c}\hfill \mathsf{E}\left[ln\left(U^{[n,\tau ]}\right)\right],\end{array}$$

(26)

is monotonically strictly increasing in n (for fixed τ).

Similarly, for any $\ell \in \mathbb{N}\cap \left[1,\frac{n}{2}-1\right]$, the negative ${\ell }^{th}$ moment of $U^{[n,\tau ]}$,

$$\begin{array}{c}\hfill \mathsf{E}\left[{\left(U^{[n,\tau ]}\right)}^{-\ell }\right],\end{array}$$

(27)

is monotonically strictly decreasing in n (for fixed τ).

Proof. 

See Appendix A.2. □

Proposition 3

(Continuity in Noncentrality Parameter). For a fixed n, the expected logarithm (26) is continuous in τ for every finite $\tau \ge 0$.

Proof. 

See Appendix A.3. □

For completeness, we present here the positive integer moments of the noncentral ${\chi }^2$ distribution.

Proposition 4

(Positive Integer Moments). For any $\ell \in \mathbb{N}$, the positive ${\ell }^{th}$ moment of $U^{[n,\tau ]}$ is given recursively as

$$\begin{array}{ccc}\hfill \mathsf{E}\left[{\left(U^{[n,\tau ]}\right)}^{\ell }\right]& =& 2^{\ell -1}(\ell -1)!(n+\ell \tau )+\sum _{j=1}^{\ell -1}\frac{(\ell -1)!2^{j-1}}{(\ell -j)!}·(n+j\tau )·\mathsf{E}\left[{\left(U^{[n,\tau ]}\right)}^{\ell -j}\right].\hfill \end{array}$$

(28)

Thus, the first two moments are

$$\begin{array}{ccc}\hfill \mathsf{E}\left[U^{[n,\tau ]}\right]& =& n+\tau ,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}\hfill \mathsf{E}\left[{\left(U^{[n,\tau ]}\right)}^2\right]& =& {(n+\tau )}^2+2n+4\tau .\hfill \end{array}$$

(30)

The corresponding expressions for $V^{[m,\lambda ]}$ follow directly from Lemma 1.

Proof. 

See, e.g., [10]. □

For the special case of the central ${\chi }^2$ distribution (i.e., the case when $\tau =\lambda =0$), it is straightforward to compute the expected logarithm and the negative integer moments by evaluating the corresponding integrals.

Proposition 5

(Expected Logarithm and Negative Integer Moments for Central ${\chi }^2$ Distribution). For any $n,m\in \mathbb{N}$, we have

$$\begin{array}{ccc}\hfill \mathsf{E}\left[ln\left(U^{[n,0]}\right)\right]& =& ln(2)+\psi \left(\frac{n}{2}\right),\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill \mathsf{E}\left[ln\left(V^{[m,0]}\right)\right]& =& \psi (m),\hfill \end{array}$$

(32)

where $\psi (·)$ denotes the digamma function ([8], Sec. 8.36) (see also (37) and (51) below).

Moreover, for any $n,m,\ell \in \mathbb{N}$,

$$\begin{array}{ccc}\hfill \mathsf{E}\left[{\left(U^{[n,0]}\right)}^{-\ell }\right]& =& \left\{\begin{array}{cc}{\displaystyle \frac{2^{-\ell }\Gamma \left(\frac{n}{2}-\ell \right)}{\Gamma \left(\frac{n}{2}\right)}}\hfill & ifn\ge 2\ell +1,\hfill \\ {\displaystyle \infty }\hfill & ifn\le 2\ell ,\hfill \end{array}\right.\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill \mathsf{E}\left[{\left(V^{[m,0]}\right)}^{-\ell }\right]& =& \left\{\begin{array}{cc}{\displaystyle \frac{\Gamma (m-\ell )}{\Gamma (m)}}\hfill & ifm\ge \ell +1,\hfill \\ {\displaystyle \infty }\hfill & ifm\le \ell .\hfill \end{array}\right.\hfill \end{array}$$

(34)

Proof. 

These results follow directly by evaluating the corresponding integrals using the PDF (8) and (12), respectively. See also (A2) in Appendix A.1 and (A46) in Appendix B. □

3. Two Families of Functions

3.1. The Family of Functions $g_m(·)$

The following family of functions will be essential for the expected logarithm and the negative integer moments of a noncentral ${\chi }^2$-distributed RV of even degrees of freedom.

Definition 3.

([1,2]) For an arbitrary $m\in \mathbb{N}$, we define the function $g_m:{\mathbb{R}}_0^{+}\to \mathbb{R}$,

$$\begin{array}{c}\hfill \xi \mapsto g_m(\xi )\triangleq \left\{\begin{array}{cc}{\displaystyle ln(\xi )-Ei\left(-\xi \right)+\sum _{j=1}^{m-1}{(-1)}^j\left[e^{-\xi }(j-1)!-\frac{(m-1)!}{j(m-1-j)!}\right]{\left(\frac{1}{\xi }\right)}^j}\hfill & if\xi >0,\hfill \\ {\displaystyle \psi (m)}\hfill & if\xi =0.\hfill \end{array}\right.\end{array}$$

(35)

Here, $Ei\left(·\right)$ denotes the exponential integral function ([8], Sec. 8.21)

$$\begin{array}{c}\hfill Ei\left(-x\right)\triangleq -{\int }_x^{\infty }\frac{e^{-t}}{t}\mathrm{d}t,x>0,\end{array}$$

(36)

and $\psi (·)$ is the digamma function ([8], Sec. 8.36) that for natural values takes the form

$$\begin{array}{c}\hfill \psi (m)\triangleq -\gamma +\sum _{j=1}^{m-1}\frac{1}{j},\end{array}$$

(37)

with $\gamma \approx 0.577$ being Euler’s constant.

Note that in spite of the case distinction in its definition, $g_m(\xi )$ actually is continuous for all $\xi \ge 0$. In particular,

$$\begin{array}{ccc}\hfill \underset{\xi \downarrow 0}{lim}\left\{ln(\xi )-Ei\left(-\xi \right)+\sum _{j=1}^{m-1}{(-1)}^j\left[e^{-\xi }(j-1)!-\frac{(m-1)!}{j(m-1-j)!}\right]{\left(\frac{1}{\xi }\right)}^j\right\}& =& \psi (m)\hfill \end{array}$$

(38)

for all $m\in \mathbb{N}$. This will follow from Proposition 3 once we have shown the connection between $g_m(·)$ and the expected logarithm (see Theorem 1).

Therefore, its first derivative is defined for all $\xi \ge 0$ and can be evaluated to

$$\begin{array}{c}\hfill g_m^{(1)}(\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{{(-1)}^m(m-1)!}{{\xi }^m}\left(e^{-\xi }-\sum _{j=0}^{m-1}\frac{{(-1)}^j}{j!}{\xi }^j\right)}\hfill & \mathrm{if}\xi >0,\hfill \\ {\displaystyle \frac{1}{m}}\hfill & \mathrm{if}\xi =0.\hfill \end{array}\right.\end{array}$$

(39)

Using the following expression for the incomplete Gamma function [11]

$$\begin{array}{c}\hfill \Gamma (m,z)=(m-1)!e^{-z}\sum _{j=0}^{m-1}\frac{z^j}{j!},\end{array}$$

(40)

the expression (39) can also be rewritten as

$$\begin{array}{c}\hfill g_m^{(1)}(\xi )=\left\{\begin{array}{cc}{\displaystyle {(-1)}^m{\xi }^{-m}{e}^{-\xi }\left(\Gamma (m)-\Gamma (m,-\xi )\right)}\hfill & \mathrm{if}\xi >0,\hfill \\ {\displaystyle \frac{1}{m}}\hfill & \mathrm{if}\xi =0.\hfill \end{array}\right.\end{array}$$

(41)

Note that also $g_m^{(1)}(·)$ is continuous and that in particular

$$\begin{array}{c}\hfill \underset{\xi \downarrow 0}{lim}\left\{\frac{{(-1)}^m(m-1)!}{{\xi }^m}\left(e^{-\xi }-\sum _{j=0}^{m-1}\frac{{(-1)}^j}{j!}{\xi }^j\right)\right\}=\frac{1}{m}.\end{array}$$

(42)

This can be checked directly using the series expansion of the exponential function to write

$$\begin{array}{ccc}\hfill g_m^{(1)}(\xi )& =& \frac{{(-1)}^m(m-1)!}{{\xi }^m}\sum _{j=m}^{\infty }\frac{{(-1)}^j}{j!}{\xi }^j\hfill \end{array}$$

(43)

$$\begin{array}{ccc}& =& (m-1)!\sum _{k=0}^{\infty }\frac{{(-1)}^k}{(k+m)!}{\xi }^k\hfill \end{array}$$

(44)

and plug $\xi =0$ in, or it follows from (63a) in Theorem 3, which shows that $g_m^{(1)}(\xi )$ can be written as a difference of two continuous functions.

Figure 1 and Figure 2 depict $g_m(·)$ and $g_m^{(1)}(·)$, respectively, for various values of m.

Note that the ${\ell }^{th}$ derivative of $g_m(·)$ can be expressed as a finite sum of $g_{m+j}(·)$ or of $g_{m+j}^{(1)}(·)$, see Corollary 2 in Section 5.

3.2. The Family of Functions $h_n(·)$

The following family of functions will be essential for the expected logarithm and the negative integer moments of a noncentral ${\chi }^2$-distributed RV of odd degrees of freedom.

Definition 4.

For an arbitrary odd $n\in {\mathbb{N}}^{\mathrm{odd}}$, we define the function $h_n:{\mathbb{R}}_0^{+}\to \mathbb{R}$,

$$\begin{array}{c}\hfill \xi \mapsto h_n(\xi )\triangleq \left\{\begin{array}{c}{\displaystyle -\gamma -2ln(2)+2\xi ·{}_2{F}_2\left(1,1;\frac{3}{2},2;-\xi \right)}\hfill \\ {\displaystyle +\sum _{j=1}^{\frac{n-1}{2}}{(-1)}^{j-1}\Gamma \left(j-\frac{1}{2}\right)·\left[\sqrt{\xi }{e}^{-\xi }erfi\left(\sqrt{\xi }\right)+\sum _{i=1}^{j-1}\frac{{(-1)}^i{\xi }^i}{\Gamma \left(i+\frac{1}{2}\right)}\right]{\left(\frac{1}{\xi }\right)}^j}\hfill & if\xi >0,\hfill \\ {\displaystyle \psi \left(\frac{n}{2}\right)}\hfill & if\xi =0.\hfill \end{array}\right.\end{array}$$

(45)

Here $\gamma \approx 0.577$ denotes Euler’s constant, $\psi (·)$ is the digamma function ([8], Sec. 8.36), ${}_2{\mathsf{F}}_2(·)$ is a generalized hypergeometric function ([8], Sec. 9.14)

$$\begin{array}{c}\hfill {}_2{\mathsf{F}}_2\left(1,1;\frac{3}{2},2;-\xi \right)\triangleq \frac{\sqrt{\pi }}{2}\sum _{k=0}^{\infty }\frac{{(-1)}^k}{\Gamma \left(\frac{3}{2}+k\right)(k+1)}{\xi }^k,\end{array}$$

(46)

and $erfi\left(·\right)$ denotes the imaginary error function [9]

$$\begin{array}{c}\hfill erfi\left(\xi \right)\triangleq \frac{2}{\sqrt{\pi }}{\int }_0^{\xi }{e}^{t^2}\mathrm{d}t.\end{array}$$

(47)

Note that one can also use Dawson’s function [12]

$$\begin{array}{c}\hfill \mathsf{D}(\xi )\triangleq e^{-{\xi }^2}{\int }_0^{\xi }{e}^{t^2}\mathrm{d}t\end{array}$$

(48)

to write

$$\begin{array}{c}\hfill e^{-\xi }erfi\left(\sqrt{\xi }\right)=\frac{2}{\sqrt{\pi }}\mathsf{D}\left(\sqrt{\xi }\right).\end{array}$$

(49)

This often turns out to be numerically more stable.

Note that $h_n(\xi )$ is continuous for all $\xi \ge 0$; in particular,

$$\begin{array}{c}\hfill \underset{\xi \downarrow 0}{lim}{h}_n(\xi )=\psi \left(\frac{n}{2}\right)\end{array}$$

(50)

for all $n\in {\mathbb{N}}^{\mathrm{odd}}$. This will follow from Proposition 3 once we have shown the connection between $h_n(·)$ and the expected logarithm (see Theorem 1).

Moreover, note that ([8], Eq. 8.366-3)

$$\begin{array}{c}\hfill h_n(0)=\psi \left(\frac{n}{2}\right)=-\gamma -2ln(2)+\sum _{j=1}^{\frac{n-1}{2}}\frac{1}{j-\frac{1}{2}}.\end{array}$$

(51)

The first derivative of $h_n(·)$ is defined for all $\xi \ge 0$ and can be evaluated to

$$\begin{array}{c}\hfill h_n^{(1)}(\xi )=\left\{\begin{array}{cc}{\displaystyle \frac{{(-1)}^{\frac{n-1}{2}}\Gamma \left(\frac{n}{2}\right)}{{\xi }^{\frac{n}{2}}}\left(e^{-\xi }erfi\left(\sqrt{\xi }\right)+\sum _{j=1}^{\frac{n-1}{2}}\frac{{(-1)}^j}{\Gamma \left(j+\frac{1}{2}\right)}{\xi }^{j-\frac{1}{2}}\right)}\hfill & \mathrm{if}\xi >0,\hfill \\ {\displaystyle \frac{2}{n}}\hfill & \mathrm{if}\xi =0.\hfill \end{array}\right.\\ \hfill \end{array}$$

(52)

Note that also $h_n^{(1)}(·)$ is continuous and that in particular

$$\begin{array}{c}\hfill \underset{\xi \downarrow 0}{lim}{h}_n^{(1)}(\xi )=\frac{2}{n}\end{array}$$

(53)

for all $n\in {\mathbb{N}}^{\mathrm{odd}}$. Checking this directly is rather cumbersome. It is easier to deduce this from (76a) in Theorem 5, which shows that $h_n^{(1)}(\xi )$ can be written as a difference of two continuous functions.

Figure 1 and Figure 2 depict $h_n(·)$ and $h_n^{(1)}(·)$, respectively, for various values of n.

Note that the ${\ell }^{th}$ derivative of $h_n(·)$ can be expressed as a finite sum of $h_{n+j}(·)$ or of $h_{n+j}^{(1)}(·)$, see Corollary 6 in Section 5.

4. Expected Logarithm and Negative Integer Moments

We are now ready for our main results. We will show how the functions $h_n(·)$ and $g_m(·)$ from Section 3 are connected to the expected logarithm and the negative integer moments of noncentral ${\chi }^2$-distributed random variables.

Theorem 1

(Expected Logarithm). For some $n\in \mathbb{N}$ and $\tau \ge 0$, let $U^{[n,\tau ]}$ be as in Definition 1. Then

$$\begin{array}{c}\hfill \mathsf{E}\left[ln\left(U^{[n,\tau ]}\right)\right]=\left\{\begin{array}{cc}{\displaystyle ln(2)+h_n\left(\frac{\tau }{2}\right)}\hfill & ifn\in {\mathbb{N}}^{\mathrm{odd}},\hfill \\ {\displaystyle ln(2)+g_{n/2}\left(\frac{\tau }{2}\right)}\hfill & ifn\in {\mathbb{N}}^{\mathrm{even}}.\hfill \end{array}\right.\end{array}$$

(54)

Similarly, for some $m\in \mathbb{N}$ and $\lambda \ge 0$, let $V^{[m,\lambda ]}$ be as in Definition 2. Then

$$\begin{array}{c}\hfill \mathsf{E}\left[ln\left(V^{[m,\lambda ]}\right)\right]=g_m(\lambda ).\end{array}$$

(55)

Theorem 2

(Negative Integer Moments). For some $n\in \mathbb{N}$ and $\tau \ge 0$, let $U^{[n,\tau ]}$ be as in Definition 1. Then, for any $\ell \in \mathbb{N}$,

$$\begin{array}{c}\hfill \mathsf{E}\left[{\left(U^{[n,\tau ]}\right)}^{-\ell }\right]=\left\{\begin{array}{cc}{\displaystyle \frac{{(-1)}^{\ell -1}}{(\ell -1)!2^{\ell }}·h_{n-2\ell }^{(\ell )}\left(\frac{\tau }{2}\right)}\hfill & ifn\ge 2\ell +1\mathit{and}n\in {\mathbb{N}}^{\mathrm{odd}},\hfill \\ {\displaystyle \frac{{(-1)}^{\ell -1}}{(\ell -1)!2^{\ell }}·g_{\frac{n}{2}-\ell }^{(\ell )}\left(\frac{\tau }{2}\right)}\hfill & ifn\ge 2\ell +1\mathit{and}n\in {\mathbb{N}}^{\mathrm{even}},\hfill \\ {\displaystyle \infty }\hfill & \mathit{if}n\le 2\ell .\hfill \end{array}\right.\end{array}$$

(56)

Similarly, for some $m\in \mathbb{N}$ and $\lambda \ge 0$, let $V^{[m,\lambda ]}$ be as in Definition 2. Then, for any $\ell \in \mathbb{N}$,

$$\begin{array}{c}\hfill \mathsf{E}\left[{\left(V^{[m,\lambda ]}\right)}^{-\ell }\right]=\left\{\begin{array}{cc}{\displaystyle \frac{{(-1)}^{\ell -1}}{(\ell -1)!}·g_{m-\ell }^{(\ell )}(\lambda )}\hfill & \mathit{if}m\ge \ell +1,\hfill \\ {\displaystyle \infty }\hfill & \mathit{if}m\le \ell .\hfill \end{array}\right.\end{array}$$

(57)

In particular, this means that for any $n\ge 3$,

$$\begin{array}{c}\hfill \mathsf{E}\left[\frac{1}{U^{[n,\tau ]}}\right]=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}{h}_{n-2}^{(1)}\left(\frac{\tau }{2}\right)}\hfill & \mathit{if}n\in {\mathbb{N}}^{\mathrm{odd}},\hfill \\ {\displaystyle \frac{1}{2}{g}_{\frac{n}{2}-1}^{(1)}\left(\frac{\tau }{2}\right)}\hfill & \mathit{if}n\in {\mathbb{N}}^{\mathrm{even}},\hfill \end{array}\right.\end{array}$$

(58)

and for any $m\ge 2$,

$$\begin{array}{c}\hfill \mathsf{E}\left[\frac{1}{V^{[m,\lambda ]}}\right]=g_{m-1}^{(1)}(\lambda ).\end{array}$$

(59)

A proof for these two main theorems can be found in Appendix B.

5. Properties

We next investigate the two families of functions $g_m(·)$ and $h_n(·)$ more closely and state some useful properties.

5.1. Properties of $g_m(·)$

Proposition 6.

For any $m\in \mathbb{N}$, the function $\xi \mapsto g_m(\xi )$ is monotonically strictly increasing and strictly concave ($\xi \in {\mathbb{R}}_0^{+}$).

Proof. 

Using the expression (A104), we have

$$\begin{array}{ccc}\hfill g_m^{(1)}(\xi )& =& e^{-\xi }\sum _{k=0}^{\infty }\frac{{\xi }^k}{k!}\frac{1}{k+m},\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill g_m^{(2)}(\xi )& =& -e^{-\xi }\sum _{k=0}^{\infty }\frac{{\xi }^k}{k!}\frac{1}{(k+m)(k+m+1)},\hfill \end{array}$$

(61)

i.e., the first derivative of $g_m(·)$ is positive and the second derivative is negative. □

Proposition 7.

For any $\xi \ge 0$, the function $m\mapsto g_m(\xi )$ is monotonically strictly increasing ($m\in \mathbb{N}$).

Proof. 

This follows directly from Theorem 1 and Proposition 2. □

Proposition 8.

For any $m\in \mathbb{N}$, the function $\xi \mapsto g_m^{(1)}(\xi )$ is positive, monotonically strictly decreasing, and strictly convex ($\xi \in {\mathbb{R}}_0^{+}$).

Proof. 

The positivity and the monotonicity follow directly from (60) and (61). To see the convexity, use (A104) to write

$$\begin{array}{c}\hfill g_m^{(3)}(\xi )=e^{-\xi }\sum _{k=0}^{\infty }\frac{{\xi }^k}{k!}\frac{2}{(k+m)(k+m+1)(k+m+2)},\end{array}$$

(62)

which is positive. □

Proposition 9.

For any $\xi \ge 0$, the function $m\mapsto g_m^{(1)}(\xi )$ is monotonically strictly decreasing ($m\in \mathbb{N}$).

Proof. 

This follows directly from Theorem 2 and Proposition 2. □

Theorem 3.

For all $m\in \mathbb{N}$, $\xi \ge 0$, and $\ell \in \mathbb{N}$, we have the following relations:

$$\begin{array}{ccc}\hfill g_{m+1}(\xi )& =& g_m(\xi )+g_m^{(1)}(\xi ),\hfill \end{array}$$

(63a)

$$\begin{array}{ccc}\hfill g_{m+1}^{(\ell )}(\xi )& =& g_m^{(\ell )}(\xi )+g_m^{(\ell +1)}(\xi ).\hfill \end{array}$$

(63b)

Proof. 

See Appendix C.1. □

Corollary 1.

For any $m>1$,

$$\begin{array}{c}\hfill \mathsf{E}\left[\frac{1}{V^{[m,\lambda ]}}\right]=g_m(\lambda )-g_{m-1}(\lambda ).\end{array}$$

(64)

Proof. 

This follows directly from (63a) and (59). □

Corollary 2.

The $\ell th$ derivative $g_m^{(\ell )}(·)$ can be written with the help of either the first derivative $g_m^{(1)}(·)$ or of $g_m(·)$ in the following ways:

$$\begin{array}{ccc}\hfill g_m^{(\ell )}(\xi )& =& \sum _{j=0}^{\ell -1}{(-1)}^{j+\ell -1}\left(\genfrac{}{}{0pt}{}{\ell -1}{j}\right)g_{m+j}^{(1)}(\xi )\hfill \end{array}$$

(65)

$$\begin{array}{ccc}& =& \sum _{j=0}^{\ell }{(-1)}^{j+\ell }\left(\genfrac{}{}{0pt}{}{\ell }{j}\right)g_{m+j}(\xi ).\hfill \end{array}$$

(66)

Proof. 

Using $g_m^{(0)}(·)$ as an equivalent expression for $g_m(·)$, we rewrite (63) as

$$\begin{array}{c}\hfill g_m^{(\ell )}(\xi )=g_{m+1}^{(\ell -1)}(\xi )-g_m^{(\ell -1)}(\xi ),\end{array}$$

(67)

and recursively apply this relation. □

Corollary 3.

For all $m\in \mathbb{N}$ and $\xi \ge 0$,

$$\begin{array}{ccc}\hfill g_m(\xi )& =& g_1(\xi )+\sum _{j=1}^{m-1}{g}_j^{(1)}(\xi )\hfill \end{array}$$

(68)

$$\begin{array}{ccc}& =& ln(\xi )-Ei\left(-\xi \right)+\sum _{j=1}^{m-1}{g}_j^{(1)}(\xi ).\hfill \end{array}$$

(69)

Proof. 

We recursively apply (63a) to obtain the relation

$$\begin{array}{ccc}\hfill g_m(\xi )& =& g_{m-1}(\xi )+g_{m-1}^{(1)}(\xi )\hfill \end{array}$$

(70)

$$\begin{array}{ccc}& =& g_{m-2}(\xi )+g_{m-2}^{(1)}(\xi )+g_{m-1}^{(1)}(\xi )\hfill \\ & ⋮& \hfill \end{array}$$

(71)

$$\begin{array}{ccc}& =& g_1(\xi )+\sum _{j=1}^{m-1}{g}_j^{(1)}(\xi ).\hfill \\ \hfill & & \end{array}$$

(72)

 □

Theorem 4.

We have the following relation:

$$\begin{array}{c}\hfill g_{m+1}^{(1)}(\xi )=\frac{1}{\xi }-\frac{m}{\xi }{g}_m^{(1)}(\xi )\end{array}$$

(73)

for all $m\in \mathbb{N}$ and all $\xi \ge 0$.

Proof. 

See Appendix C.2. □

Corollary 4.

For any $m>1$,

$$\begin{array}{c}\hfill \mathsf{E}\left[\frac{1}{V}\right]=\frac{1-\lambda g_m^{(1)}(\lambda )}{m-1}.\end{array}$$

(74)

Proof. 

This follows directly from (73) and (59). □

5.2. Properties of $h_n(·)$

Proposition 10.

For any $n\in {\mathbb{N}}^{\mathrm{odd}}$, the function $\xi \mapsto h_n(\xi )$ is monotonically strictly increasing and strictly concave ($\xi \in {\mathbb{R}}_0^{+}$).

Proof. 

From (A95) and (A98) we see that $h_n^{(1)}(\xi )>0$ and $h_n^{(2)}(\xi )<0$. □

Proposition 11.

For any $\xi \ge 0$, the function $n\mapsto h_n(\xi )$ is monotonically strictly increasing ($n\in {\mathbb{N}}^{\mathrm{odd}}$).

Proof. 

This follows directly from Theorem 1 and Proposition 2. □

Proposition 12.

For any $n\in {\mathbb{N}}^{\mathrm{odd}}$, the function $\xi \mapsto h_n^{(1)}(\xi )$ is positive, monotonically strictly decreasing, and strictly convex ($\xi \in {\mathbb{R}}_0^{+}$).

Proof. 

The positivity and the monotonicity follow directly from (A95) and (A98). To see the convexity, we use (A99) to write

$$\begin{array}{c}\hfill h_n^{(3)}(\xi )=e^{-\xi }\sum _{k=0}^{\infty }\frac{{\xi }^k}{k!}\frac{2}{\left(k+\frac{n}{2}\right)\left(k+1+\frac{n}{2}\right)\left(k+2+\frac{n}{2}\right)},\end{array}$$

(75)

which is positive. □

Proposition 13.

For any $\xi \ge 0$, the function $n\mapsto h_n^{(1)}(\xi )$ is monotonically strictly decreasing ($n\in {\mathbb{N}}^{\mathrm{odd}}$).

Proof. 

This follows directly from Theorem 2 and Proposition 2. □

Theorem 5.

For all $n\in {\mathbb{N}}^{\mathrm{odd}}$, $\xi \ge 0$, and $\ell \in \mathbb{N}$, we have the following relations:

$$\begin{array}{ccc}\hfill h_{n+2}(\xi )& =& h_n(\xi )+h_n^{(1)}(\xi ),\hfill \end{array}$$

(76a)

$$\begin{array}{ccc}\hfill h_{n+2}^{(\ell )}(\xi )& =& h_n^{(\ell )}(\xi )+h_n^{(\ell +1)}(\xi ).\hfill \end{array}$$

(76b)

Proof. 

See Appendix C.1. □

Corollary 5.

For any $n\in {\mathbb{N}}^{\mathrm{odd}}$, $n\ge 3$,

$$\begin{array}{c}\hfill \mathsf{E}\left[\frac{1}{U^{[n,\tau ]}}\right]=\frac{1}{2}{h}_n\left(\frac{\tau }{2}\right)-\frac{1}{2}{h}_{n-2}\left(\frac{\tau }{2}\right).\end{array}$$

(77)

Proof. 

This follows directly from (76a) and (58). □

Corollary 6.

The ${\ell }^{th}$ derivative $h_n^{(\ell )}(·)$ can be written with the help of either the first derivative $h_n^{(1)}(·)$ or of $h_n(·)$ in the following ways:

$$\begin{array}{ccc}\hfill h_n^{(\ell )}(\xi )& =& \sum _{j=0}^{\ell -1}{(-1)}^{j+\ell -1}\left(\genfrac{}{}{0pt}{}{\ell -1}{j}\right)h_{n+2j}^{(1)}(\xi )\hfill \end{array}$$

(78)

$$\begin{array}{ccc}& =& \sum _{j=0}^{\ell }{(-1)}^{j+\ell }\left(\genfrac{}{}{0pt}{}{\ell }{j}\right)h_{n+2j}(\xi ).\hfill \end{array}$$

(79)

Proof. 

We rewrite (76) as

$$\begin{array}{c}\hfill h_n^{(\ell )}(\xi )=h_{n+2}^{(\ell -1)}(\xi )-h_n^{(\ell -1)}(\xi )\end{array}$$

(80)

(where $h_n^{(0)}$ is understood as being equivalent to $h_n$) and recursively apply this relation. □

Corollary 7.

For all $n\in {\mathbb{N}}^{\mathrm{odd}}$ and $\xi \ge 0$,

$$\begin{array}{ccc}\hfill h_n(\xi )& =& h_1(\xi )+\sum _{j=1}^{\frac{n-1}{2}}{h}_{2j-1}^{(1)}(\xi )\hfill \end{array}$$

(81)

$$\begin{array}{ccc}& =& -\gamma -2ln(2)+2\xi ·{}_2{F}_2\left(1,1;\frac{3}{2},2;-\xi \right)+\sum _{j=1}^{\frac{n-1}{2}}{h}_{2j-1}^{(1)}(\xi ).\hfill \end{array}$$

(82)

Proof. 

This follows by recursive application of (76a) in the same way as Corollary 3 follows from (63a). □

Theorem 6.

We have the following relation:

$$\begin{array}{c}\hfill h_{n+2}^{(1)}(\xi )=\frac{1}{\xi }-\frac{n}{2\xi }{h}_n^{(1)}(\xi )\end{array}$$

(83)

for all $n\in {\mathbb{N}}^{\mathrm{odd}}$ and all $\xi \ge 0$.

Proof. 

See Appendix C.2. □

Corollary 8.

For any $n\in {\mathbb{N}}^{\mathrm{odd}}$, $n\ge 3$,

$$\begin{array}{c}\hfill \mathsf{E}\left[\frac{1}{U}\right]=\frac{1-\frac{\tau }{2}{h}_n^{(1)}\left(\frac{\tau }{2}\right)}{n-2}.\end{array}$$

(84)

Proof. 

This follows directly from (83) and (58). □

5.3. Additional Properties

Proposition 14.

For all $\xi \ge 0$, if $n\in {\mathbb{N}}^{\mathrm{odd}}$,

$$\begin{array}{c}\hfill h_n(\xi )\le g_{\frac{n+1}{2}}(\xi ),\end{array}$$

(85)

and if $n\in {\mathbb{N}}^{\mathrm{even}}$,

$$\begin{array}{c}\hfill g_{\frac{n}{2}}(\xi )\le h_{n+1}(\xi ).\end{array}$$

(86)

Similarly, for all $\xi \ge 0$, if $n\in {\mathbb{N}}^{\mathrm{odd}}$,

$$\begin{array}{c}\hfill h_n^{(1)}(\xi )\ge g_{\frac{n+1}{2}}^{(1)}(\xi ),\end{array}$$

(87)

and if $n\in {\mathbb{N}}^{\mathrm{even}}$,

$$\begin{array}{c}\hfill g_{\frac{n}{2}}^{(1)}(\xi )\ge h_{n+1}^{(1)}(\xi ).\end{array}$$

(88)

Proof. 

The relations (85) and (86) follow from (54) and Proposition 2; and (87) and (88) follow from (58) and Proposition 2. See also Figure 1 and Figure 2 for a graphical representation of this relationship. □

Lemma 2.

For any $m\in \mathbb{N}$, the function $\xi \mapsto g_m\left(\frac{1}{\xi }\right)$ is monotonically strictly decreasing and convex. Similarly, for any $n\in {\mathbb{N}}^{\mathrm{odd}}$, the function $\xi \mapsto h_n\left(\frac{1}{\xi }\right)$ is monotonically strictly decreasing and convex.

Proof. 

Since

$$\begin{array}{c}\hfill \frac{\partial }{\partial \xi }{g}_m\left(\frac{1}{\xi }\right)=-\frac{1}{{\xi }^2}{g}_m^{(1)}\left(\frac{1}{\xi }\right)\end{array}$$

(89)

and because (by Proposition 8) $g_m^{(1)}(·)>0$, we conclude that $g_m\left(\frac{1}{\xi }\right)$ is monotonically strictly decreasing.

To check convexity, we use Theorem 3 to rewrite (89) as

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial \xi }{g}_m\left(\frac{1}{\xi }\right)& =& -\frac{1}{{\xi }^2}{g}_{m+1}\left(\frac{1}{\xi }\right)+\frac{1}{{\xi }^2}{g}_m\left(\frac{1}{\xi }\right)\hfill \end{array}$$

(90)

such that

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial {\xi }^2}{g}_m\left(\frac{1}{\xi }\right)& =& \frac{2}{{\xi }^3}{g}_{m+1}\left(\frac{1}{\xi }\right)+\frac{1}{{\xi }^4}{g}_{m+1}^{(1)}\left(\frac{1}{\xi }\right)-\frac{2}{{\xi }^3}{g}_m\left(\frac{1}{\xi }\right)-\frac{1}{{\xi }^4}{g}_m^{(1)}\left(\frac{1}{\xi }\right)\hfill \end{array}$$

(91)

$$\begin{array}{ccc}& =& \frac{2}{{\xi }^3}{g}_m^{(1)}\left(\frac{1}{\xi }\right)+\frac{1}{{\xi }^4}\left(\xi -m\xi g_m^{(1)}\left(\frac{1}{\xi }\right)\right)-\frac{1}{{\xi }^4}{g}_m^{(1)}\left(\frac{1}{\xi }\right)\hfill \end{array}$$

(92)

$$\begin{array}{ccc}& =& g_m^{(1)}\left(\frac{1}{\xi }\right)·\frac{2\xi -m\xi -1}{{\xi }^4}+\frac{1}{{\xi }^3}\hfill \end{array}$$

(93)

$$\begin{array}{ccc}& \ge & \frac{1}{\frac{1}{\xi }+m}·\frac{2\xi -m\xi -1}{{\xi }^4}+\frac{1}{{\xi }^3}\hfill \end{array}$$

(94)

$$\begin{array}{ccc}& =& \frac{2}{{\xi }^2(m\xi +1)}>0.\hfill \end{array}$$

(95)

Here, in the second equality we use Theorems 3 and 4; and the inequality follows from the lower bound (105) in Theorem 8 below. (Note that while the derivation of the bounds in Section 6 do strongly rely on the properties derived in Section 5, the results of this Lemma 2 are not needed there.)

The derivation for $h_n\left(\frac{1}{\xi }\right)$ is completely analogous. In particular, using Theorems 5 and 6 one shows that

$$\begin{array}{ccc}\hfill \frac{{\partial }^2}{\partial {\xi }^2}{h}_n\left(\frac{1}{\xi }\right)& =& h_n^{(1)}\left(\frac{1}{\xi }\right)·\frac{2\xi -\frac{n}{2}\xi -1}{{\xi }^4}+\frac{1}{{\xi }^3}\hfill \end{array}$$

(96)

$$\begin{array}{ccc}& \ge & \frac{1}{\frac{1}{\xi }+\frac{n}{2}}·\frac{2\xi -\frac{n}{2}\xi -1}{{\xi }^4}+\frac{1}{{\xi }^3}\hfill \end{array}$$

(97)

$$\begin{array}{ccc}& =& \frac{2}{{\xi }^2\left(\frac{n}{2}\xi +1\right)}>0,\hfill \end{array}$$

(98)

where the inequality follows from (117) in Theorem 10 below. □

6. Bounds

Finally, we derive some elementary upper and lower bounds on $g_m(·)$ and $h_n(·)$ and their first derivative.

6.1. Bounds on $g_m(·)$ and $g_m^{(1)}(·)$

Theorem 7.

For any $m\in \mathbb{N}$ and $\xi \in {\mathbb{R}}_0^{+}$, $g_m(\xi )$ is lower-bounded as follows:

$$\begin{array}{ccc}\hfill g_m(\xi )& \ge & ln(\xi +m-1),\hfill \end{array}$$

(99)

$$\begin{array}{ccc}\hfill g_m(\xi )& \ge & ln(\xi +m)-ln(m)+\psi (m),\hfill \end{array}$$

(100)

and upper-bounded as follows:

$$\begin{array}{ccc}\hfill g_m(\xi )& \le & ln(\xi +m),\hfill \end{array}$$

(101)

$$\begin{array}{ccc}\hfill g_m(\xi )& \le & \frac{m+1}{m}ln\left(\frac{\xi +m+1}{m+1}\right)+\psi (m).\hfill \end{array}$$

(102)

Proof. 

See Appendix D.1. □

Note that the bounds (101) and (99) are tighter for larger values of $\xi$, and they are exact asymptotically when $\xi \to \infty$:

$$\begin{array}{c}\hfill \underset{\xi \to \infty }{lim}\left\{ln(\xi +m)-ln(\xi +m-1)\right\}=0.\end{array}$$

(103)

In contrast, the bounds (102) and (100) are better for small values of $\xi$ and are exact for $\xi =0$:

$$\begin{array}{c}\hfill \underset{\xi \downarrow 0}{lim}\frac{\frac{m+1}{m}ln\left(\frac{\xi +m+1}{m+1}\right)+\psi (m)}{ln(\xi +m)-ln(m)+\psi (m)}=1.\end{array}$$

(104)

In general, the tightness of the bounds increases with increasing m.

The bounds of Theorem 7 are depicted in Figure 3 and Figure 4 for the cases of $m=1$, $m=2$, and $m=5$.

Theorem 8.

For any $m\in \mathbb{N}$ and $\xi \in {\mathbb{R}}_0^{+}$, $g_m^{(1)}(\xi )$ is lower-bounded as follows:

$$\begin{array}{c}\hfill g_m^{(1)}(\xi )\ge \frac{1}{\xi +m},\end{array}$$

(105)

and upper-bounded as follows:

$$\begin{array}{ccc}\hfill g_m^{(1)}(\xi )& \le & \frac{m+1}{m(\xi +m+1)},\hfill \end{array}$$

(106)

$$\begin{array}{ccc}\hfill g_m^{(1)}(\xi )& \le & \frac{1}{\xi +m-1}.\hfill \end{array}$$

(107)

Proof. 

See Appendix D.1. □

Note that the lower bound (105) is exact for $\xi =0$ and asymptotically when $\xi \to \infty$. The upper bound (106) is tighter for small values of $\xi$ and is exact for $\xi =0$, while (107) is better for larger values of $\xi$ and is exact asymptotically when $\xi \to \infty$. Concretely, we have

$$\begin{array}{c}\hfill \underset{\xi \to \infty }{lim}\left\{\frac{1}{\xi +m-1}-\frac{1}{\xi +m}\right\}=0\end{array}$$

(108)

and

$$\begin{array}{c}\hfill \underset{\xi \downarrow 0}{lim}\frac{\frac{m+1}{m(\xi +m+1)}}{\frac{1}{\xi +m}}=1.\end{array}$$

(109)

In general, also here it holds that the tightness of the bounds increases with increasing m.

The bounds of Theorem 8 are depicted in Figure 5 for the cases of $m=1$, $m=3$, and $m=8$.

6.2. Bounds on $h_n(·)$ and $h_n^{(1)}(·)$

Theorem 9.

For any $\xi \in {\mathbb{R}}_0^{+}$, $h_n(\xi )$ is lower-bounded as follows:

$$\begin{array}{cccc}\hfill h_n(\xi )& \ge & ln\left(\xi +\frac{n}{2}-1\right)\hfill & (n\in {\mathbb{N}}^{\mathrm{odd}},n\ge 3),\end{array}$$

(110a)

$$\begin{array}{cccc}\hfill h_1(\xi )& \ge & ln\left(\xi +\frac{1}{2}\right)-\frac{2}{\xi }\left(1-e^{-\xi }\right)\hfill & (n=1).\end{array}$$

(110b)

Moreover, for any $n\in {\mathbb{N}}^{\mathrm{odd}}$,

$$\begin{array}{ccc}\hfill h_n(\xi )& \ge & ln\left(\xi +\frac{n}{2}\right)-ln\left(\frac{n}{2}\right)+\psi \left(\frac{n}{2}\right).\hfill \end{array}$$

(111)

For any $\xi \in {\mathbb{R}}_0^{+}$ and any $n\in {\mathbb{N}}^{\mathrm{odd}}$, $h_n(\xi )$ is upper-bounded as follows:

$$\begin{array}{ccc}\hfill h_n(\xi )& \le & ln\left(\xi +\frac{n}{2}\right),\hfill \end{array}$$

(112)

$$\begin{array}{ccc}\hfill h_n(\xi )& \le & \frac{n+2}{n}ln\left(\frac{\xi +\frac{n}{2}+1}{\frac{n}{2}+1}\right)+\psi \left(\frac{n}{2}\right).\hfill \end{array}$$

(113)

Proof. 

See Appendix D.2. □

Note that the bounds (112) and (110) are tighter for larger values of $\xi$, and they are exact asymptotically when $\xi \to \infty$:

$$\begin{array}{c}\hfill \underset{\xi \to \infty }{lim}\left\{ln\left(\xi +\frac{n}{2}\right)-ln\left(\xi +\frac{n}{2}-1\right)\right\}=0\end{array}$$

(114)

and

$$\begin{array}{c}\hfill \underset{\xi \to \infty }{lim}\left\{ln\left(\xi +\frac{1}{2}\right)-ln\left(\xi +\frac{1}{2}\right)+\frac{2}{\xi }\left(1-e^{-\xi }\right)\right\}=0,\end{array}$$

(115)

respectively.

In contrast, the bounds (113) and (111) are better for small values of $\xi$ and are exact for $\xi =0$:

$$\begin{array}{c}\hfill \underset{\xi \downarrow 0}{lim}\frac{\frac{n+2}{n}ln\left(\frac{\xi +\frac{n}{2}+1}{\frac{n}{2}+1}\right)+\psi \left(\frac{n}{2}\right)}{ln\left(\xi +\frac{n}{2}\right)-ln\left(\frac{n}{2}\right)+\psi \left(\frac{n}{2}\right)}=1.\end{array}$$

(116)

In general, the tightness of the bounds increases with increasing n.

The bounds of Theorem 9 are depicted in Figure 6 and Figure 7 for the cases of $n=1$, $n=3$, and $n=9$.

Theorem 10.

For any $n\in {\mathbb{N}}^{\mathrm{odd}}$ and $\xi \in {\mathbb{R}}_0^{+}$, $h_n^{(1)}(\xi )$ is lower-bounded as follows:

$$\begin{array}{c}\hfill h_n^{(1)}(\xi )\ge \frac{1}{\xi +\frac{n}{2}},\end{array}$$

(117)

and upper-bounded as follows:

$$\begin{array}{ccc}\hfill h_n^{(1)}(\xi )& \le & \frac{n+2}{n\left(\xi +\frac{n}{2}+1\right)}.\hfill \end{array}$$

(118)

Moreover,

$$\begin{array}{cccc}\hfill h_n^{(1)}(\xi )& \le & \frac{1}{\xi +\frac{n}{2}-1}\hfill & (n\in {\mathbb{N}}^{\mathrm{odd}},n\ge 3),\end{array}$$

(119a)

$$\begin{array}{cccc}\hfill h_1^{(1)}(\xi )& \le & \frac{2}{\xi }\left(1-e^{-\xi }\right)\le \frac{2}{\xi }\hfill & (n=1).\end{array}$$

(119b)

Proof. 

See Appendix D.2. □

Note that the lower bound (117) is exact for $\xi =0$ and asymptotically when $\xi \to \infty$. The upper bound (118) is tighter for small values of $\xi$ and is exact for $\xi =0$, while (119) is better for larger values of $\xi$ and is exact asymptotically when $\xi \to \infty$. Concretely, we have

$$\begin{array}{c}\hfill \underset{\xi \to \infty }{lim}\left\{\frac{1}{\xi +\frac{n}{2}-1}-\frac{1}{\xi +\frac{n}{2}}\right\}=0\end{array}$$

(120)

or

$$\begin{array}{c}\hfill \underset{\xi \to \infty }{lim}\left\{\frac{2}{\xi }-\frac{1}{\xi +\frac{n}{2}}\right\}=0,\end{array}$$

(121)

respectively, and

$$\begin{array}{c}\hfill \underset{\xi \downarrow 0}{lim}\frac{\frac{n+2}{n\left(\xi +\frac{n}{2}+1\right)}}{\frac{1}{\xi +\frac{n}{2}}}=1.\end{array}$$

(122)

In the special case $n=1$, the improved version of (119b) is exact also for $\xi =0$, but it is still less tight for low $\xi$ than (118).

In general, also here it holds that the tightness of the bounds increases with increasing n.

These bounds are depicted in Figure 8 and Figure 9 for the cases $n=1$, $n=3$, and $n=9$.

7. Discussion

We have shown that the expected logarithm and the negative integer moments of a noncentral ${\chi }^2$-distributed RV can be expressed with the help of two families of functions $g_m(·)$ and $h_n(·)$, depending on whether the degrees of freedom are even or odd. While these two families of functions are very similar in many respects, they are actually surprisingly different in their description. The case of odd degrees of freedom thereby turns out to be quite a bit more complicated than the situation of even degrees of freedom (which explains why $g_m(·)$ was defined in [1] already, while $h_n(·)$ is newly introduced in this work).

We have also provided a whole new set of properties of both family of functions and derived new tight upper and lower bounds that are solely based on elementary functions.

It is intuitively pleasing that $U^{-\ell }$—being proportional to the $\ell th$ derivative of $ln(U)$—has an expectation that is related to the ${\ell }^{th}$ derivative of the function describing the expectation of the logarithm.

The recently proposed trick of representing the logarithm by an integral [7] turned out to be very helpful in the proof of the continuity of the expected logarithm (see Appendix A.3). While in general very well behaved, the logarithmic function nevertheless is a fickle beast due to its unboundedness both at zero and infinity.