Keywords. Real-order moments; survival functions; tail probabilities; negative moments; logarithmic means; Laplace transforms.
Mathematics Subject Classification (2010). 60E05; 62H99; 62N05.

1 Introduction↩︎

Moments are among the most fundamental characteristics of probability distributions and play a central role in probability theory, statistics, reliability analysis, actuarial science, stochastic modeling, and risk theory. Besides summarizing important distributional features, moments are frequently used in asymptotic analysis, statistical inference, and the study of tail behavior. Consequently, representations of moments in terms of distribution functions or tail probabilities have attracted considerable attention in the probability literature.

For a nonnegative random variable \(X\) and a real number \(p>0\), the classical tail-integral representation \[\mathbb{E}[X^p] = p\int_0^\infty x^{p-1}\mathbb{P}(X>x){\rm d}x\] provides a direct connection between moments and survival probabilities. This identity is a standard tool in probability theory and appears, explicitly or implicitly, in classical references such as [1], [2], and [3]. It has proved particularly useful in the study of heavy-tailed distributions, regular variation, risk measures, and moment existence criteria.

Although tail representations are well understood for nonnegative random variables, corresponding formulas for arbitrary real-valued random variables are less frequently presented within a unified framework. In particular, treatments covering continuous, discrete, and mixed distributions simultaneously are comparatively scarce. Moreover, negative-order and fractional moments are often considered separately, despite their natural connection with classical tail-integral methods.

The main purpose of this paper is to develop a unified framework for real-order moments of arbitrary random variables supported on the whole real line. We derive general representations involving cumulative distribution functions and survival functions that extend the classical formulas for nonnegative random variables. The resulting identities apply equally to continuous, discrete, and mixed distributions and provide a common treatment of positive, fractional, and negative moments.

Particular attention is devoted to discrete distributions. Exploiting the stepwise structure of cumulative distribution functions, we obtain explicit series representations for real-order moments in terms of cumulative probabilities. These formulas yield simple and practical criteria for moment existence and naturally connect moment finiteness with the asymptotic behavior of distribution tails. Applications are presented for the zeta distribution, recovering the classical condition for the existence of power moments, and for the Skellam distribution, where the proposed representations admit a natural geometric interpretation.

A further contribution concerns logarithmic moments. By investigating the limiting behavior of real-order moments as the order tends to zero, we derive an integral representation for \(\mathbb{E}[\log(X)]\) in terms of the Laplace transform of a positive random variable. This identity establishes a direct connection between logarithmic moments, Laplace transforms, and the classical Frullani identity [4]. Moreover, it provides a convenient tool for comparing logarithmic moments through the Laplace-transform order.

The paper is organized as follows. Section 2 develops general representations for real-order moments and establishes necessary and sufficient conditions for moment existence. Special cases involving negative moments, absolutely continuous distributions, and discrete distributions are examined in detail. Applications to the zeta and Skellam distributions are presented, and a representation for logarithmic moments in terms of Laplace transforms is derived. Concluding remarks are given in Section 3.

2 sec:Real-order32moments32and32the32logarithmic32mean↩︎

In this section, we derive formulas for real-order moments of arbitrary random variables, including discrete, absolutely continuous, and mixed distributions. We also discuss a connection with the logarithmic mean.

2.1 General moments of a random variable↩︎

Let \((\Omega, \mathcal{F}, \mathbb{P})\) be a probability space, \(X: \Omega \to \mathbb{R}\) a random variable (discrete, absolutely continuous, or mixed) taking values in a nonempty set \(S\subset\mathbb{R}\), and \(F_X\) its cumulative distribution function. Assuming that \(X^p\), with \(p>0\), is integrable, the \(p\)-th real moment of \(X\) is given by \[\begin{align} \label{main-eq} \mathbb{E}[X^p] = \mathbb{E}\!\left[X^p\mathbb{1}_{\{X>0\}}\right] + \mathbb{E}\!\left[X^p\mathbb{1}_{\{X<0\}}\right], \end{align}\tag{1}\] where \(\mathbb{1}_A\) denotes the indicator function of an event \(A\). Whenever the negative part \(\mathbb{E}\!\left[X^p\mathbb{1}_{\{X<0\}}\right]\) is involved, we assume that \(p=m/n\in\mathbb{Q}\) is expressed in lowest terms with \(n\) odd, ensuring that \(x^p\in\mathbb{R}\) for all \(x\in\mathbb{R}\).

On the one hand, we may write \[\begin{align} \mathbb{E}[X^p\mathbb{1}_{\{X>0\}}] = \int_{0}^{\infty} x^p {\rm d}F_X(x) &= p\int_{0}^{\infty}\left[\int_{0}^{\infty} \mathbb{1}_{\{x>t\}} t^{p-1}{\rm d}t\right] {\rm d}F_X(x) \nonumber \\[0,2cm] &= p\int_{0}^{\infty} t^{p-1} \left[\int_{0}^{\infty} \mathbb{1}_{\{x>t\}}{\rm d}F_X(x) \right]{\rm d}t \nonumber \\[0,2cm] &= p\int_{0}^{\infty} t^{p-1} \left[\int_{\Omega} \mathbb{1}_{\{X>t\}}(\omega) {\rm d}\mathbb{P}(\omega) \right]{\rm d}t \nonumber \\[0,2cm] &= p\int_{0}^{\infty} t^{p-1} \mathbb{P}(X>t) {\rm d}t \label{positiva}, \end{align}\tag{2}\] where, in the third identity, we applied Fubini’s theorem to interchange the order of integration, and in the fourth identity, we used a change of variables in the Lebesgue-Stieltjes integral.

On the other hand, using 2 , we obtain \[\begin{align} \label{negativa} \mathbb{E}[X^{p}\mathbb{1}_{\{X<0\}}] &= (-1)^{p} \mathbb{E}[(-X)^{p}\mathbb{1}_{\{-X>0\}}] \nonumber \\[0,2cm] &\stackrel{\eqref{positiva}}{=} -p\int_{0}^{\infty} (-t)^{p-1} \mathbb{P}(X<-t) {\rm d}t \nonumber \\[0,2cm] &= -p\int_{-\infty}^0 t^{p-1} \mathbb{P}(X<t) {\rm d}t. \end{align}\tag{3}\]

Substituting 2 and 3 into 1 , we obtain \[\begin{align} \label{main-formula} \mathbb{E}[X^p] = p\int_{0}^{\infty} t^{p-1} [1-F_X(t)] {\rm d}t - p\int_{-\infty}^{0} t^{p-1} F_X(t^-) {\rm d}t . \end{align}\tag{4}\]

In particular case, \[\begin{align} \mathbb{E}[X^p] = \begin{cases} \displaystyle p\int_{0}^{\infty} t^{p-1} [1-F_X(t)] {\rm d}t, &\text{if} \;X>0, \\[0,5cm] \displaystyle -p\int_{-\infty}^{0} t^{p-1} F_X(t^-) {\rm d}t, &\text{if} \;X<0. \end{cases} \end{align}\]

2.1.1 Existence of real-order moments↩︎

The representation 4 immediately yields a necessary and sufficient criterion for the existence of real-order moments.

2.1.2 Negative-order moments↩︎

The general representation 4 also yields a convenient expression for negative-order moments.

2.1.3 Moments of an absolutely continuous random variable↩︎

Let \(X:\Omega\to\mathbb{R}\) be an absolutely continuous random variable with cumulative distribution function \(F_X\). From formula 4 , we have \[\begin{align} \label{exp-figure} \mathbb{E}[X^p] = p\int_{0}^{\infty} t^{p-1} [1-F_X(t)] {\rm d}t - p\int_{-\infty}^{0} t^{p-1} F_X(t) {\rm d}t . \end{align}\tag{5}\]

In particular case, \[\begin{align} \mathbb{E}[X^p] = \begin{cases} \displaystyle p\int_{0}^{\infty} t^{p-1} [1-F_X(t)] {\rm d}t, & \text{if} \;X>0, \\[0,5cm] \displaystyle -p\int_{-\infty}^{0} t^{p-1} F_X(t) {\rm d}t, & \text{if} \;X<0. \end{cases} \end{align}\]

In Figure 1, we consider the cumulative distribution function of \(X\sim \mathrm{SN}(\alpha)\). From 5 and Figure 1, we see that \(\mathbb{E}[X]\) is obtained by subtracting the blue area from the red area.

2.1.4 Moments of a discrete random variable↩︎

Let \(X:\Omega\to\mathbb{R}\) be a discrete random variable taking values in the set \[S=\{\ldots,x_{-2},x_{-1},x_{0},x_{1},x_{2},\ldots\},\] such that \[\begin{align} \cdots<x_{-2}<x_{-1}<0\leqslant x_{0}<x_{1}<x_{2}<\cdots. \end{align}\] Note that the cumulative distribution function of \(X\) is given by \[\begin{align} F_{X}(t) = \begin{cases} \qquad \vdots & \qquad \vdots \\ F_{X}(x_{-(k+1)})=F_X((x_{-k})^-), &x_{-(k+1)}\leqslant t<x_{-k}, \\ \qquad \vdots & \qquad \vdots \\ F_X(x_{-2})=F_X((x_{-1})^-), &x_{-2}\leqslant t<x_{-1}, \\ F_X(x_{-1})=F_X(0^-), &x_{-1}\leqslant t<0, \\ F_X(0), &0\leqslant t<x_0, \\ F_X(x_0), & x_0\leqslant t<x_1, \\ F_X(x_1), & x_1\leqslant t<x_2, \\ \qquad \vdots & \qquad \vdots \\ F_X(x_k), & x_k\leqslant t<x_{k+1}, \\ \qquad \vdots & \qquad \vdots \end{cases} \end{align}\]

Using 2 and the fact that \[F_X(t)=F_X(x_k)\] is constant on the interval \(x_k\leqslant t<x_{k+1}\), we have \[\begin{align} \mathbb{E}[X^p\mathbb{1}_{\{X>0\}}] &= p\int_{0}^{\infty} t^{p-1} \mathbb{P}(X>t) {\rm d}t \nonumber \\[0,2cm] &= p \int_{0}^{x_0} t^{p-1} \mathbb{P}(X>0) {\rm d}t + p\sum_{k=0}^{\infty} \int_{x_k}^{x_{k+1}} t^{p-1} \mathbb{P}(X>x_k) {\rm d}t \nonumber \\[0,2cm] &= x_0^p\mathbb{P}(X>0) + \sum_{k=0}^{\infty} (x_{k+1}^p-x_k^p) \mathbb{P}(X>x_k). \label{positiva-discrete} \end{align}\tag{6}\]

On the other hand, using 3 and the fact that \[\mathbb{P}(X<t)=F_X(x_{-k}^-)\] is constant on the interval \(x_{-(k+1)}\leqslant t<x_{-k}\), we have \[\begin{align} \label{negativa-discrete} \mathbb{E}[X^{p}\mathbb{1}_{\{X<0\}}] &= -p\int_{-\infty}^0 t^{p-1} \mathbb{P}(X<t) {\rm d}t \nonumber \\[0,2cm] &= -p \left[ \int_{x_{-1}}^0 t^{p-1} \mathbb{P}(X<0) {\rm d}t + \sum_{k=1}^{\infty} \int_{x_{-(k+1)}}^{x_{-k}} t^{p-1} \mathbb{P}(X<x_{-k}) {\rm d}t \right] \nonumber \\[0,2cm] &= -\left[ - (x_{-1})^p \mathbb{P}(X<0) + \sum_{k=1}^{\infty} [x_{-k}^p-x_{-(k+1)}^p] \mathbb{P}(X<x_{-k}) \right]. \end{align}\tag{7}\]

Substituting 6 and 7 into 1 , we obtain \[\begin{align} \mathbb{E}[X^p] = x_0^p[1-F_X(0)] + \sum_{k=0}^{\infty} (x_{k+1}^p-x_k^p) [1-F_X(x_k)] - \sum_{k=0}^{\infty} [x_{-k}^p-x_{-(k+1)}^p] F_X(x_{-k}^-) , \end{align}\] where, in the above identity, more precisely in the series on the right-hand side, we adopt the convention that \(x_{-0}\equiv 0\).

Since \(F_X(x_{-k}^-)=F_X(x_{-(k+1)})\), the previous identity can be written as \[\begin{align} \mathbb{E}[X^p] = x_0^p[1-F_X(0)] + \sum_{k=0}^{\infty} (x_{k+1}^p-x_k^p) [1-F_X(x_k)] - \sum_{k=0}^{\infty} [x_{-k}^p-x_{-(k+1)}^p] F_X(x_{-(k+1)}) . \end{align}\] Using a change of variables, we rewrite the above identity as \[\begin{align} \label{general-p} \mathbb{E}[X^p] = x_0^p[1-F_X(0)] + \sum_{k=0}^{\infty} (x_{k+1}^p-x_k^p) [1-F_X(x_k)] - \sum_{k=1}^{\infty} [x_{-(k-1)}^p-x_{-k}^p] F_X(x_{-k}) . \end{align}\tag{8}\]

In particular case, \[\begin{align} \mathbb{E}[X^p] = \begin{cases} \displaystyle x_0^p[1-F_X(0)] + \sum_{k=0}^{\infty} (x_{k+1}^p-x_k^p) [1-F_X(x_k)], & \text{if} \;X>0, \\[0,5cm] \displaystyle -\sum_{k=1}^{\infty} [x_{-(k-1)}^p-x_{-k}^p] F_X(x_{-k}), & \text{if} \;X<0. \end{cases} \end{align}\]

In the Figure 2, we consider a discrete random variable \[X\in\{x_{-2},x_{-1},x_0,x_1\}, \quad x_{-2}<x_{-1}<0\leqslant x_{0}<x_{1},\] with probabilities \(p_{x_{-2}}\), \(p_{x_{-1}}\), \(p_{x_0}\), and \(p_{x_1}\), respectively. From 8 and Figure 2, we see that \(\mathbb{E}[X]\) is obtained by subtracting the blue area from the red area.

Special case: \(S=\mathbb{Z}\).

In the particular case \(x_k=k\), \(k\in\mathbb{Z}\), equation 8 reduces to \[\begin{align} \label{general-p-1} \mathbb{E}[X^p] &= \sum_{k=0}^{\infty} \bigl[(k+1)^p-k^p\bigr] \bigl[1-F_X(k)\bigr] + (-1)^p \sum_{k=1}^{\infty} \bigl[k^p-(k-1)^p\bigr] F_X(-k). \end{align}\tag{9}\]

As particular cases, we obtain \[\begin{align} \mathbb{E}[X^p] = \begin{cases} \displaystyle \sum_{k=0}^{\infty} \bigl[(k+1)^p-k^p\bigr] \bigl[1-F_X(k)\bigr], & \text{if} \;X>0, \\[0.5cm] \displaystyle (-1)^p \sum_{k=1}^{\infty} \bigl[k^p-(k-1)^p\bigr] F_X(-k), & \text{if} \;X<0. \end{cases} \end{align}\]

Setting \(p=1\) in 9 , we obtain \[\begin{align} \label{general-p-2} \mathbb{E}[X] = \sum_{k=0}^{\infty} \bigl[1-F_X(k)\bigr] - \sum_{k=1}^{\infty} F_X(-k). \end{align}\tag{10}\]

In particular, \[\begin{align} \mathbb{E}[X] = \begin{cases} \displaystyle \sum_{k=0}^{\infty} \bigl[1-F_X(k)\bigr], & \text{if} \;X>0, \\[0.5cm] \displaystyle - \sum_{k=1}^{\infty} F_X(-k), & \text{if} \;X<0. \end{cases} \end{align}\]

In Figure 3, we plot the cumulative distribution function of \(X\sim \mathrm{Skellam}(\lambda_1,\lambda_2).\) From 10 and Figure 3, we see that \(\mathbb{E}[X]\) is obtained by subtracting the blue area from the red area.

2.1.5 The logarithmic mean↩︎

Let \(X:\Omega\to\mathbb{R}\) be a positive absolutely continuous random variable with cumulative distribution function \(F_X\).

Since \[\begin{align} \label{id-log} x^r\log(x) = \lim_{p\to 0^+} x^{p}\log(x) = \lim_{p\to 0^+} \, {{\rm d}{x^{p}}\over {\rm d}p}, \quad x>0; \;p>0, \end{align}\tag{11}\] we have \[\begin{align} \mathbb{E}[ \log(X)] = \lim_{p\to 0^+} \, {{\rm d}\over {\rm d}p} \mathbb{E}[X^p]. \end{align}\]

From formula 5 , we have \[\begin{align} \label{mom-1} \mathbb{E}[X^p] &= p\int_{0}^{\infty} t^{p-1} [1-F_X(t)] {\rm d}t \nonumber \\[0,2cm] &= p\int_{1}^{\infty} t^{p-1} [1-F_X(t)] {\rm d}t + 1 - p\int_{0}^{1} t^{p-1} F_X(t) {\rm d}t. \end{align}\tag{12}\]

Combining 11 and 12 gives \[\begin{align} \mathbb{E}[\log(X)] &= \lim_{p\to 0^+} {{\rm d}\over {\rm d}p} \mathbb{E}[X^p] \\[0,2cm] &= \lim_{p\to 0^+} \int_{1}^{\infty} [t^{p-1}+p(p-1)t^{p-2}] [1-F_X(t)] {\rm d}t - \lim_{p\to 0^+} \int_{0}^{1} [t^{p-1}+p(p-1)t^{p-2}] F_X(t) {\rm d}t \\[0,2cm] &= \int_{1}^{\infty} {1\over t} \, [1-F_X(t)] {\rm d}t - \int_{0}^{1} {1\over t} \, F_X(t) {\rm d}t. \end{align}\] Using the identity \[\begin{align} {1\over t}=\int_0^{\infty} \exp\{-tx\}{\rm d}x, \quad t>0, \end{align}\] we have \[\begin{align} \mathbb{E}[\log(X)] &= \int_{0}^{\infty} \left[ \int_{1}^{\infty} \exp\{-tx\} \, [1-F_X(t)] {\rm d}t - \int_{0}^{1} \exp\{-tx\} \, F_X(t) {\rm d}t \right] {\rm d}x \\[0,2cm] &= \int_{0}^{\infty} \left[ \int_{1}^{\infty} \exp\{-tx\} [1-F_X(t)]{\rm d}t - \int_{0}^{1} \exp\{-tx\} F_X(t){\rm d}t \right] {\rm d}x \\[0,2cm] &= \int_{0}^{\infty} \frac{\exp\{-x\}-\mathbb{E}[\exp\{-xX\}]}{x} \, {\rm d}x, \end{align}\] where in the third identity we used integration by parts to obtain \[\int_{0}^{\infty} \exp\{-tx\} F_X(t){\rm d}t = \frac{1}{x}\, \mathbb{E}[\exp\{-xX\}],\] since \(X\) is positive.

Hence, we have the following identity for the logarithmic mean: \[\begin{align} \label{id-exp-log} \mathbb{E}[\log(X)] = \int_{0}^{\infty} \frac{\exp\{-x\}-\mathcal{L}_X(x)}{x} {\rm d}x, \end{align}\tag{13}\] where \(\mathcal{L}_X(x)\) denotes the Laplace transform of \(X\).

Formula 13 can be used to compare logarithmic moments through Laplace transforms. Indeed, if \(X\) and \(Y\) are positive random variables such that \(\mathcal{L}_X(s)\geqslant \mathcal{L}_Y(s), \;s>0,\) then \[\begin{align} \label{logarithmic-mean} \mathbb{E}[\log(X)] = \int_0^\infty \frac{\exp\{-s\}-\mathcal{L}_X(s)}{s}{\rm d}s \leqslant \int_0^\infty \frac{\exp\{-s\}-\mathcal{L}_Y(s)}{s} {\rm d}s = \mathbb{E} [\log(Y)]. \end{align}\tag{14}\] Hence, the Laplace-transform order implies the ordering of logarithmic moments.

3 sec:Concluding32remarks↩︎

In this paper, we established a unified approach to real-order moments of arbitrary random variables. The main contribution is the derivation of general representations expressing moments directly in terms of cumulative distribution functions and survival functions, without requiring the existence of densities or restricting attention to nonnegative random variables. The resulting formulas apply equally to continuous, discrete, and mixed distributions and provide a common framework for positive, fractional, and negative moments.

For discrete distributions, explicit summation formulas were obtained in terms of cumulative probabilities. These representations lead naturally to criteria for moment existence based on tail behavior and recover classical results for heavy-tailed models. The examples involving the zeta and Skellam distributions illustrate both the analytical and geometric interpretations of the proposed formulas.

Another contribution is the derivation of an integral representation for logarithmic moments in terms of the Laplace transform. This identity reveals a direct connection between logarithmic means, Laplace-transform ordering, and the classical Frullani integral, thereby placing logarithmic moments within the same general framework.

The results presented here suggest several directions for future research. Possible extensions include analogous representations for truncated moments, conditional moments, multivariate distributions, dependence measures, and stochastic processes. It would also be of interest to investigate applications to heavy-tailed models, regular variation, risk theory, and probabilistic inequalities.

Acknowledgements

The research was supported in part by CNPq and CAPES grants from the Brazilian government.

Disclosure statement

There are no conflicts of interest to disclose.

References↩︎