Exact Results for the Distribution of Randomly Weighted Sums

Abstract

Dependent random variables play a crucial role in various fields, from finance and statistics to engineering and environmental sciences. Often, interest lies in understanding the aggregate sum of a collection of dependent variables with random weights. In this paper, we provide a comprehensive study of the distribution of the aggregate sum with random weights. Expressions derived include those for the cumulative distribution function, probability density function, conditional expectation, moment generating function, characteristic function, cumulant generating function, moments, variance, skewness, kurtosis, cumulants, value at risk and the expected shortfall. Real data applications are discussed.

1. Introduction

There have been many papers studying the distribution of randomly weighted sums. Papers published since 2020 have studied: inequalities for sums of randomly weighted random variables [1,2,3]; randomly weighted sums of conditionally dependent and dominated varying-tailed increments [4]; second-order tail behavior of randomly weighted heavy-tailed sums [5]; complete and complete moment convergence for randomly weighted sums [6]; approximations for the tail behavior of bidimensional randomly weighted sums [7,8]; complete convergence for randomly weighted sums of random variables satisfying some moment inequalities [9]; complete convergence and complete moment convergence for maximal randomly weighted sums [10]; asymptotic distributions of randomly weighted sums [11]; complete moment convergence for randomly weighted sums of extended negatively dependent sequences [12]; complete convergence for randomly weighted sums [13,14]; complete f-moment convergence for randomly weighted sums [15]; tail asymptotics of randomly weighted sums of dependent strong subexponential random variables [16]; sums of two dependent randomly weighted random variables [17]; tail behavior of randomly weighted sums of dependent subexponential random variables [18]; randomly weighted sums for multivariate Dirichlet distributions [19]; complete convergence and complete integral convergence for randomly weighted sums [20]; complete moment convergence for randomly weighted sums of negatively superadditive-dependent random variables [21]; complete moment convergence for the maximum of randomly weighted sums [22]; the Baum–Katz theorem for randomly weighted sums [23]; asymptotics for the joint tail probability of bidimensional randomly weighted sums [24]; complete moment convergence for randomly weighted sums [25]. Other highly cited papers studying the distribution of randomly weighted sums include [26,27,28,29,30].

But, none of these papers study the exact distribution of randomly weighted sums without considering particular distributions. In this paper, we provide a comprehensive study of the exact distribution of $Z=W_1{X}_1+\cdots +W_n{X}_n$, where $X_1,\dots ,X_n$ are dependent with a copula-induced dependency and the weights $W_1,\dots ,W_n$ are random and subject to the condition that they sum to 1. We consider four situations: (i) $\left(X_1,\dots ,X_n\right)$ are dependent and $\left(W_1,\dots ,W_{n-1}\right)$ are independent; (ii) $\left(X_1,\dots ,X_N\right)$ are dependent conditioned on N, $\left(W_1,\dots ,W_{N-1}\right)$ are independent conditioned on N and N itself is a random variable; (iii) $\left(X_1,\dots ,X_n\right)$ are dependent and $\left(W_1,\dots ,W_{n-1}\right)$ are dependent too; (iv) $\left(X_1,\dots ,X_N\right)$ are dependent conditioned on N, $\left(W_1,\dots ,W_{N-1}\right)$ are dependent too conditioned on N and N itself is a random variable. For each of these situations, we derive formulae for the cumulative distribution function, probability density function, conditional expectation, moment generating function, characteristic function, cumulant generating function, moments, variance, skewness, kurtosis, cumulants, value at risk and the expected shortfall of Z.

Section 2 gives preliminary results on the distribution of Z. Formulae on the distribution of Z are given in Section 3 and Appendix A for situation (i). Formulae on the distribution of Z are given in Section 4 and Appendix B for situation (ii). Formulae on the distribution of Z are given in Section 5 and Appendix C for situation (iii). Formulae on the distribution of Z are given in Section 6 and Appendix D for situation (iv). Real data applications of the results in Section 3, Section 4, Section 5 and Section 6 are discussed in Section 7. Some conclusions are given in Section 8.

2. Preliminary Results

Similarly to [17], we make use of the following assumptions and notations: assume that $\left(X_1,\dots ,X_p\right)$ and $\left(W_1,\dots ,W_p\right)$ are absolutely continuous; let $F_{X_i}$ denote the cumulative distribution function of $X_i$; let $f_{X_i}$ denote the probability density function of $X_i$; let $F_{W_i}$ denote the cumulative distribution function of $W_i$; let $f_{W_i}$ denote the probability density function of $W_i$; let $H_X\left(x_1,\dots ,x_p\right)$ denote the joint cumulative distribution function of $\left(X_1,\dots ,X_p\right)$; let $H_W\left(w_1,\dots ,w_p\right)$ denote the joint cumulative distribution function of $\left(W_1,\dots ,W_p\right)$; let $S_p$ denote the p-dimensional simplex defined by

$$\begin{array}{c}\hfill {\displaystyle S_p=\left\{\left(w_1,\dots ,w_p\right):w_1\ge 0,\dots ,w_p\ge 0,w_1+\cdots +w_p=1\right\};}\end{array}$$

let

$$\begin{array}{c}\hfill {\displaystyle T_p=\left\{\left(v_1,\dots ,v_{p-1}\right):F_{W_1}^{-1}\left(v_1\right)\ge 0,\dots ,F_{W_{p-1}}^{-1}\left(v_{p-1}\right)\ge 0,F_{W_1}^{-1}\left(v_1\right)+\cdots +F_{W_{p-1}}^{-1}\left(v_{p-1}\right)\le 1\right\};}\end{array}$$

let

$${\displaystyle A_z=A_{z,p}=\left\{\left(x_1,\dots ,x_p\right)\in R^p:w_1{x}_1+\left(1-\sum _{i=1}^{p-1}{w}_i\right)x_p\le z\right\};}$$

let

$$\begin{array}{c}\hfill {\displaystyle B_z=B_{z,p}=\left\{\left(u_1,\dots ,u_p\right)\in {[0,1]}^p:w_1{F}_{X_1}^{-1}\left(u_1\right)+\cdots +\left(1-\sum _{i=1}^{p-1}{w}_i\right)w_p{F}_{X_p}^{-1}\left(u_p\right)\le z\right\}.}\end{array}$$

We use the concept of copulas introduced by [31] to describe the dependence among $\left(X_1,\dots ,X_p\right)$ and that among $\left(W_1,\dots ,W_p\right)$.

Definition 1.

A p-dimensional function $C:{[0,1]}^p\to [0,1]$ is a copula if

$$\begin{array}{c}\hfill {\displaystyle C\left(u_1,\dots ,u_{i-1},0,u_{i+1},\dots ,u_p\right)=0,}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle C(1,\dots ,1,u,1,\dots ,1)=u}\end{array}$$

and C is p non-decreasing; that is, for every $B={\prod }_{i=1}^p\left[x_i,y_i\right]\subseteq {[0,1]}^p$,

$$\begin{array}{c}\hfill {\displaystyle {\int }_{B}dC\left(u_1,\dots ,u_p\right)\ge 0.}\end{array}$$

The fundamental theorem of copulas due to [31] is the following.

Theorem 1.

Let H be a p-dimensional cumulative distribution function with marginal distribution functions $F_{Z_i}$, $i=1,\dots ,p$. Then, there exists a copula C such that

$${\displaystyle H\left(z_1,\dots ,z_p\right)=C\left(F_{Z_1}\left(z_1\right),\dots ,F_{Z_p}\left(z_p\right)\right).}$$

Conversely, if $F_{Z_i}$, $i=1,\dots ,p$ are univariate cumulative distribution functions and C is any copula, then H is a p-dimensional cumulative distribution function with marginals $F_{Z_i}$, $i=1,\dots ,p$. Further, if all of the marginals are continuous, then C is unique.

Finally, we introduce the two most popular measures of financial risk highlighted in [17]. They are value at risk and expected shortfall, defined by

$${\displaystyle {\mathrm{VaR}}_p\left(Z\right)=F_Z^{-1}\left(p\right)=\inf \left\{z\in \mathbb{R}:F_Z\left(z\right)\ge p\right\}}$$

and

$${\displaystyle E{S}_p\left(Z\right)=E\left[Z|Z\le {\mathrm{VaR}}_p\left(Z\right)\right],}$$

respectively.

3. Weighted Distribution Results I

In this section, we assume that $X_1,\dots ,X_n$ are dependent random variables; $W_1,\dots ,W_n$ are independent random variables on an n-dimensional simplex; $X_1,\dots ,X_n$ are independent of $W_1,\dots ,W_n$. We study the distribution of $Z=W_1{X}_1+\cdots +W_n{X}_n$.

Theorem 2.

With the stated assumptions, the cumulative distribution function of Z can be expressed as

$${\displaystyle F_Z\left(z\right)=\left\{\begin{array}{c}{\displaystyle {\int }_0^1{F}_{X_1}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1,if n=1,}\hfill \\ \\ \\ {\displaystyle {\int }_{S_n}{\int }_{B_z}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n{f}_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ {\displaystyle d{w}_1\cdots d{w}_{n-1},if n\ge 2,}\hfill \end{array}\right.}$$

where $c_X$ is the density of a copula.

Proof. 

The case $n=1$ is obvious. If $n\ge 2$, then we can write

$$\begin{array}{cc}\hfill {\displaystyle F_Z\left(z\right)}& {\displaystyle =P(Z\le z)}\hfill \\ \hfill & {\displaystyle =P\left(W_1{X}_1+\cdots +W_n{X}_n\le z\right)}\hfill \\ \hfill & {\displaystyle =P\left(W_1{X}_1+\cdots +\left(1-W_1-\cdots -W_{n-1}\right)X_n\le z\right)}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}P\left(W_1{X}_1+\cdots +\left(1-\sum _{i=1}^{n-1}{W}_i\right)X_n\le z|W_1=w_1,\dots ,W_{n-1}=w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}P\left(w_1{X}_1+\dots +\left(1-\sum _{i=1}^{n-1}{w}_i\right)X_n\le z\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{w}_1\cdots d{w}_{n-1},}\hfill \end{array}$$

(1)

where the last step follows because of the independence of $X_1,\dots ,X_n$ and $W_1,\dots ,W_n$. Writing

$$\begin{array}{c}\hfill {\displaystyle P\left(w_1{X}_1+\dots +\left(1-\sum _{i=1}^{n-1}{w}_i\right)X_n\le z\right)={\int }_{A_z}\frac{{\partial }^n}{\partial x_1\cdots \partial x_n}{H}_X\left(x_1,\dots ,x_n\right)d{x}_1\cdots d{x}_n,}\end{array}$$

and using Theorem 1, we can rewrite (1) as

$$\begin{array}{cc}\hfill & {\displaystyle {\int }_{S_n}{\int }_{A_z}\frac{{\partial }^n}{\partial x_1\cdots \partial x_n}{H}_X\left(x_1,\dots ,x_n\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{x}_1\cdots d{x}_{n}d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}{\int }_{A_z}\frac{{\partial }^n}{\partial x_1\cdots \partial x_n}{C}_X\left(F_{X_1}\left(x_1\right),\dots ,F_{X_n}\left(x_n\right)\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{x}_1\cdots d{x}_{n}d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}{\int }_{A_z}{c}_X\left(F_{X_1}\left(x_1\right),\dots ,F_{X_n}\left(x_n\right)\right)f_{X_1}\left(x_1\right)\cdots f_{X_n}\left(x_n\right)}\hfill \\ \hfill & {\displaystyle f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{x}_1\cdots d{x}_{n}d{w}_1\cdots d{w}_{n-1}.}\hfill \end{array}$$

(2)

The result follows by substituting $u_1=F_{X_1}\left(x_1\right),\dots ,u_n=F_{X_n}\left(x_n\right)$ into (2). □

Theorem 3.

With the stated assumptions, the probability density function of Z can be expressed as

$$\begin{array}{c}\hfill {\displaystyle f_Z\left(z\right)=\left\{\begin{array}{c}{\displaystyle {\int }_0^1\frac{1}{w_1}{f}_{X_1}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1,if n=1,}\hfill \\ \\ \\ {\displaystyle {\int }_{S_n}\frac{\partial }{\partial z}\left[{\int }_{B_z}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n\right]}\hfill \\ {\displaystyle f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1},if n\ge 2,}\hfill \end{array}\right.}\end{array}$$

where $c_X$ is the density of a copula.

Proof. 

Immediate from Theorem 2. □

Theorem 4.

With the stated assumptions, the mode of the probability density function of Z is the root of

$$\begin{array}{c}\hfill {\displaystyle {\int }_0^1\frac{1}{w_1^2}{f}_{X_1}^{{}^{\prime }}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1=0}\end{array}$$

if $n=1$ and

$$\begin{array}{c}\hfill {\displaystyle {\int }_{S_n}\frac{{\partial }^2}{\partial z^2}\left[{\int }_{B_z}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n\right]f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}=0}\end{array}$$

if $n\ge 2$, where $c_X$ is the density of a copula.

Proof. 

Immediate from Theorem 3. □

Theorem 5.

With the stated assumptions, the conditional expectation of Z can be expressed as

$${\displaystyle E(Z\mid Z\le z)=\left\{\begin{array}{c}{\displaystyle \frac{1}{F_Z\left(z\right)}{\int }_0^1{\int }_{-\infty }^{\frac{z}{w_1}}{w}_1{x}_1{f}_{X_1}\left(x_1\right)f_{W_1}\left(w_1\right)d{x}_{1}d{w}_1,if n=1,}\hfill \\ \\ \\ {\displaystyle \frac{1}{F_Z\left(z\right)}{\int }_{S_n}{\int }_{B_z}\left[w_1{F}_{X_1}^{-1}\left(u_1\right)+\cdots +\left(1-\sum _{i=1}^{n-1}{w}_i\right)F_{X_n}^{-1}\left(u_n\right)\right]}\hfill \\ {\displaystyle c_X\left(u_1,\dots ,u_n\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ {\displaystyle d{u}_1\cdots d{u}_{n}d{w}_1\cdots d{w}_{n-1},if n\ge 2,}\hfill \end{array}\right.}$$

where $c_X$ is the density of a copula.

Proof. 

The case $n=1$ is obvious. If $n\ge 2$, then we can write

$$\begin{array}{cc}\hfill {\displaystyle E\left(Z\right|Z\le z)}& {\displaystyle ={\int }_{S_n}E\left(Z|Z\le z,W_1=w_1,\dots ,W_n=w_n\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}E\left(W_1{X}_1+\cdots +W_n{X}_n\mid W_1{X}_1+\cdots +W_n{X}_n\le z,W_1=w_1,\dots ,W_n=w_n\right)}\hfill \\ \hfill & {\displaystyle f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle =\frac{1}{F_Z\left(z\right)}{\int }_{S_n}{\int }_{A_z}\left(w_1{x}_1+\cdots +w_n{x}_n\right)\frac{{\partial }^n}{\partial x_1\cdots \partial x_n}{H}_X\left(x_1,\dots ,x_n\right)}\hfill \\ \hfill & {\displaystyle d{x}_1\cdots d{x}_n{f}_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle =\frac{1}{F_Z\left(z\right)}{\int }_{S_n}{\int }_{A_z}\left(w_1{x}_1+\cdots +w_n{x}_n\right)\frac{{\partial }^n}{\partial x_1\cdots \partial x_n}{C}_X\left(F_{X_1}\left(x_1\right),\dots ,F_{X_n}\left(x_n\right)\right)}\hfill \\ \hfill & {\displaystyle d{x}_1\cdots d{x}_n{f}_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle =\frac{1}{F_Z\left(z\right)}{\int }_{S_n}{\int }_{A_z}\left(w_1{x}_1+\cdots +w_n{x}_n\right)}\hfill \\ \hfill & {\displaystyle c_X\left(F_{X_1}\left(x_1\right),\dots ,F_{X_n}\left(x_n\right)\right)f_{X_1}\left(x_1\right)\cdots f_{X_n}\left(x_n\right)d{x}_1\cdots d{x}_n}\hfill \\ \hfill & {\displaystyle f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}.}\hfill \end{array}$$

(3)

The result follows by substituting $u_1=F_{X_1}\left(x_1\right),\dots ,u_n=F_{X_n}\left(x_n\right)$ into (3). □

4. Weighted Distribution Results II

In this section, we assume that $X_1,\dots ,X_N$ are dependent random variables for a fixed N; $W_1,\dots ,W_N$ are independent random variables on an N-dimensional simplex for a fixed N; $X_1,\dots ,X_N$ are independent of $W_1,\dots ,W_N$ for a fixed N; N is a positive-integer-valued random variable independent of $X_1,X_2,\dots$ and $W_1,W_2,\dots$. We study the distribution of $Z=W_1{X}_1+\cdots +W_N{X}_N$.

Theorem 6.

With the stated assumptions, the cumulative distribution function of Z can be expressed as

$$\begin{array}{cc}\hfill {\displaystyle F_Z\left(z\right)}& {\displaystyle =P(N=1){\int }_0^1{F}_{X_1}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1}\hfill \\ \hfill & {\displaystyle +\sum _{n=2}^{\infty }P(N=n){\int }_{S_n}{\int }_{B_{z,n}}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n}\hfill \\ \hfill & {\displaystyle f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1},}\hfill \end{array}$$

where $c_X$ is the density of a copula.

Proof. 

Write

$$\begin{array}{c}\hfill {\displaystyle F_Z\left(z\right)=P(Z\le z)=\sum _{n=1}^{\infty }P\left(Z\le z\mid N=n\right)P(N=n)}\end{array}$$

and use Theorem 2. □

Theorem 7.

With the stated assumptions, the probability density function of Z can be expressed as

$$\begin{array}{cc}\hfill {\displaystyle f_Z\left(z\right)}& {\displaystyle =P(N=1){\int }_0^1\frac{1}{w_1}{f}_{X_1}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1}\hfill \\ \hfill & {\displaystyle +\sum _{n=2}^{\infty }P(N=n){\int }_{S_n}\frac{\partial }{\partial z}\left\{{\int }_{B_{z,n}}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n\right\}}\hfill \\ \hfill & {\displaystyle f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1},}\hfill \end{array}$$

where $c_X$ is the density of a copula.

Proof. 

Immediate from Theorem 6. □

Theorem 8.

With the stated assumptions, the mode of the probability density function of Z is the root of

$$\begin{array}{cc}\hfill & {\displaystyle P(N=1){\int }_0^1\frac{1}{w_1^2}{f}_{X_1}^{{}^{\prime }}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1}\hfill \\ \hfill & {\displaystyle +\sum _{n=2}^{\infty }P(N=n){\int }_{S_n}\frac{{\partial }^2}{\partial z^2}\left\{{\int }_{B_{z,n}}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n\right\}{f}_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}=0,}\hfill \end{array}$$

where $c_X$ is the density of a copula.

Proof. 

Immediate from Theorem 7. □

Theorem 9.

With the stated assumptions, the conditional expectation of Z can be expressed as

$$\begin{array}{cc}\hfill & {\displaystyle E\left(Z\mid Z\le z\right)=\frac{P(N=1)}{F_Z\left(z\right)}{\int }_0^1{\int }_{-\infty }^{\frac{z}{w_1}}{w}_1{x}_1{f}_{X_1}\left(x_1\right)f_{W_1}\left(w_1\right)d{x}_{1}d{w}_1}\hfill \\ \hfill & {\displaystyle +\frac{1}{F_Z\left(z\right)}\sum _{n=2}^{\infty }P(N=n){\int }_{S_n}{\int }_{B_{z,n}}\left[w_1{F}_{X_1}^{-1}\left(u_1\right)+\cdots +\left(1-\sum _{i=1}^{n-1}{w}_i\right)F_{X_n}^{-1}\left(u_n\right)\right]}\hfill \\ \hfill & {\displaystyle c_X\left(u_1,\dots ,u_n\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{u}_1\cdots d{u}_{n}d{w}_1\cdots d{w}_{n-1},}\hfill \end{array}$$

where $c_X$ is the density of a copula.

Proof. 

Write

$$\begin{array}{c}\hfill {\displaystyle E\left(Z\mid Z\le z\right)=\sum _{n=1}^{\infty }E\left(Z\mid Z\le z,N=n\right)P(N=n)}\end{array}$$

and use Theorem 5. □

5. Weighted Distribution Results III

In this section, we assume that $X_1,\dots ,X_n$ are dependent random variables; $W_1,\dots ,W_n$ are dependent random variables on an n-dimensional simplex; $X_1,\dots ,X_n$ are independent of $W_1,\dots ,W_n$. We study the distribution of $Z=W_1{X}_1+\cdots +W_n{X}_n$.

Theorem 10.

With the stated assumptions, the cumulative distribution function of Z can be expressed as

$${\displaystyle F_Z\left(z\right)=\left\{\begin{array}{c}{\displaystyle {\int }_0^1{F}_{X_1}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1,if n=1,}\hfill \\ \\ \\ {\displaystyle {\int }_{T_n}{\int }_{B_z}{c}_X\left(u_1,\dots ,u_n\right)c_W\left(v_1,\dots ,v_{n-1}\right)d{u}_1\cdots d{u}_{n}d{v}_1\cdots d{v}_{n-1},if n\ge 2,}\hfill \end{array}\right.}$$

where $c_X$ and $c_W$ are densities of copulas.

Proof. 

The case $n=1$ is obvious. If $n\ge 2$, then we can write

$$\begin{array}{cc}\hfill {\displaystyle F_Z\left(z\right)}& {\displaystyle =P(Z\le z)}\hfill \\ \hfill & {\displaystyle =P\left(W_1{X}_1+\cdots +W_n{X}_n\le z\right)}\hfill \\ \hfill & {\displaystyle =P\left(W_1{X}_1+\cdots +\left(1-W_1-\cdots -W_{n-1}\right)X_n\le z\right)}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}P\left(W_1{X}_1+\cdots +\left(1-\sum _{i=1}^{n-1}{W}_i\right)X_n\le z|W_1=w_1,\dots ,W_{n-1}=w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle \frac{{\partial }^n}{\partial w_1\cdots \partial w_{n-1}}{H}_W\left(w_1,\dots ,w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}P\left(W_1{X}_1+\cdots +\left(1-\sum _{i=1}^{n-1}{W}_i\right)X_n\le z|W_1=w_1,\dots ,W_{n-1}=w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle \frac{{\partial }^n}{\partial w_1\cdots \partial w_{n-1}}{C}_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}P\left(W_1{X}_1+\cdots +\left(1-\sum _{i=1}^{n-1}{W}_i\right)X_n\le z|W_1=w_1,\dots ,W_{n-1}=w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle c_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}{\int }_{A_z}\frac{{\partial }^n}{\partial x_1\cdots \partial x_n}{H}_X\left(x_1,\dots ,x_n\right)}\hfill \\ \hfill & {\displaystyle c_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{x}_1\cdots d{x}_{n}d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}{\int }_{A_z}\frac{{\partial }^n}{\partial x_1\cdots \partial x_n}{C}_X\left(F_{X_1}\left(x_1\right),\dots ,F_{X_n}\left(x_n\right)\right)}\hfill \\ \hfill & {\displaystyle c_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{x}_1\cdots d{x}_{n}d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}{\int }_{A_z}{c}_X\left(F_{X_1}\left(x_1\right),\dots ,F_{X_n}\left(x_n\right)\right)f_{X_1}\left(x_1\right)\cdots f_{X_n}\left(x_n\right)}\hfill \\ \hfill & {\displaystyle c_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{x}_1\cdots d{x}_{n}d{w}_1\cdots d{w}_{n-1}.}\hfill \end{array}$$

(4)

The result follows by substituting $u_1=F_{X_1}\left(x_1\right),\dots ,u_n=F_{X_n}\left(x_n\right)$ and $v_1=F_{W_1}\left(w_1\right),\dots ,v_{n-1}=F_{W_{n-1}}\left(w_{n-1}\right)$ into (4). □

Theorem 11.

With the stated assumptions, the probability density function of Z can be expressed as

$${\displaystyle f_Z\left(z\right)=\left\{\begin{array}{c}{\displaystyle {\int }_0^1\frac{1}{w_1}{f}_{X_1}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1,if n=1,}\hfill \\ \\ \\ {\displaystyle {\int }_{T_n}\frac{\partial }{\partial z}\left[{\int }_{B_z}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n\right]c_W\left(v_1,\dots ,v_{n-1}\right)d{v}_1\cdots d{v}_{n-1},if n\ge 2,}\hfill \end{array}\right.}$$

where $c_X$ and $c_W$ are densities of copulas.

Proof. 

Immediate from Theorem 10. □

Theorem 12.

With the stated assumptions, the mode of the probability density function of Z is the root of

$$\begin{array}{c}\hfill {\displaystyle {\int }_0^1\frac{1}{w_1^2}{f}_{X_1}^{{}^{\prime }}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1=0}\end{array}$$

if $n=1$ and

$$\begin{array}{c}\hfill {\displaystyle {\int }_{T_n}\frac{{\partial }^2}{\partial z^2}\left[{\int }_{B_z}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n\right]c_W\left(v_1,\dots ,v_{n-1}\right)d{v}_1\cdots d{v}_{n-1}=0}\end{array}$$

if $n\ge 2$, where $c_X$ and $c_W$ are densities of copulas.

Proof. 

Immediate from Theorem 11. □

Theorem 13.

With the stated assumptions, the conditional expectation of Z can be expressed as

$${\displaystyle E(Z\mid Z\le z)=\left\{\begin{array}{c}{\displaystyle \frac{1}{F_Z\left(z\right)}{\int }_0^1{\int }_{-\infty }^{\frac{z}{w_1}}{w}_1{x}_1{f}_{X_1}\left(x_1\right)f_{W_1}\left(w_1\right)d{x}_{1}d{w}_1,if n=1,}\hfill \\ \\ \\ {\displaystyle \frac{1}{F_Z\left(z\right)}{\int }_{T_n}{\int }_{B_z}\left[F_{W_1}^{-1}\left(v_1\right)F_{X_1}^{-1}\left(u_1\right)+\cdots +\left(1-\sum _{i=1}^{n-1}{F}_{W_i}^{-1}\left(v_i\right)\right)F_{X_n}^{-1}\left(u_n\right)\right]}\hfill \\ {\displaystyle c_X\left(u_1,\dots ,u_n\right)c_W\left(v_1,\dots ,v_{n-1}\right)d{u}_1\cdots d{u}_{n}d{v}_1\cdots d{v}_{n-1},}\hfill \\ {\displaystyle if n\ge 2,}\hfill \end{array}\right.}$$

where $c_X$ and $c_W$ are densities of copulas.

Proof. 

The case $n=1$ is obvious. If $n\ge 2$, then we can write

$$\begin{array}{cc}\hfill {\displaystyle E\left(Z\right|Z\le z)}& {\displaystyle ={\int }_{S_n}E\left(Z|Z\le z,W_1=w_1,\dots ,W_n=w_n\right)\frac{{\partial }^{n-1}}{\partial w_1\cdots \partial w_{n-1}}{H}_W\left(w_1,\dots ,w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}E\left(W_1{X}_1+\cdots +W_n{X}_n\mid W_1{X}_1+\cdots +W_n{X}_n\le z,W_1=w_1,\dots ,W_n=w_n\right)}\hfill \\ \hfill & {\displaystyle \frac{{\partial }^{n-1}}{\partial w_1\cdots \partial w_{n-1}}{H}_W\left(w_1,\dots ,w_{n-1}\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}E\left(W_1{X}_1+\cdots +W_n{X}_n\mid W_1{X}_1+\cdots +W_n{X}_n\le z,W_1=w_1,\dots ,W_n=w_n\right)}\hfill \\ \hfill & {\displaystyle \frac{{\partial }^{n-1}}{\partial w_1\cdots \partial w_{n-1}}{C}_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle ={\int }_{S_n}E\left(W_1{X}_1+\cdots +W_n{X}_n\mid W_1{X}_1+\cdots +W_n{X}_n\le z,W_1=w_1,\dots ,W_n=w_n\right)}\hfill \\ \hfill & {\displaystyle c_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle =\frac{1}{F_Z\left(z\right)}{\int }_{S_n}{\int }_{A_z}\left(w_1{x}_1+\cdots +w_n{x}_n\right)}\hfill \\ \hfill & {\displaystyle \frac{{\partial }^n}{\partial x_1\cdots \partial x_n}{H}_X\left(x_1,\dots ,x_n\right)d{x}_1\cdots d{x}_n}\hfill \\ \hfill & {\displaystyle c_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle =\frac{1}{F_Z\left(z\right)}{\int }_{S_n}{\int }_{A_z}\left(w_1{x}_1+\cdots +w_n{x}_n\right)}\hfill \\ \hfill & {\displaystyle \frac{{\partial }^n}{\partial x_1\cdots \partial x_n}{C}_X\left(F_{X_1}\left(x_1\right),\dots ,F_{X_n}\left(x_n\right)\right)d{x}_1\cdots d{x}_n}\hfill \\ \hfill & {\displaystyle c_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{w}_1\cdots d{w}_{n-1}}\hfill \\ \hfill & {\displaystyle =\frac{1}{F_Z\left(z\right)}{\int }_{S_n}{\int }_{A_z}\left(w_1{x}_1+\cdots +w_n{x}_n\right)}\hfill \\ \hfill & {\displaystyle c_X\left(F_{X_1}\left(x_1\right),\dots ,F_{X_n}\left(x_n\right)\right)f_{X_1}\left(x_1\right)\cdots f_{X_n}\left(x_n\right)d{x}_1\cdots d{x}_n}\hfill \\ \hfill & {\displaystyle c_W\left(F_{W_1}\left(w_1\right),\dots ,F_{W_{n-1}}\left(w_{n-1}\right)\right)f_{W_1}\left(w_1\right)\cdots f_{W_{n-1}}\left(w_{n-1}\right)}\hfill \\ \hfill & {\displaystyle d{w}_1\cdots d{w}_{n-1}.}\hfill \end{array}$$

(5)

The result follows by substituting $u_1=F_{X_1}\left(x_1\right),\dots ,u_n=F_{X_n}\left(x_n\right)$ and $v_1=F_{W_1}\left(w_1\right),\dots ,v_{n-1}=F_{W_{n-1}}\left(w_{n-1}\right)$ into (5). □

6. Weighted Distribution Results IV

In this section, we assume that $X_1,\dots ,X_N$ are dependent random variables for a fixed N; $W_1,\dots ,W_N$ are dependent random variables on an N-dimensional simplex for a fixed N; $X_1,\dots ,X_N$ are independent of $W_1,\dots ,W_N$ for a fixed N; N is a positive-integer-valued random variable independent of $X_1,X_2,\dots$ and $W_1,W_2,\dots$. We study the distribution of $Z=W_1{X}_1+\cdots +W_N{X}_N$.

Theorem 14.

With the stated assumptions, the cumulative distribution function of Z can be expressed as

$$\begin{array}{cc}\hfill & {\displaystyle F_Z\left(z\right)=P(N=1){\int }_0^1{F}_{X_1}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1}\hfill \\ \hfill & {\displaystyle +\sum _{n=2}^{\infty }P(N=n){\int }_{T_n}{\int }_{B_{z,n}}{c}_X\left(u_1,\dots ,u_n\right)c_W\left(v_1,\dots ,v_{n-1}\right)d{u}_1\cdots d{u}_{n}d{v}_1\cdots d{v}_{n-1},}\hfill \end{array}$$

where $c_X$ and $c_W$ are densities of copulas.

Proof. 

Write

$$\begin{array}{c}\hfill {\displaystyle F_Z\left(z\right)=P(Z\le z)=\sum _{n=1}^{\infty }P\left(Z\le z\mid N=n\right)P(N=n)}\end{array}$$

and use Theorem 10. □

Theorem 15.

With the stated assumptions, the probability density function of Z can be expressed as

$$\begin{array}{cc}\hfill & {\displaystyle f_Z\left(z\right)=P(N=1){\int }_0^1\frac{1}{w_1}{f}_{X_1}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1}\hfill \\ \hfill & {\displaystyle +\sum _{n=2}^{\infty }P(N=n){\int }_{T_n}\frac{\partial }{\partial z}\left\{{\int }_{B_{z,n}}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n\right\}{c}_W\left(v_1,\dots ,v_{n-1}\right)d{v}_1\cdots d{v}_{n-1},}\hfill \end{array}$$

where $c_X$ and $c_W$ are densities of copulas.

Proof. 

Immediate from Theorem 14. □

Theorem 16.

With the stated assumptions, the mode of the probability density function of Z is the root of

$$\begin{array}{cc}\hfill & {\displaystyle P(N=1){\int }_0^1\frac{1}{w_1^2}{f}_{X_1}^{{}^{\prime }}\left(\frac{z}{w_1}\right)f_{W_1}\left(w_1\right)d{w}_1}\hfill \\ \hfill & {\displaystyle +\sum _{n=2}^{\infty }P(N=n){\int }_{T_n}\frac{{\partial }^2}{\partial z^2}\left\{{\int }_{B_{z,n}}{c}_X\left(u_1,\dots ,u_n\right)d{u}_1\cdots d{u}_n\right\}{c}_W\left(v_1,\dots ,v_{n-1}\right)d{v}_1\cdots d{v}_{n-1}=0,}\hfill \end{array}$$

where $c_X$ and $c_W$ are densities of copulas.

Proof. 

Immediate from Theorem 15. □

Theorem 17.

With the stated assumptions, the conditional expectation of Z can be expressed as

$$\begin{array}{cc}\hfill & {\displaystyle E(Z\mid Z\le z)=\frac{P(N=1)}{F_Z\left(z\right)}{\int }_0^1{\int }_{-\infty }^{\frac{z}{w_1}}{w}_1{x}_1{f}_{X_1}\left(x_1\right)f_{W_1}\left(w_1\right)d{x}_{1}d{w}_1}\hfill \\ \hfill & {\displaystyle +\frac{1}{F_Z\left(z\right)}\sum _{n=2}^{\infty }P(N=n){\int }_{T_n}{\int }_{B_{z,n}}\left[F_{W_1}^{-1}\left(v_1\right)F_{X_1}^{-1}\left(u_1\right)+\cdots +\left(1-\sum _{i=1}^{n-1}{F}_{W_i}^{-1}\left(v_i\right)\right)F_{X_n}^{-1}\left(u_n\right)\right]}\hfill \\ \hfill & {\displaystyle c_X\left(u_1,\dots ,u_n\right)c_W\left(v_1,\dots ,v_{n-1}\right)d{u}_1\cdots d{u}_{n}d{v}_1\cdots d{v}_{n-1},}\hfill \end{array}$$

where $c_X$ and $c_W$ are densities of copulas.

Proof. 

Write

$$\begin{array}{c}\hfill {\displaystyle E\left(Z\mid Z\le z\right)=\sum _{n=1}^{\infty }E\left(Z\mid Z\le z,N=n\right)P(N=n)}\end{array}$$

and use Theorem 13. □

7. Real Data Applications

In this section, we discuss real examples where the results of Section 3, Section 4, Section 5 and Section 6 can be applied. Section 7.1 and Section 7.3 use data on the gross domestic product and population of various countries from 1960 to 2022. These data were obtained from the World Bank. We could not obtain real data to illustrate the situations described in Section 7.2 and Section 7.4.

7.1. Weighted Distribution I

Consider the five largest economics in the world: the United States of America, China, Japan, Germany and India. Let $X_1=$ gross domestic product for the United States of America, $X_2=$ gross domestic product for China, $X_3=$ gross domestic product for Japan, $X_4=$ gross domestic product for Germany, $X_5=$ gross domestic product for India, $W_1=$ population of the United States of America/total population of the five largest economies, $W_2=$ population of China/total population of the five largest economies, $W_3=$ population of Japan/total population of the five largest economies, $W_4=$ population of Germany/total population of the five largest economies and $W_5=$ population of India/total population of the five largest economies. Then, $Z=W_1{X}_1+\cdots +W_5{X}_5$ will denote the gross domestic product per person in the five largest economies. In this case, $\left(X_1,\dots ,X_5\right)$ are dependent because of the trade among the largest economies. $\left(W_1,\dots ,W_4\right)$ are not likely to be dependent because there is not so much immigration between India and China or India and Japan.

The marginal distributions of $\left(X_1,\dots ,X_5\right)$ and $\left(W_1,\dots ,W_5\right)$ were modeled using non-parametric kernel densities. The joint distribution of $\left(X_1,\dots ,X_5\right)$ was modeled using the Clayton copula [32]. Figure 1 shows the fitted quantile function of Z.

7.2. Weighted Distribution II

Suppose that there are N accidents at a busy intersection on a given day. Clearly, N is a random variable. Possible models for N could include the Poisson distribution. Let $X_1,\dots ,X_N$ denote the damages caused by the N accidents. Some of the accidents could involve more than one individual or more than one vehicle. Let $W_1,\dots ,W_N$ denote scale factors corresponding to one individual or one vehicle. Then, $Z=W_1{X}_1+\cdots +W_N{X}_N$ will denote the average damage for the individual or vehicle. In this case, $\left(X_1,\dots ,X_N\right)$ given N could be dependent because of the possibility of serial accidents. But, $\left(W_1,\dots ,W_{N-1}\right)$ given N are likely to be independent.

7.3. Weighted Distribution III

Consider the countries in the Indian subcontinent: India, Nepal, Bangladesh, Pakistan, Bhutan, Sri Lanka and the Maldives. Let $X_1=$ gross domestic product for India, $X_2=$ gross domestic product for Nepal, $X_3=$ gross domestic product for Bangladesh, $X_4=$ gross domestic product for Pakistan, $X_5=$ gross domestic product for Bhutan, $X_6=$ gross domestic product for Sri Lanka, $X_7=$ gross domestic product for the Maldives, $W_1=$ population of India/total population of the Indian subcontinent, $W_2=$ population of Nepal/total population of the Indian subcontinent, $W_3=$ population of Bangladesh/total population of the Indian subcontinent, $W_4=$ population of Pakistan/total population of the Indian subcontinent, $W_5=$ population of Bhutan/total population of the Indian subcontinent, $W_6=$ population of Sri Lanka/total population of the Indian subcontinent and $W_7=$ population of the Maldives/total population of the Indian subcontinent. Then, $Z=W_1{X}_1+\cdots +W_7{X}_7$ will denote the gross domestic product per person in the Indian subcontinent. In this case, $\left(X_1,\dots ,X_7\right)$ are dependent because of the trade among the countries. Likewise, $\left(W_1,\dots ,W_6\right)$ are dependent because of immigration among neighboring countries.

The marginal distributions of $\left(X_1,\dots ,X_7\right)$ and $\left(W_1,\dots ,W_7\right)$ were modeled using non-parametric kernel densities. The joint distribution of $\left(X_1,\dots ,X_7\right)$ and the joint distribution of $\left(W_1,\dots ,W_7\right)$ were modeled using the Clayton copula. Figure 2 shows the fitted quantile function of Z. There appears to be a shift at around $p=0.5$.

7.4. Weighted Distribution IV

Let N denote the number of goals scored in a football match. Let $X_i$, $i=1,\dots ,N$ denote the amount paid out by a betting company if i goals were scored. Let $W_i$, $i=1,\dots ,N$ denote the odds of there being i goals scored determined by the betting company. Then, $Z=W_1{X}_1+\cdots +W_N{X}_N$ will denote the average amount paid out by the company. In this case, $\left(X_1,\dots ,X_N\right)$ will be dependent and so will $\left(W_1,\dots ,W_{N-1}\right)$.

8. Conclusions

Let $X_1,\dots ,X_n$ be random variables and let $W_1,\dots ,W_n$ be random weights, independent of $X_1,\dots ,X_n$ taking values on the unit simplex. Let $Z=W_1{X}_1+\cdots +W_n{X}_n$. In this paper, we have derived formulae for the cumulative distribution function, probability density function, conditional expectation, moment generating function, characteristic function, cumulant generating function, mth moment, value at risk and expected shortfall of Z, considering four possible situations. We have discussed real situations where these formulas can be applied. Future work is to derive exact results for the distribution of $\mathbf{Z}={\mathbf{W}}_1{\mathbf{X}}_1+\cdots +\left(\mathbf{I}-{\mathbf{W}}_1-\cdots -{\mathbf{W}}_{n-1}\right){\mathbf{X}}_n$, where ${\mathbf{X}}_1,\dots ,{\mathbf{X}}_n$ are dependent multivariate random variables.