The characteristic function of order statistics

[EngWiPy]

Suppose that $Y=\sum_{k=1}^KX_{(k)}$, where $X_{(1)}\leq X_{(2)}\leq\cdots X_{(N)}$ and ($N\geq K$). I want to find the characteristic function of $Y$ as

$$\phi(jvY)=E\left[e^{jvY}\right]=E\left[e^{jv\sum_{k=1}^KX_{(k)}}\right]$$

In the case where $\{X\}$ are i.i.d random variables, the above characteristic function will be

$$\phi(jvY)=\prod_{k=1}^K\phi(jvX_k)$$

but when the random variables are ordered, they are no longer independent. How can the characteristic function be found in this case?

 

[mathman]

, Are the random variables independently chosen and then ordered? If so it doesn't matter, since the definition of Y doesn't depend on the order. . On the other hand if your question means choose a random variable, then a second variable greater than the first, etc., it is a different problem.

 

[EngWiPy]

, Are the random variables independently chosen and then ordered? If so it doesn't matter, since the definition of Y doesn't depend on the order. . On the other hand if your question means choose a random variable, then a second variable greater than the first, etc., it is a different problem.

The selection is like this: I have $N$ i.i.d random variables. Then $Y$ is the sum of the $K$ smallest random variables of the $N$ random variables.

 

[EngWiPy]

I found that the joint PDF

$f_{X_{(1)},\,X_{(2)},\ldots,\,X_{(K)}}(x_1,\,x_2,\,\dots,\,x_K)=K!\prod_{k=1}^Kf_X(x_k)$

for $0<x_1<x_2<\cdots<x_K<\infty$, where $f_X(x_k)$ is the PDF of the original random variables before ordering. Then we can write the characteristic function as

$$\phi(jvY)=\int_{x_1=0}^{\infty}\int_{x_2>x_1}\cdots\int_{x_K>x_{K-1}}K!\prod_{k=1}^Ke^{jvx_k}f_X(x_k)\,dx_1\,dx_2\cdots\,dx_K$$

which is the multiplication of $K$ integrals

$$\phi(jvY)=\left(\int_{x_1=0}^{\infty}e^{jvx_1}f_X(x_1)\,dx_1\right)\left(\int_{x_2>x_1}e^{jvx_2}f_X(x_2)\,dx_2\right)\cdots \left(\int_{x_K>x_{K-1}}e^{jvx_K}f_X(x_K)\,dx_K\right)$$

Is this right?