Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with Time-Varying Dimension

This paper focuses mainly on the problem of computing the ${\gamma }^{\mathrm{th}}$, , moment of a random variable $Y_n:={\sum }_{i=1}^n{\alpha }_i{X}_i$ in which the
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org/1998/Math/MathML" display="inline">αi’s are positive real numbers and the $X_i$’s are independent and distributed according to noncentral chi-square distributions.Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of , especially when not all ${\alpha }_i$’s are equal.

We analytically solve this problem by showing that the ${\gamma }^{\mathrm{th}}$ moment of can be expressed in terms of generalized hypergeometric functions.Additionally, we extend our result to computing the ${\gamma }^{\mathrm{th}}$ moment of when $X_i$ is a combination of statistically independent and $G_i$ in which the ’s are distributed according to normal or Maxwell–Boltzmann distributions and the $G_i$’s are distributed according to gamma, Erlang, or exponential distributions.
Short-term wind power combination forecasting method based on wind speed correction of numerical weather prediction
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Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding ${\gamma }^{\mathrm{th}}$ conditional moment.Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests.