Since $X(\bx)$ is a random variable, it induces the definition of an expectation value, which we denote by $\EE[\cdot]$. The spatial Fourier transform of $X(\bx)$ is again denoted by $X(\bk)$. By $X^{(i)}(\bx)$, we denote an individual independent realization of $X(\bx)$, that is, an individual map.

The bispectrum is the next higher-order statistic beyond the power spectrum, and is defined as $$\begin{aligned}
B(\bk,\bl,\bmm=\bk+\bl) &= h_k^2 \, \EE[X(\bk) X(\bl) X^*(\bmm) ]. \label{eqBispecDiscrete}\end{aligned}$$ The three-point correlation function is the Fourier transform of the bispectrum, just as the two-point (auto-)correlation function is the Fourier transform of the power spectrum: $$\begin{aligned}
\EE[X(\bx_0 + \bx) X(\bx_0 + \by)X(\bx_0)] 
&= h_k^4 \sum_{\bk,\bl} B(\bk,\bl,\bk+\bl) 
\,e^{i\bk \cdot \bx + i\bl \cdot \by}, \label{eqE3Bispec}\end{aligned}$$ where the expectation value on the left-hand side of Equation  does not depend on $\bx_0$ due to the assumed translation invariance of $X(\bx)$.

Similarly, the skewness of the random field $X(\bx)$, $\skewTrue$, which is the third-order moment normalized by the variance, is the sum of the variance-normalized bispectrum, $$\begin{aligned}
\skewTrue &= \frac{\EE[X(\bx_0)^3]}{\sigma^3} = \frac{h_k^4}{\sigma^3} \sum_{\bk,\bl}  B(\bk,\bl,\bk+\bl) \, , \label{eqSkewnBispec}\\
\sigma^2 &= \EE[X(\bx_0)^2] = h_k^2 \sum_\bk P(\bk), \label{eqSigmaDef}\end{aligned}$$ where the power spectral density $P$ is defined by $$\begin{aligned}
P(\bk) &= h_k^2 \, \EE[|X(\bk)|^2]. \label{eqPowerDef}\end{aligned}$$ Table \[tabFormulas\] provides 