# Third term in expansion of multivariate cumulant generating function

I'm doing a Taylor series expansion around $$\mathbf{0}$$ on $$\log{\text{E}[2\mathbf{x}^\intercal\Sigma_{xy}\mathbf{Y}]}$$ where the expectation is over random vector $$\mathbf{Y}$$.

The derivatives evaluated at $$\mathbf{0}$$ are the cumulants for $$\mathbf{Y}$$ so we get $$\log{\text{E}[2\mathbf{x}^\intercal\Sigma_{xy}\mathbf{Y}]} \approx 2\mathbf{x}^\intercal \text{E}[Y] + 2\mathbf{x}^\intercal\Sigma_{xy}\text{Cov}[\mathbf{Y}]\Sigma_{xy}^\intercal\mathbf{x} + \ldots$$

However, I want the third term of this expansion...and I am absolutely lost in both how I would write the next term down and also how I would compute this term from a sample of $$\mathbf{Y}$$ (e.g. the expectation is simply the column means of my sample, the covariance is the covariance matrix)

Any help would be appreciated!

EDIT:

Would this be correct code for computing the sample third central moment from a sample (code in R)

x <- matrix(rnorm(10*100), 100,10) #sample data
y <-array(0,dim=c(10,10,10)) #third central moment
m<-colMeans(x)
for(i in 1:10){
for(j in 1:10){
for(k in 1:10){
y[i,j,k] <- mean((x[,i]-m[i]) * (x[,j]-m[j]) * (x[,k]-m[k]))
}
}
}