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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]))
    }
  }
}
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