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!


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
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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