If \(Z\) is a standard real Gaussian random variable, then \(\mathbb E[Z^4]=3\).

If \(Z\) is a standard real Gaussian random variable, then \(\mathbb E[Z^3]=0\).

If \(Z\) is a standard real Gaussian random variable, then \(\mathbb E[Z^2]=1\).

A standard real Gaussian random variable satisfies the invariance moment conditions: it has finite fourth absolute moment, its first three moments are \(0,1,0\), and its fourth moment is at most \(9\).

Let \(X\) be a real random variable on a probability space, measurable up to a null set, with \(\mathbb EX=0\) and \(X(\omega )^2=1\) for every \(\omega \). Then \(X\) has finite fourth absolute moment and satisfies \(\mathbb EX^2=1\), \(\mathbb EX^3=0\), and \(\mathbb EX^4\le 9\), so it satisfies the invariance moment conditions.

Let \(B\) be a uniform bit and put \(X=1\) when \(B\) is false and \(X=-1\) when \(B\) is true. Then \(X\) satisfies the invariance moment conditions, including finite fourth absolute moment, first three moments \(0,1,0\), and fourth moment at most \(9\).