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In this paper, we consider a target random variable $Y \sim \CVG$ distributedaccording to a centered Variance--Gamma distribution. For a generic randomelement $F=I_2(f)$ in the second Wiener chaos with $\E[F^2]= \E[Y^2]$ weestablish a non-asymptotic optimal bound on the distance between $F$ and $Y$ interms of the maximum of difference of the first six cumulants. This six momenttheorem extends the celebrated optimal fourth moment theorem of I.\ Nourdin \&G.\ Peccati for normal approximation. The main body of our analysis constitutesa splitting technique for test functions in the Banach space of Lipschitzfunctions relying on the compactness of the Stein operator. The recentdevelopments around Stein method for Variance--Gamma approximation by R.\ Gauntplay a significant role in our study. As an application we consider thegeneralized Rosenblatt process at the extreme critical exponent, first studiedby S.\ Bai \& M.\ Taqqu.

Optimal Variance--Gamma approximation on the second Wiener chaos