Let Y ∼ T N d (0 , Σ , − l, + ∞ ) if Y = Z | Z ≥ − l with Z ∼ N d (0 , Σ) i.e. a multivariate normal distribution with mean vector 0 and variance-covariance matrix Σ truncated below − l Let L = chol (Σ) that is LL T = Σ so Y ≥ − l ⇒ LX ≥ − l , X i i.i.d. ∼ N (0 , 1) So we can write LX ≥ − l as: L 1 , 1 X 1 ≥ − l 1 , L 2 , 1 X 1 + L 2 , 2 X 2 ≥ − l 2 , . . . L d, 1 X 1 + L d, 2 X 2 + · · · + Ld, dX d ≥ − l d That is X 1 ≥ − l 1 L 1 , 1 , X 2 ≥ − l 2 + L 2 , 1 X 1 L 2 , 2 , . . . X d ≥ − l d + L d, 1 X 1 + L d, 2 X 2 + · · · + L d,d − 1 X d − 1 L d,d So we can sample X i | X 1 , X 2 , . . . , X i − 1 ∼ T N ( 0 , 1 , − l i + L i, 1 X 1 + L i, 2 X 2 + · · · + L i,i − 1 X i − 1 L i,i , + ∞ ) and set Y = LX