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Let Y ∼ T Nd (0, Σ, −l, +∞) if Y = Z|Z ≥ −l with Z ∼ Nd (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 LLT = Σ so i.i.d. Y ≥ −l ⇒ LX ≥ −l , Xi ∼ N (0, 1) . So we can write LX ≥ −l as: L1,1 X1 ≥ −l1 , L2,1 X1 + L2,2 X2 ≥ −l2 , ... Ld,1 X1 + Ld,2 X2 + · · · + Ld, dXd ≥ −ld . That is l1 X1 ≥ − , L1,1 l2 + L2,1 X1 X2 ≥ − , L2,2 ... ld + Ld,1 X1 + Ld,2 X2 + · · · + Ld,d−1 Xd−1 Xd ≥ − . Ld,d So we can sample   li + Li,1 X1 + Li,2 X2 + · · · + Li,i−1 Xi−1 Xi |X1 , X2 , . . . , Xi−1 ∼ T N 0, 1, − , +∞ Li,i and set Y = LX.