If an heir decides to renounce their inheritance in Luxembourg, they must formally declare their renunciation before a notary. By doing so, they waive their rights to any part of the estate, including both assets and liabilities. The renounced portion of the inheritance is then redistributed according to the legal succession order, which may result in other heirs receiving a larger share. It is important for heirs to consider the implications of renunciation, as it is a definitive decision and cannot be reversed.