Andrew Affleck, 32, was given a life sentence and ordered to serve at least 23 years for killing Amanda Cooper, 20, and sisters Anna Teraysa Murray, 18, and Carrie Marie Murray, 12, who had been visiting the flat in Sanderson Avenue, Irvine.
Affleck was found guilty at a trial in 2003 of stuffing burning paper through the letterbox during a hate campaign against Diane Docherty, 20, the woman who occupied the flat and who had a relationship with a convicted rapist.
An appeal was rejected but Affleck, of Irvine, continued to protest his innocence, and in 2008 the Scottish Criminal Cases Review Commission referred the case back to the Court of Criminal Appeal, on the ground that he may have suffered a miscarriage of justice.
The commission said the Crown had failed to disclose to the defence that a key witness, Scott Devine, who said he had seen Affleck running from the scene, had been charged with drugs-supplying offences, for which Affleck was a witness.
After giving evidence at Affleck's trial, some of the charges against Devine had been dropped.
Yesterday, Lord Clarke, sitting with Lords Mackay and Malcolm, announced that the second appeal had been refused.
He said the defence had known there were proceedings against Devine, and added: "We are of the view that, in that situation, the matter could have been explored by counsel for the defence with the witness, and the fact that there was no disclosure of the precise actual nature and full extent of the charges did not result in a real risk of prejudice to the defence with the result that the trial was unfair."
They concluded that there had not been a miscarriage of justice in the case.