MELBOURNE, Australia —More than 30 years after the body of Scott Johnson, a 27-year-old mathematician from the United States, was found at the bottom of a Sydney cliff, the man accused in the death was convicted on Thursday of murder.

A surprising admission from the defendant, Scott White, 50, earlier in the week cleared the way for the judge to convict him in the 1988 case, bringing to an end a decades-long fight by his brother, Steve Johnson, to find the perpetrators behind his brother’s death. “It’s over, finally,” Steve Johnson, an American tech entrepreneur, said in a telephone interview. “This 33-year odyssey has come to a conclusion.” Scott Johnson, who was gay, was a doctoral student at the Australian National University in Canberra when he was killed on Dec.

8, 1988. He had moved to Australia to be with his partner and had applied for permanent residency. When his body was found in an area frequented by gay men, the Australian police initially ruled the death a suicide.