Robert B. Bush Jr., a 40-year-old construction worker, was shot to death Oct. 14 on Martin Luther King Avenue in Southeast Washington.

When he died, his former wife, Linda, and their three young children lost the $150 in child support Bush had been paying each month. The family was also saddled with an $1,800 funeral bill.

In years past, that would have been the end of the story. Linda Bush, 35, an unemployed Loveville, Md., resident, said she and her children, aged 16, 12, and 11, "would have struggled on through the best we could" each month on $306 in federal food stamps and the $411 they receive from Maryland's Department of Social Services.

But because of a D.C. law that went into effect Oct. 1, just two weeks before Bush's death, his family may be able to recover the funeral costs and the lost child support payments. Linda Bush's claim is likely to be the first one honored under the District's new Crime Victims Compensation Program, according to John Dean, the Employment Services Department official who oversees the program.

"There's no guarantee that the claim will be approved, but it is one of five cases we now have in the pipeline," Dean said last week. "It looks solid."

Approved last February by the District's City Council, the D.C. victims compensation program provides a maximum $25,000 payment to innocent victims of violent crime or, in the event of their death, to their dependents. The program was established primarily to help the poor, and eligibility is based on need. Victims with insurance, large savings or other available assets may not be eligible.

The program is designed to assist victims in some cases by providing medical and funeral expenses and economic support, sometimes including replacment of lost earnings. The District has a $500,000 pool to dispense to crime victims.

"It's the oldest kind of compensation in the world, one that goes back to tribal society," said Dean. "In those days, if you killed someone or hurt a family member, you would have to make reparation to that family. In the 12th century, all that changed: you had to make your payment to the kings."

By the 1950s, however, prominent European jurists were calling for a return to victim compensation; England enacted a victim rights program in 1964, and California followed suit a year later. Now 37 states, including Maryland and Virginia, have some system for compensating victims of violent crime.

In 1978, Virginia's victims compensation law came to the aid of Linda Bush's brother, Roosevelt Dickerson, who was shot in the shoulder during a confrontation in the parking lot of a Richmond hospital, she said. The shooting, which injured his spine, left him paralyzed from the waist down. The state paid for his hospital stay and other expenses, Bush said.

The District has reciprocal agreements with Maryland and Virginia that allow residents of those states to file claims if they or their relatives are injured or killed while in Washington. Maryland's criminal compensation board limits grants to victims' dependents to $45,000 in fatal cases; Virginia's crime victim program allows up to $10,000 to victims or their dependents.

In filing her claim, Bush had to list her mortgage and living expenses, food costs and other monthly bills, and was required to give Dean a copy of her husband's death certificate. Bush, who quit her job as a corrections officer in August, had not kept up the payments on her former husband's life insurance policy.

"Without a program like D.C.'s, it would have been real tough on this family, going without the child support," Bush said. "The victims' program is a good idea. In my case, my husband was alive one minute and the next minute he's dead. What are the children supposed to do about that?"

D.C. police said last week they have charged one man, whom they identified as Douglas Michael Wright, in connection with Bush's killing.

Although interest in the victims compensation program is running high -- Dean said he gets about 15 telephone inquiries a day -- few cases are actually being processed during the program's early stages, he said. The victims compensation office has no investigative staff, only Dean and two clerks. But Dean said his budget will provide for at least two claims examiners. He will soon hire the first of those, he said.

''The program is a response to a need that hadn't been filled before,'' said Dean, a municipal government veteran who has worked in New York City. ''How many times have we heard about somebody walking along and becoming a victim of crime? When that happens, you can incur expenses which could literally take you under''

The victims compensation program, which was proposed to the City Council by District Mayor Marion S. Barry Jr., was also designed to improve relations between the public and police, officials said. In return for compensation, victims are obligated to testify in court unless excused by Dean or the police, Dean said. Victims who do not co-operate with police and prosecutors are ineligible for compensation.

''The whole point is that the compensation is there for people,'' said Dean. ''In return, we expect their help.''