TWO YEARS AGO this month my brother Ben ran in the Boston Marathon. It was a triumphant day for him, and my father arranged for the whole family to stay at the Ritz Hotel so that we could watch the race and have a gala congratualtory dinner. We all had things to celebrate: my parents, the success of my father's collected short stories; me, the imminent publication of my first novel; and Ben, the marathon. After a slow start he ran well, crossing the finish line at the Prudential Center just over three hours after the start in Hopkinton 26 miles away. Wrapped in a silver mylar blanket and the warmth of his own achievement, he walked back to the Ritz for a hot bath. We were still out watching the race, but there was a stack of messages from newspapers and wire services waiting at the desk; more messages from radio and television networks were pinned to the door of the room. The attention was absolutely consistent with Ben's euphoric sense of accomplishment, but he knew that something else was going on. And it was. My father had just won the Pulitzer Prize.

What a grand party we had then! The room filled with flowers and the telephone rang and rang and the Ritz chef made a special baked Alaska. The next morning was cold for April, but clear, and we all walked out to the newsstand because People magazine had just published a story on my father and his picture was in The New York Times and The Boston Globe. After that he took us to see Trinity Church and showed us the Augustus Saint-Gaudens statue of Phillips Brooks receiving a special benediction from his good friend Jesus Christ.

It's not quite like anything else, being the child of a famous man, as Kathy Cronkite explains very well in her book, On The Edge of the Spotlight. It's not like anything else at all. Whether it's me or Kathy or Paul Newman's son Scott, whose death at the age of 28 jolted her into investigating the phenomenon and writing this book, or Arlo Guthrie or Bing Crosby's daughter Mary, or Christie Hefner or Jack and Debby Erhard, Werner's children, we all carry names which are associated with accomplishments and personalities that are not our own. The first question people ask us is not, How are you? but, Are you related to? And although, as Cronkite conscientiously points out, the questions that follow are usually meant in a friendly way, they are often tedious, embarrassing and prurient. Sometimes fans will babble about how much they enjoyed a book my father didn't write -- I've never decided if it's better to correct them gently, or just to hoist my phony smile a little higher on my face. Frequently, Cronkite writes, admirers will corner her father and blurt out in their confusion, "Oh! Walter Cronkite! You're my favorite fan!" Nothing is more rewarding than honest praise from a stranger for work well done. This is not that.

"One of my greatest fears," Cronkite writes, "is the way my father's death will ultimately be handled. I am afraid of hearing the news of that event on a TV bulletin . . . I am afraid of not being allowed to grieve, of having to wear a public face, of cameras and microphones being thrust at me and intruding upon me; I am afraid of having to walk to his funeral through a gauntlet of shoving fans."

But for the most part, Cronkite's point of view, and those of the 26 celebrities' children she interviewed, are definitely sunny side up. She thinks her father is great. Christopher Buckley thinks his father (William F.) is great. Jack Ford thinks his father (Gerald R.) is great, and so do Polly Styron (William), Jennifer Buckwald (Art), Monte Schulz (Charles) and Amy Wallace (irving). Arlo Guthrie speaks for all of them, especially those who have chosen their parents' careers, when he tells Cronkite, "It suddenly occurred to me that I wasn't following in his footsteps [just] because I'm his son and doing what he did. It suddenly occurred to me that where I'm going there aren't any footsteps. If there were footsteps it would be a hell of a lot easier. But what I do, no one has done. Because no one's been me."

Cronkite's interviews are studded with family detail, and organized into chapters which deal with almost every aspect of her subjects' bizarre situation: the fans; the difficulty of living a public life (celebrities choose this, their children have no choice); the struggle to create an identity in the shadow of established fame; the mistaken impression many people have that to be famous is to be rich. The writing is as clear and straightforward as the writer, and Cronkite herself seems to have made a totally satisfactory adjustment to her position as an actress and a writer who is also Walter Cronkite's daughter.

A thorough reporter, she is careful to deal with the negative possibilities inherent in the celebrity situation. The son of "one of the leading pioneers in the civil rights movement" asked that his name not be used and spoke reluctantly about his fear of being used by people -- then cut the interview short. Zsa Zsa Gabor's daughter Francesca Hilton was also elusive and reticent, Carrie Fisher turned Cronkite down cold with a sarcastic "good luck," and Scott Newman was angry enough about something to ingest the combination of drugs and alcohol that killed him. "Much of his anxiety was directly related to his father's fame," Cronkite tells us. "One evening, I heard Scott drunkenly accuse someone of being interested in him only because of his father, when in fact the 'antagonist' did not even know who his father was. Later, at the same party, he said belligerently to someone else, 'Don't you know who my father is?'" But Cronkite's style is to take the problems with wit and good humor. In one lovely anecdote she tells about the response of a teacher in a course she was taking under her ex-husband's name when her father's name came up.

"'I just think Walter Cronkite's the greatest,' the teacher said.

"'Yeah, so do I,' I agreed. 'He's my father.' It was the first time I remember overcoming my embarrassment enough for me to volunteer the fact.

"'Yes,' was the response, 'the nation's father figure.'

"'No, I mean, he's really my father.'

"'Oh I know,' my teacher persisted, 'a lot of people feel that way.'

'Yeah,' I said, giving up. 'I guess they do.'"

There are some built-in difficulties in a book like this. The connection between father and son or father and daughter is so dark and deep that it is impossible to know how much celebrity can change it. "Ultimately the major influence on our lives is not our parents fame, but our parents," Cronkite concludes. Most of those interviewed in the book are in their twenties, most are much, much less famous than their parents, and most have come to upbeat terms with the situation -- at least for the moment. It would be interesting to know more about how the children of the famous feel as they grow into their thirties and forties and have children and careers of their own. I wonder what happens when the fame of the children begins to eclipse the fame of the parents.

Maybe someday there will be another book about parents and children, exploring more deeply the competitive nature of the family and the odd mixture of pride and abashment that most of us have felt about the old folks from time to time. Maybe there will be another book about the powerful force that publicly acknowledged success can exert on a personality. Let's hope that it is as sensitive and well-written as this one.