I don't know if there's an answer that could serve all cases; at least I'm not aware of such. And I don't know if history writing in the ancient world was meant for earning a living in general, regardless the fact that could lead to some fame later. Probably as human-beings are different, there would be different or altered motivations for their actions. And possibly this question should be answered only with examples with each ancient historian speaking for their own part, sometimes compared with their own lives; though this last part I think it's difficult.

As you've mentioned Herodotus, Livy and Arrian in the description of your question, here're how they themselves presented some of their works:

Herodotus [1.1] wrote that he wanted the events not to be forgotten, so to serve probably humans' memory, but also mentioning the glory that the narrated deeds were including. So most possibly the deeds themselves were some inspiration.

Livy [1.pr.3] spoke of pleasure-satisfaction in studying the deeds of the greatest of the people in the world [=principis terrarum populi]. Thus, his own people, Romans themselves, should be his own inspiration.

Arrian in the preface of his Anabasis of Alexander [Arr. An. 1.pr] mentioned the big quantity of the existing narratives around Alexander's life, while he underlined his preference on some sources based on their direct knowledge combined with the fact that they wrote after Alexander's death, thus with no existing dependency, so with less bias. This would indicate an intention to clear the truth up, probably on a topic that had interested him.

All the above may show some personal interest on some part of history that could be vaguely considered their own somehow. What they seem to have in common is that the authors were writing about events of the past. And possibly in the prologue of each historic work [or in some other part] we can find some similar statement or description.

However, here're two cases, where the historians were contemporary with the events.

For example Thucydides is mentioning [Thuc. 1.1] that he started writing about the almost 30-year Peloponnesian War shortly after it broke out with the thought that it would be really big and considerable compared to the previous ones. So it seems to be underlined the expected events' importance as his motivation. But I'm wondering if it's implied an expected interest by the future readers, too.

And Polybius [1.1] is writing that many previous chroniclers had indicated the value of history, as truly educative and exercise for political action; thus it wasn't necessary for him to repeat. However, he adds asking if there could be anyone who wouldn't be interested in reading on such great events, such as the Roman conquest of the 2nd c BCE. A more clear reference to future readers.

I hope that these could give a general picture.

However, I'm sure there're more cases to be added.