When are a prince's stats decided?

[Ironside]

The Muslim factions are all somewhat disadvantaged by this situation being hard-coded for them. (Or is it that, when I play as them, I just think it's completely absurd for them to request marriage to a Catholic\Orthodox princess?) One way or another, the princes don't get given access to the harem until they accede to the throne, which doesn't seem realistic to me.

Rather the opposite, the muslim hiers does always get access to the harem. The problem is that it can only exist 6 male hiers at the same time, so a king with 5 brothers can only have 1 son and cannot have children before he becomes king.

In my current English campaign, I have a king aged 59, whose influence has gone up substantially during his reign (from ~5 to 9) but whose command stars have not increased at all (from 4). His son is 41; command 3; married and has under-age children.

Based on what macsen and Phill have been saying and, since the heir has been practically inactive for much of his life, when his eldest son comes of age, in about 4 years' time, he will have lesser stats, like command 2, rather than stats based on what his grandfather has acquired. All assuming that my current king is still alive when the grandson comes of age, that is.

The hier's stats are only affected by the current king, no matter the true genetical relationships.