If a girl is underweight, then she won't have her periods/they will be irregular. I'm 7 stone and mine crop up whever they feel like it, normally about once every three months. If I was heavier they'd be regular, but I find it hard to put on weight.
I don't think there is a link between the type of period and weight, though. Things like length of time and period pain are individual, and are affected (I think) by hormonal rather than weight changes. I'm very slim, but my periods have always been fairly heavy and accompanied by cramps, though they've not been regular. I've always been under the impression, like No Future said, that period type is more genetic than anything else - unless you're severely underweight, in which case they're very light/non-existant, because your body couldn't cope with a child.