They are independent only if you explicitly state that you are computing them for best/average/worst case independently. Is it commonly done? Sure. But there is nothing in the definition that forces us to do that.
If the things were as you state would you have any use for Omega and Omicron then? Wouldn't just Theta suffice?
> But there is nothing in the definition that forces us to do that.
That's true. You're right.
> If the things were as you state would you have any use for Omega and Omicron then? Wouldn't just Theta suffice?
Couldn't there be cases where we don't have a known tight asymptotic bound but do have an upper and/or lower bound? And although it's an abuse of notation, you do often see big-O used in place of Theta. From CLRS:
"In the literature, we sometimes find O-notation informally describing asymptotically tight bounds, that is, what we have defined using Theta-notation."
They are independent only if you explicitly state that you are computing them for best/average/worst case independently. Is it commonly done? Sure. But there is nothing in the definition that forces us to do that.
If the things were as you state would you have any use for Omega and Omicron then? Wouldn't just Theta suffice?