By definition the power dissipation is the average value of the instantaneous power dissipation - v(t)*i(t), i.e,
Pav=lim 1/T ∫ v(t)i(t) dt as T->∞
in the usual case where v(t)=VDD this reduces to
Pav=VDD lim 1/T ∫ i(t) dt=VDD*Iav as T->∞
where Iav is the AVERAGE current.
People sometimes confuse this because of sinusoidal regime, were if
v(t)=Vm Cos(wt) and i(t)=Im Cos(wt),
we have
Pav=lim 1/T ∫ v(t)i(t) dt as T->∞ =
= 1/T ∫ v(t)i(t) dt (T is the period: w=2Pi/T)=
=VmIm/2= Vm/√2*Im/√2=Vrms*Irms