  Recent experimental observations have found two different kinds of ``strange
kinetic behaviors" in individual semiconductor nanocrystals (or quantum dots).
Fluorescence intermittency observed in the quantum dots shows power--law
statistics in both {\it on} and {\it off} times. Spectral diffusion of the
quantum dots is also described by power--law statistics in the sojourn times.
Motivated by these experimental observations we consider two different but
related problems: (a) a stochastic lineshape theory for the Kubo-Anderson
oscillator whose frequency modulation follows power-law statistics and (b)
photon counting statistics of quantum dots whose intensity fluctuation is
characterized by power-law kinetics. In the first problem, we derive an
analytical expression for the lineshape formula and find rich type of behaviors
when compared with the standard theory. For example, new type of resonances and
narrowing behavior have been found. We show that the lineshape is extremely
sensitive to the way the system is prepared at time $t=0$ and discuss the
problem of stationarity. In the second problem, we use semiclassical photon
counting statistics to characterize the fluctuation of the photon counts
emitted from quantum dots. We show that the photon counting statistics problem
can be mapped onto a L\'evy walk process. We find unusually large fluctuations
in the photon counts that have not been encountered previously. In particular,
we show that $Q$ may increase in time even in the long time limit.
