Motivated by recent experimental observations of power-law statistics both in spectral diffusion process and fluorescence intermittency of individual semiconductor nanocrystals (quantum dots), we consider two different but related problems: (a) a stochastic lien shape theory for the Kubo-Anderson oscillator whose frequency modulation follows power-law statistics and (b) photon counting statistics of quantum dots shows intensity fluctuation is characterized by power-law kinetics. In the first problem, we derive an analytical expression for the lienshape 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 lienshape 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 Levy walk process. We find unusually large fluctuations in the photon counts that have not been encountered previously. In particular, we show that Mandel's Q parameter may increase in time even in the long time limit.
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