Let a finite alphabet Ω. We consider a sequence of letters from Ω generated by a discrete time semi-Markov process We derive the probability of a word occurrence in the sequence. We also obtain results for the mean and variance of the number of overlapping occurrences of a word in a finite discrete time semi-Markov sequence of letters under certain conditions.
We investigate convergence and divergence of specific subsequences of partial sums with respect to the Walsh system on martingale Hardy spaces. By using these results we obtain a relationship of the ratio of convergence of the partial sums of the Walsh series and the modulus of continuity of the martingale. These conditions are in a sense necessary and sufficient.
Let X be the unique normal martingale such that X0=0 and d[X]t=(1−t−Xt−) dXt+dt and let Yt:=Xt+t for all t≥0; the semimartingale Y arises in quantum probability, where it is the monotone-independent analogue of the Poisson process. The trajectories of Y are examined and various probabilistic properties are derived; in particular, the level set {t≥0: Yt=1} is shown to be non-empty, compact, perfect and of zero Lebesgue measure. The local times of Y are found to be trivial except for that at level...
We study the statistical properties of the solutions of the Kadomstev-Petviashvili equations (KP-I and KP-II) on the torus when the initial datum is a random variable. We give ourselves a random variable with values in the Sobolev space with big enough such that its Fourier coefficients are independent from each other. We assume that the laws of these Fourier coefficients are invariant under multiplication by for all . We investigate about the persistence of the decorrelation between the...