Large deviations for longest runs in Markov chains

We continue our investigation on general large deviation principles (LDPs) for longest runs. Previously, a general LDP for the longest success run in a sequence of independent Bernoulli trails was derived in [Z. Liu and X. Yang, A general large deviation principle for longest runs, Statist. Probab. Lett. 110 2016, 128–132]. In the present note, we establish a general LDP for the longest success run in a two-state (success or failure) Markov chain which recovers the previous result in the aforementioned paper. The main new ingredient is to implement suitable estimates of the distribution function of the longest success run recently established in [Z. Liu and X. Yang, On the longest runs in Markov chains, Probab. Math. Statist. 38 2018, 2, 407–428].

Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions

We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube I 2 in R 2 . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total I 2 (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.