We discuss the order of the variance on a lattice analogue of the Hammersley process with boundaries, for which the environment on each site has independent, Bernoulli distributed values. The last passage time is the maximum number of Bernoulli points that can be collected on a piecewise linear path, where each segment has strictly positive but finite slope.We show that along characteristic directions the order of the variance of the last passage time is of order N 2/3 in the model with boundary. These characteristic directions are restricted in a cone starting at the origin, and along any direction outside the cone, the order of the variance changes to O(N ) in the boundary model and to O(1) for the nonboundary model. This behaviour is the result of the two flat edges of the shape function.

The increased availability of multivariate time-series asks for the development of suitable methods able to holistically analyse them. To this aim, we propose a novel flexible method for data-mining, forecasting and causal patterns detection that leverages the coupling of Hidden Markov Models and Gaussian Graphical Models. Given a multivariate non-stationary time-series, the proposed method simultaneously clusters time points while understanding probabilistic relationships among variables. The clustering divides the time points into stationary sub-groups whose underlying distribution can be inferred through a graphical model. Such coupling can be further exploited to build a time-varying regression model which allows to both make predictions and obtain insights on the presence of causal patterns. We extensively validate the proposed approach on synthetic data showing that it has better performance than the state of the art on clustering, graphical models inference and prediction. Finally, to demonstrate the applicability of our approach in real-world scenarios, we exploit its characteristics to build a profitable investment portfolio. Results show that we are able to improve the state of the art, by going from a -%20 profit to a noticeable 80%.

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We prove a large deviation principle and give an expression for the rate function, for the last passage time in a Bernoulli environment. The model is exactly solvable and its invariant version satisfies a Burke-type property. Finally, we compute explicit limiting logarithmic moment generating functions for both the classical and the invariant models. The shape function of this model exhibits a flat edge in certain directions, and we also discuss the rate function and limiting log-moment generating functions in those directions.