Moderate deviation principle for $ m $-dependent random variables under the sub-linear expectation

<abstract><p>Let $ \{X_n, n\geq1\} $ be a sequence of $ m $-dependent strictly stationary random variables in a sub-linear expectation $ (\Omega, \mathcal{H}, \mathbb{E}) $. In this article, we give the definition of $ m $-dependent sequence of random variables under sub-linear expectation spaces taking values in $ \mathbb{R} $. Then we establish moderate deviation principle for this kind of sequence which is strictly stationary. The results in this paper generalize the result that in the case of independent identically distributed samples. It provides a basis to discuss the moderate deviation principle for other types of dependent sequences.</p></abstract>

Further Spitzer’s law for widely orthant dependent random variables

The Spitzer’s law is obtained for the maximum partial sums of widely orthant dependent random variables under more optimal moment conditions.

On the comparison of Shapley values for variance and standard deviation games

Motivated by the problem of variance allocation for the sum of dependent random variables, Colini-Baldeschi, Scarsini and Vaccari (2018) recently introduced Shapley values for variance and standard deviation games. These Shapley values constitute a criterion satisfying nice properties useful for allocating the variance and the standard deviation of the sum of dependent random variables. However, since Shapley values are in general computationally demanding, Colini-Baldeschi, Scarsini and Vaccari also formulated a conjecture about the relation of the Shapley values of two games, which they proved for the case of two dependent random variables. In this work we prove that their conjecture holds true in the case of an arbitrary number of independent random variables but, at the same time, we provide counterexamples to the conjecture for the case of three dependent random variables.

Simultaneous prediction of functionally dependent random variables by maximum likelihood estimation

The paper presents a fundamental parametric approach to simultaneous forecasting of a vector of functionally dependent random variables. The motivation behind the proposed method is the following: each random variable at interest is forecasted by its own model and then adjusted in accordance with the functional link. The method incorporates the assumption that models’ errors are independent or weekly dependent. Proposed adjustment is explicit and extremely easy-to-use. Not only does it allow adjusting point forecasts, but also it is possible to adjust the expected variance of errors, that is useful for computation of confidence intervals. Conducted thorough simulation and empirical testing confirms, that proposed method allows to achieve a steady decrease in the mean-squared forecast error for each of predicted variables.