Exponential Growth of Products of Non-Stationary Markov-Dependent Matrices

Let $(\xi _j)_{j\ge 1} $ be a non-stationary Markov chain with phase space $X$ and let $\mathfrak {g}_j:\,X\mapsto \textrm {SL}(m,{\mathbb {R}})$ be a sequence of functions on $X$ with values in the unimodular group. Set $g_j=\mathfrak {g}_j(\xi _j)$ and denote by $S_n=g_n\ldots g_1$, the product of the matrices $g_j$. We provide sufficient conditions for exponential growth of the norm $\|S_n\|$ when the Markov chain is not supposed to be stationary. This generalizes the classical theorem of Furstenberg on the exponential growth of products of independent identically distributed matrices as well as its extension by Virtser to products of stationary Markov-dependent matrices.

Application of Markov processes for analysis and control of aircraft maintainability

The process of aircraft operation involves constant effects of various factors on its components leading to accidental or systematic changes in their technical condition. Markov processes are a particular case of stochastic processes, which take place during aeronautical equipment operation. The relationship of the reliability characteristics with the cost recovery of the objects allows us to apply the analytic apparatus of Markov processes for the analysis and optimization of maintainability factors. The article describes two methods of the analysis and control of object maintainability based on stationary and non-stationary Markov chains. The model of a stationary Markov chain is used for the equipment with constant in time intensity of the events. For the objects with time-varying events intensity, a non-stationary Markov chain is used. In order to reduce the number of the mathematical operations for the analysis of aeronautical engineering maintainability by using non-stationary Markov processes an algorithm for their optimization is presented. The suggested methods of the analysis by means of Markov chains allow to execute comparative assessments of expected maintenance and repair costs for one or several one-type objects taking into account their original conditions and operation time. The process of maintainability control using Markov chains includes search of the optimal strategy of maintenance and repair considering each state of an object under which maintenance costs will be minimal. The given approbation of the analysis methods and maintainability control using Markov processes for an object under control allowed to build a predictive-controlled model in which the expected costs for its maintenance and repair are calculated as well as the required number of spare parts for each specified operating time interval. The possibility of using the mathematical apparatus of Markov processes for a large number of objects with different reliability factors distribution is shown. The software implementation of the described methods as well as the usage of tabular adapted software will contribute to reducing the complexity of the calculations and improving data visualization.

Characterising regime behaviour in the stably stratified nocturnal boundary layer on the basis of stationary Markov chains

Recent research has demonstrated that hidden Markov model (HMM) analysis is an effective tool to classify regimes of the stratified nocturnal boundary layer (SBL) at different tower sites. Here we analyse if SBL regime statistics (the occurrence of regime transitions, subsequent transitions after the first, and very persistent nights) in observations match theoretical calculations obtained from a stationary Markov chain with the goal of developing the foundations of novel Markov-chain-based boundary layer schemes which capture the effects of SBL regime dynamics. The regime statistics of a stationary Markov chain using the best estimate transition probabilities from the HMM analyses generally overestimate occurrence probabilities of regime transitions, resulting in an underestimation of persistent nights. Across the locations considered, sensitivity analyses of transition probability matrices in the HMM and the stationary Markov chain reveal that regimes are generally required to be more persistent in the stationary Markov chain in order to simulate observations accurately. A range of transition probability matrices allowing for a relatively accurate description of the occurrence of at least one transition within a night, multiple transitions, and the mean event durations is identified. The occurrence of very persistent nights (nights without regime transitions) is found to depend highly on the season. Therefore, for better representations of very persistent nights a nonstationary Markov chain linked to external drivers is likely appropriate. The observed transition probability maximum between one and two hours after a previous transition cannot be accounted for by two-state Markov processes (stationary or not). The use of these results in the development of SBL turbulence parameterisations is discussed.