In this paper, we are to establish a class of strong deviation theorems for the random fields relative to m th-order nonhomogeneous Markov chains indexed by an m rooted Cayley tree. As corollaries, we obtain the strong law of large numbers and Shannon-McMillan theorem for m th-order nonhomogeneous Markov chains indexed by that tree. 2000 Mathematics Subject Classification: 60F15; 60J10.

To achieve efficient vineyard grape picking, a vision-based information processing framework integrating two-stage segmentation with morphological perception is proposed. In the first stage, an improved YOLOv8s-seg model is employed for coarse segmentation, incorporating two key enhancements: first, a dynamic deformation feature aggregation module (DDFAM), which facilitates the extraction of complex structural and morphological features; and second, an efficient asymmetric decoupled head (EADHead), which improves boundary awareness while reducing parameter redundancy. Compared with mainstream segmentation models, the improved model achieves superior performance, attaining the highest mAP@0.5 of 86.75%, a lightweight structure with 10.34 M parameters, and a real-time inference speed of 10.02 ms per image. In the second stage, the fine segmentation of fruit stems is performed using an improved OTSU thresholding algorithm, which is applied to a single-channel image derived from the hue component of the HSV color space, thereby enhancing robustness under complex lighting conditions. Morphological features extracted from the preprocessed fruit stem, including centroid coordinates and a skeleton constructed via medial axis transform (MAT), are further utilized to establish the spatial relationships with a picking point and cutting axis. The visualization analysis confirms the high feasibility and adaptability of the proposed framework, providing essential technical support for the automation of grape harvesting.

In this paper, we give the definition of tree-indexed Markov chains in random environment with countable state space, and then study the realization of Markov chain indexed by a tree in random environment. Finally, we prove the strong law of large numbers and Shannon–McMillan theorem for Markov chains indexed by a Cayley tree in a Markovian environment with countable state space.

This paper gives the definition of tree-indexed Markov chains in random environment with discrete state space, and then studies some equivalent theorems of tree-indexed Markov chains in random environment. Finally, we give the equivalence on tree-indexed Markov chains in Markov environment and double Markov chains indexed by a tree.