The asymptotic equipartition property (AEP) plays an important role in establishing lossless source coding theorems and asymptotic coding theorems through the concepts of typical sets and typical sequences in information theory. In this paper, we shall study the strong law of large numbers and the AEP for hidden Markov tree models. First, we give a strict definition of hidden Markov tree models, and study their key properties and equivalent characterizations. In fact, hidden Markov tree models are closely related to the tree-indexed Markov chains, therefore, we also introduce the definition of tree-indexed Markov chains and their equivalent properties. We also establish a strong law of large numbers for hidden Markov tree models indexed by a Cayley tree. As corollaries, we obtain some strong laws of large numbers for the parameters and the AEP for these models.

Efficient and precise open-vocabulary 3D scene segmentation remains a critical challenge in computer vision. While current leading methods encode CLIP language features into 3D Gaussians to achieve high segmentation accuracy and fast inference speeds, they suffer from point ambiguity issues caused by separately training on multi-level 2D semantic masks. This approach not only compromises time and space efficiency but also degrades accuracy when selecting optimal semantic levels. To overcome these limitations, we propose Voxel-Aware Fusion Language Gaussian Splatting (VaF-LangSplat), a novel framework that jointly optimizes geometric and semantic representations. Our approach first voxelizes 3D Gaussians using sparse point clouds and lightweight MLP decoders, effectively disentangling language features from geometric attributes. This enables simultaneous training across arbitrary semantic levels with minimal overhead. Crucially, we introduce Fusion Language Splatting, which aligns geometric and multi-level semantic distributions to sharpen boundary definitions while eliminating redundant Gaussian expansions. The voxel-aware representation further enhances robustness against motion blur and lighting variations. Experiments on open-vocabulary 3D localization and segmentation tasks demonstrate that VaF-LangSplat outperforms LangSplat (the prior state-of-the-art) with significant improvements in both segmentation/localization accuracy and efficiency: 4X faster training and 15X reduced storage requirements.

The drones formation control in 5G wireless sensor network is discussed. The base station (BS) is used to receive backhaul position signals from the lead drone in formation and launches the beam to the lead one as the fronthaul flying signal enhancement. It is a promising approach to raise the formation strength of drones during flight control. The BS can transform the direction of the antennas and transmit energy to the lead drone that could widely enlarge the number of the receivers and increase the transmission speed of the data links. The millimeter‐Wave (mmWave) communication system offers new opportunities to meet this requirement owing to the tremendous amount of available spectrums. However, the massive non‐line‐of‐sight (NLoS) transmission and the site constraints in urban environment are severely challenging the conventional deploying terrestrial low power nodes (LPNs). Simulation experiments have been performed to verify the availability and effectiveness of mmWave in 5G wireless sensor network.

Guyon (Guyon J. Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann Appl Probab, 2007, 17: 1538-1569) introduced an important model for homogeneous bifurcating Markov chains indexed by a binary tree taking values in general state space and studied their limit theorems. The results were applied to detect cellular aging. In this paper, we define a discrete form of nonhomogeneous bifurcating Markov chains indexed by a binary tree and discuss the equivalent properties for them. The strong law of large numbers and the entropy ergodic theorem are studied for these Markov chains with finite state space. In contrast to previous work, we use a new approach to prove the main results of this paper.

In this correspondence, we first establish a strong limit theorem for countable nonhomogeneous Markov chains indexed by a homogeneous tree. As corollaries, we obtain some strong limit theorems for frequencies of occurrence of states and ordered couple of states for these Markov chains. Finally, we prove the strong law of large numbers and the asymptotic equipartition property (AEP) for finite nonhomogeneous Markov chains indexed by a homogeneous tree.

In this correspondence, we first establish a limit theorem for averages of the functions of m+1 variables of mth-order nonhomogeneous Markov information sources. As corollaries, we obtain several limit theorems for frequency of occurrence of the states and a limit theorem of the entropy density for these information sources. Finally, we prove the asymptotic equipartition property (AEP) for a class of nonhomogeneous Markov information sources.

We study the strong law of large numbers and the Shannon-McMillan theorem for Markov chain fields on trees. First, we prove the strong law of large numbers for the frequencies of occurrence of states and ordered couples of states for Markov chain fields on trees. Then, we prove the Shannon-McMillan theorem with almost everywhere (a.e.) convergence for Markov chain fields on trees. We prove the results on a Bethe tree and then just state the analogous results on a rooted Cayley tree. In the proof, a new technique for establishing the strong limit theorem in probability theory is applied.