In this paper, we consider a size-dependent renewal risk model with stopping time claim-number process. In this model, we do not make any assumption on the dependence structure of claim sizes and inter-arrival times. We study large deviations of the aggregate amount of claims. For the subexponential heavy-tailed case, we obtain a precise large-deviation formula; our method substantially relies on a martingale for the structure of our models.

Aiming at the shortcomings of the existing lossless digital watermarking algorithm based on frequency domain in reversibility and embedding capacity, this study proposes a lossless digital image watermarking algorithm based on fractional wavelet transform, which is used for large-capacity reversible information hiding of images. First, the image is transformed by LeGall5/3 fractional wavelet, and then, the watermark is embedded in the high-frequency subband by the histogram shift method. In order to obtain maximum embedding capacity and reduce image distortion, the methods of selecting embedding parameters and stopping parameters are proposed, respectively. At the same time, in order to prevent overflow and reduce additional information, a new method of generating position map is proposed. The experimental results show that Lena is the result of multilayer embedding based on the algorithm in this study. In order to better observe the distortion phenomenon and enlarge the image, the Lena test image is the watermark image obtained after two and three layers of embedding, and its embedding capacity can be 2.7 bpp. It is proved that wavelet transform is suitable for encrypted images to implement covert communication.

In this paper, we consider a discrete-time risk model with the claim number following a Poisson ARCH process. In this model, the mean of the current claim number depends on the previous observations. We study the large deviations for the aggregate amount for claims. For a heavy-tailed case, we obtain a precise large deviation formula, which agrees with existing ones in the literature. In computing the moderate deviation principle required by the structure of the claim-number process, our treatment substantially relies on an algorithm specifically designed for the autoregressive structure of our models.

In this paper, farmers in tourist destinations are taken as an example to study the co-creation mechanism of social capital and rural farmers' value under the background of common prosperity. Through the case study and contribution study of the value co-creation mechanism of the consortium, this paper establishes the value co-creation mechanism of farmers in rural tourist destinations with social capital as a universal model, and verifies the accuracy and fitting precision of the model with existing cases. Taking the leading enterprises, the support of government policies, tourism and agricultural products as variables, this paper studies their impact on rural economic income as a whole. Through model research, this paper solves the problems of market mechanism disorder and unfair distribution in the three distributions of common prosperity, and combines the concept of rural tourism development and value co-creation. The research results are universal, providing a reference model for social capital to participate in the development of value co-creation of farmers in tourist destinations, and providing new ideas and ideas for farmers in tourist destinations to achieve common prosperity.