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Cross-component linear model (CCLM) prediction has been repeatedly proven to be effective in reducing the inter-channel redundancies in video compression. Essentially speaking, the linear model is identically trained by employing accessible luma and chroma reference samples at both encoder and decoder, elevating the level of operational complexity due to the least square regression or max-min based model parameter derivation. In this paper, we investigate the capability of the linear model in the context of sub-sampled based cross-component correlation mining, as a means of significantly releasing the operation burden and facilitating the hardware and software design for both encoder and decoder. In particular, the sub-sampling ratios and positions are elaborately designed by exploiting the spatial correlation and the inter-channel correlation. Extensive experiments verify that the proposed method is characterized by its simplicity in operation and robustness in terms of rate-distortion performance, leading to the adoption by Versatile Video Coding (VVC) standard and the third generation of Audio Video Coding Standard (AVS3).