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The answer to this question is somewhat incorrect. In particular, the convolution operator only has the translational equivariance property. In this case, several factors that can constitute the translational invariance property of CNNs. We could argue that the translational invariance is due to the usage of pooling layers which downsamples the feature maps and thus makes the model less sensitive to small translations. Or, data augmentation helps the model learn to be more robust to positional displacements. Ultimately, CNNs are not translational invariant by design, see this paper.
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