CNNs are naturally translation equivariant, meaning that if we translate the input, then the feature maps are translated the equally.
With the use of max/avg pooling layers, this translation equivariance leads to approximate translation invariance, in the sense that it gives translation invariance for small translations, but for longer translation, the max/avg values could differ due to the limitations by the size of the pooling and the module of the translation.
However, if I use the biggest size possible for the pooling, i.e. global max/avg pooling, then the CNN will be translation invariant in an exact way, for every translation of the input, no matter the strength (module) of the translation.
Is this intuition correct? I can visualize this in my head, but I can't really visualize if this effect would hold for deep networks (i.e. after many conv->global pool layers are stacked).