# Statistical benefits of K-fold cross validation

I understand how k-fold cross validation works. For each iteration, the training data is split into $$k$$ portions, and using $$k-1$$ portion of the data for training and the $$k$$-th portion of the data for testing.

Can we relate k-fold cross validation to the idea of empirical-risk minimisation ? where suppose we are able to define a probability distribution over the $$(x,y)$$ training examples. At each cross validation iteration, we sample $$(k-1)$$ portions of the $$(x,y)$$ data from $$p(x,y)$$.

Intuition tells me that since we are sampling many times from $$p(x,y)$$ for each iteration and computing validation error, it would result in a lower variance for the validation error.

Is that the main benefit of k-fold cross validation ?