  This paper studies the infinite-width limit of deep linear neural networks
initialized with random parameters. We obtain that, when the number of neurons
diverges, the training dynamics converge (in a precise sense) to the dynamics
obtained from a gradient descent on an infinitely wide deterministic linear
neural network. Moreover, even if the weights remain random, we get their
precise law along the training dynamics, and prove a quantitative convergence
result of the linear predictor in terms of the number of neurons.
  We finally study the continuous-time limit obtained for infinitely wide
linear neural networks and show that the linear predictors of the neural
network converge at an exponential rate to the minimal $\ell_2$-norm minimizer
of the risk.
