  It is well known that any positive matrix can be scaled to have prescribed
row and column sums by multiplying its rows and columns by certain positive
scaling factors (which are unique up to a positive scalar). This procedure is
known as matrix scaling, and has found numerous applications in operations
research, economics, image processing, and machine learning. In this work, we
investigate the behavior of the scaling factors and the resulting scaled matrix
when the matrix to be scaled is random. Specifically, letting
$\widetilde{A}\in\mathbb{R}^{M\times N}$ be a positive and bounded random
matrix whose entries assume a certain type of independence, we provide a
concentration inequality for the scaling factors of $\widetilde{A}$ around
those of $A = \mathbb{E}[\widetilde{A}]$. This result is employed to bound the
convergence rate of the scaling factors of $\widetilde{A}$ to those of $A$, as
well as the concentration of the scaled version of $\widetilde{A}$ around the
scaled version of $A$ in operator norm, as $M,N\rightarrow\infty$. When the
entries of $\widetilde{A}$ are independent, $M=N$, and all prescribed row and
column sums are $1$ (i.e., doubly-stochastic matrix scaling), both of the
previously-mentioned bounds are $\mathcal{O}(\sqrt{\log N / N})$ with high
probability. We demonstrate our results in several simulations.
