Order problem for canonical systems and a conjecture of Valent

A precise upper estimate for the exponential order of the monodromy matrix of $ 2\times 2 $ canonical systems is obtained. The estimate is shown to become an equality in the case of the Krein strings. As an application we prove a conjecture by Valent on the order for a certain class of indeterminate moment problems with polynomial Jacobi parameters related to birth–death processes.