  The asymptotic quantum trajectory of weak continuous measurement for the
magnetometer is investigated. The magnetometer refers to a setup where the
field-to-estimate and the measured moment are orthogonal, and the quantum state
is governed by the stochastic master equation which, in addition to a
deterministic part, depends on the measurement outcomes. We find that the
asymptotic behavior is insensitive to the initial state in the following sense:
given one realization, the quantum trajectories starting from arbitrary initial
states asymptotically converge to the {\em same} realization-specific {\em
pure} state. For single-qubit systems, we are able to prove this statement
within the framework of Probability Theory by deriving and analyzing an
effective one-dimensional stochastic equation. Numerical simulations strongly
indicate that the same statement holds for multi-qubit systems. Built upon this
conclusion, we consider the problem of real-time parameter estimation whose
feasibility hinges on the insensitivity to the initial state, and explicitly
propose and test a scheme where the quantum state and the field-to-estimate are
updated simultaneously.
