All computable increasing sequences converging to random reals converge at the same speed up to a constant factor, as this follows immediately from Bishop's upcrossing inequality.
Miller, On work of Barmpalias and Lewis-Pye: a derivation on the d.c.e
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Bishop's (up)crossing inequality and lower semicomputable random reals revisited
All computable increasing sequences converging to random reals converge at the same speed up to a constant factor, as this follows immediately from Bishop's upcrossing inequality.