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The differentiability of transition functions

1960
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Bulletin of the American Mathematical Society
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In this paper we prove that the transition functions of a denumerable Markoff chain are differentiable or equivalently : Given a matrix of real valued functions Pij{t) (i, j=l, 2, • • • ) 0^/< oo satisfying (1) P%j(t) is non-negative and continuous, i\y(0)= L. " 10 if ^ j, E^w-i. 1 y-i Our theorem is that Pij(t) has a finite continuous derivative for all *>0. This result was conjectured by Kolmogoroff in [4], Doob showed [3] that P%j{t) has a right hand derivative (possibly infinite) at 2 = 0

doi:10.1090/s0002-9904-1960-10381-3
fatcat:3efujywgxzhurjs5qrrlc72m5i