### The differentiability of transition functions

Donald Ornstein
1960 Bulletin of the American Mathematical Society
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 , Doob showed  that P%j{t) has a right hand derivative (possibly infinite) at 2 = 0
more » ... d Kolmogoroff showed  that this derivative is finite if i^j, (if i=j there are examples where it is infinite). Austin [l; 2] showed that that Pij(t) has a finite continuous derivative for />0 if either Pu{t) or Pjj{t) has a finite derivative at 0. We will now give the proof 3 of our theorem. We will think of the matrices {P»y(/)} as transformations on sequences in such a way that [Pij(t)} transforms the sequence with 1 in the rath place and 0 elsewhere into the sequence whose &th term is P m k(t). We will use letters like v to denote a sequence, T to denote a particular matrix and T(v) to denote the sequence v transformed by the matrix T. Our first step will be to show that Pn{t) has bounded variation in some interval (say from 0 to to). To do this we will estimate JliJo 1 \Pn(ito/N)-P n ((i+l)to/N)\ for a fixed integer N. The estimate will turn out to be independent of N. To simplify notation we will let T= {P i3 {t 0 /N)} and letfi = Pu(ito/N). We will first define a sequence of vectors (or sequences) V{. v 0 will be the sequence with 1 in the first place and 0 elsewhere. Let us de-1