Let X be an exchangeable increment (EI) process whose sample paths are of infinite variation. We prove that for any fixed t almost surely, lim sup hÑ0˘p X t'h´Xt q {h " 8 and lim sup This extends a celebrated result of Rogozin for Lévy processes obtained in 1968 and completes the known picture for finite-variation EI processes. Applications are numerous. For example, we deduce that both half-lines p´8, 0q and p0, 8q are visited immediately for infinite variation EI processes (called upward and

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... ownward regularity). We also generalize the zero-one law of Millar for Lévy processes by showing continuity of X when it reaches its minimum in the infinite variation EI case; an analogous result for all EI processes links right and left continuity at the minimum with upward and downward regularity. We also consider results of Durrett, Iglehart, and Miller on the weak convergence of conditioned Brownian bridges to the normalized Brownian excursion considered in [DIM77] and broadened to a subclass of Lévy processes and EI processes by Chaumont and the second author. We prove it here for all infinite variation EI processes. We furthermore obtain a description of the convex minorant known for Lévy processes found in [Ann. Prob. 40 (2012), pp. 1636-1674] and extend it to non-piecewise linear EI processes. Our main tool to study the Dini derivatives is a change of measure for EI processes which extends the Esscher transform for Lévy processes. Statement of the results Undoubtedly, Lévy processes are one of the most studied classes of stochastic processes. A less known class which contains them is that of exchangeable increment (EI) processes considered in general by Kallenberg in [Kal73]. Definition. A continuous time càdlàg R-valued stochastic process X " pX t , t P r0, 1sq has exchangeable increments if for every n ě 1, the random variables X 1{n , X 2{n´X1{n , . . . , X 1´Xpn´1q{n are exchangeable. Clearly, all Lévy processes are EI since iid random variables are exchangeable. Therefore, one can inherit results for Lévy processes from their counterparts for EI processes, as we illustrate in this paper. However, conditioning a Lévy process X by its final value (to obtain the so-called Lévy bridges as in [CUB11] and [UB14]) or