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Discrete Time Harness Processes

We study the invariant measures and fluctuation limits of discrete-time harness processes in one spatial dimension. We construct one essential ergodic (under spatial shifts) invariant measure of the increment process derived from harness process, and all other ergodic invariant measures can be obtained by adding constants. We also show that the weak limit of the one dimensional height fluctuations starting from the increments under several translation-invariant ergodic measures will obey Edwards-Wilkinson equation, and the finite-dimensional marginal convergence can be extended to a process level convergence.