A sequence of meteorological predictands of one kind (e.g., temperature) forms a discrete-time, continuous-state stochastic process, which typically is non-stationary and periodic (due to seasonality). Three contributions to the field of probabilistic forecasting of such processes are reported. First, a meta-Gaussian Markov model of the stochastic process is formulated, which provides a climatic probabilistic forecast with the lead time of n days in the form of a (prior) n-step transition distribution function. A measure of the temporal dependence of the process is the autocorrelation coefficient (which is non-stationary). Second, a Bayesian Processor of Forecast (BPF) is formulated, which fuses the climatic probabilistic forecast with an operational deterministic forecast produced by any system (e.g., a numerical weather prediction model, a human forecaster, a statistical post-processor). A measure of the predictive performance of the system is the informativeness score (which may be non-stationary). The BPF outputs a probabilistic forecast in the form of a (posterior) n-step transition distribution function, which quantifies the uncertainty about the predictand that remains, given the antecedent observation and the deterministic forecast. The working of the Markov BPF is explained on probabilistic forecasts obtained from the official deterministic forecasts of the daily maximum temperature issued by the U.S. National Weather Service with the lead times of 1, 4, and 7 days. Third, a numerical experiment demonstrates how the degree of posterior uncertainty varies with the informativeness of the deterministic forecast and the autocorrelation of the predictand series. It is concluded that, depending upon the level of informativeness, the Markov BPF is a contender for operational implementation when a rank autocorrelation coefficient is between 0.3 and 0.6, and is the preferred processor when a rank autocorrelation coefficient exceeds 0.6. Thus, the climatic autocorrelation can play a significant role in quantifying, and ultimately in reducing, the meteorological forecast uncertainty.