Each period an outcome (out of finitely many possibilities) is observed. For simplicity assume two possible outcomes, a and b. Each period, a forecaster announces the probability of a occurring next period based on the past. Consider an arbitrary subsequence of periods (e.g., odd periods, even periods, all periods in which b is observed, etc.). Given an integer n, divide any such subsequence into associated sub-subsequences in which the forecast for a is between [i/n, i+1/n), i ∈ {0, 1,..., n}. We compare the forecasts and the outcomes (realized next period) separately in each of these sub-subsequences. Given any countable partition of [0, 1] and any countable collection of subsequences, we construct a forecasting scheme such that for all infinite strings of data, the long-run average forecasts for a matches the long-run frequency of realized a's.
