Withhold a random-walk label until grouping and lookback are checked

The unaggregated daily comparison

The historical workflow took unaggregated daily prices from a broad equity index and used an interleaved split, sending successive observations into two parallel frequency histograms. Each frequency histogram was a bin count of those single prices, later turned into a density by dividing each bin by that histogram's total count.

The two densities were compared at decile prices with a chi-square test, a check of whether percentile gaps between two interleaved densities differ from an even decile baseline enough to reject a match. That comparison was not treated as significant, so the single-print series was judged apparently random.

Neighbor-sum tests on one window

The workflow then applied higher-order aggregation, replacing each single print with a sum of adjacent pairs or triples before repeating the same histogram comparison.

On one window of daily closes, adjacent pair-sums assigned to two interleaved histograms produced a chi-square treated as significant, so the pair aggregates were not judged random. In that same window, adjacent triple-sums produced a chi-square treated as evidence that the two histograms differed, so the triple aggregates were not random.

The same tests after the sample is extended

Extending the sample made the two single-price histograms essentially identical, so unaggregated daily prices were judged random over the longer window. On that longer window, the two pair-sum histograms were also essentially identical, so pair aggregates that had failed the short-window test were judged random once the lookback was lengthened.

Six-price sums in the longer sample produced odd-group and even-group histograms that were similar enough to be treated as random, even though they were not identical.

In this workflow a random-walk hypothesis test is a verdict that prices or neighbor-sum aggregates behave as if the two empirical densities are interchangeable, so no extra structure is detected at that grouping and window.

A second start for aggregated groups

Unlike the single-print test, the aggregated tests form different neighbor groups if the starting offset is shifted. A second pass with another start is part of a cautious evaluation.

Uncertainty beyond expected-return scatter

The evaluation treated uncertainty as more than the standard deviation of expected return. A series described as non-stationary, independent, and random was framed as higher uncertainty than one described as stationary, dependent, and not random, even when expected-return level and that standard deviation looked similar.

Stationarity here means whether the rules that generate the price series stay stable across the sample used for the test. Independence versus randomness keeps two series properties apart: whether one observation influences another, versus whether two interleaved densities still match.

Editorial reading

TradersWeek editorial reading: withhold a random-walk label until this same interleaved-split, density, and chi-square decile test has been rerun on raw prints, on neighbor-sum aggregates, and on more than one lookback. Randomness is then a setting of the test rather than a property of the series.

The archive facts describe that historical workflow. They do not say that any present series is random, dependent, or stationary.