Function randomWalkIndex

Random Walk Index (E. Michael Poulos, TASC 1991) — how far price actually travelled, in units of how far a random walk of the same length would be expected to travel, asked at every horizon up to period and answered with the largest reading:

${prefix}High = max over n = 2 … period of  (high − low[−n])  / (meanTR(n) · √n)
${prefix}Low  = max over n = 2 … period of  (high[−n] − low)  / (meanTR(n) · √n)
Copy

Appends two columns. A symmetric random walk covers a distance proportional to √n after n steps, and one step is estimated by the mean true range, so a reading above 1 says the move was larger than chance would produce at that horizon and a reading near or below 1 says the market is wandering. rwiHigh measures the up-move (today's high against the low n bars back) and rwiLow the down-move; the conventional reading is a trend when one is above 1 and the other below.

Neither column is bounded below by zero, which the "distance travelled" description hides: high − low[−n] is negative whenever the market fell across every horizon, and the max of negative terms is negative. On the package's oracle input rwiHigh runs −0.31 … 2.29 (measured). A negative reading simply says the move went the other way.

The multi-horizon form is the study — the G2 shape, and its cost

The corpus assessment (§5, G2) singles this out as the awkward window: the reducer needs a different rolling statistic at every horizon inside the window, which no window kernel in the package can express. The alternative — one ATR(period) in the denominator for every horizon — is a different indicator, not an optimisation: it drops the n-specific volatility estimate that makes the √n comparison meaningful. It is not what ships, and the generator measures how far away it is rather than asserting a preference.

So the study is O(N · period) in time — the horizon sweep lives in kernels/random-walk.ts, which folds one horizon at a time and keeps memory at O(N) regardless of period. scripts/perf-studies.mjs carries a period 14 and a period 50 entry side by side so the linearity in period stays visible; a study that is linear in its look-back is acceptable here for the same reason commodityChannelIndex is (the published periods are small and the inner pass is cache-local), and it is documented rather than hidden.

meanTR(n) is the n-bar mean of true range, not Wilder's ATR

Poulos writes "the average true range over the last n periods", and the structure agrees with the words: √n is a statement about n independent steps, so the denominator has to be the average step size over exactly those n bars. Wilder's recursion has infinite memory and no n-bar window at all, so at horizon n it would scale by a quantity that partly reflects bars from before the horizon being tested. Measured on the package's oracle input, running the whole study on Wilder denominators puts rwiHigh 0.190 apart at period 14 on a reading that spans −0.31 … 2.29, and the single-horizon form (only n = period) 2.374 apart (scripts/oracle/generate.py). Both are asserted. The Wilder separation shrinks with period — 0.031 at period 30, measured — because the recursion and the n-bar mean converge as n grows; it is the short horizons, which dominate a small period, where the choice actually shows.

Warm-up

Length-preserving, and strict: a bar reads a value only when every one of the period − 1 horizons has one, because a "maximum over 2 … 14" taken over the four horizons that happened to be available is not that maximum. TR[0] is undefined (no previous close), so the slowest horizon needs period true ranges and low[i − period], both of which first exist on bar period — bar 14 at the default.

Edges

• Scale- and shift-invariant. Numerator and denominator are both first-order in price, so a scale cancels; every term is a difference of prices, so a shift cancels too. Pinned as property tests.
• A zero denominator → undefined. meanTR(n) = 0 means every bar in that horizon was completely flat, but low[i − n] is the low of a bar outside that window and is not forced to equal today's high — so it is a real number over zero rather than a 0/0, and the guard is live. One flat horizon blanks the whole bar, which is the strict rule again.
• period must be at least 2 — the horizon list 2 … period is empty below that, and a "random walk index" with no horizons is not a reading.
• An interior gap costs the bar plus every horizon window that holds it — up to period bars — and then recovers. Nothing propagates to the end, because no recursion is involved.