Beyond the Sharpe Ratio: A Sequential Functional Limit Theory and Calibrated Path Monitoring

A terminal Sharpe ratio discards the path by which performance was achieved. This paper develops one statistical consequence of that observation. For independent and identically distributed returns with a finite fourth moment, the entire path of prefix Sharpe estimates, suitably scaled, converges weakly to a nonstationary Gaussian process on any horizon bounded away from the start of the sample. The limit is a Brownian motion divided by elapsed time, so its variance falls in proportion to the fraction of the sample used. It is not a Brownian bridge, and treating it as one misstates both the shape of the resulting band and its width. A uniform-prefix lemma establishes the required asymptotic linearization, and an explicit influence function identifies the limiting scale as a fourth-moment functional of the return distribution, which is where the theorem’s boundary originates. The closest methodological relatives are the recursive-estimates fluctuation and sequential monitoring tests of structural-break econometrics; what is new is not the shape of that argument but the object, a nonlinear ratio whose scale inherits a moment condition a linear-model fluctuation test does not carry. Monte Carlo evidence converts the limit into simultaneous prefix bands, using ten thousand paths per design and a critical value simulated on the same grid over which the bands are evaluated. Two volatility designs that share the persistence of a heavy-tailed stress design, but satisfy the fourth-moment condition, cover at the nominal rate, whereas the stress design covers about ninety percent; the failure is therefore attributable to the moment condition rather than to near-integrated volatility. The paper also gives a conditional asymptotic confidence sequence under a named strong-approximation assumption, with explicit boundary slack and a deterministic burn-in limit, so that compact-horizon convergence is not mislabelled as anytime validity. Independently calibrated monitoring simulations compare path and terminal alarms under the same heavy-tail stress. The result is deliberately narrow: a functional limit, an honest failure boundary, and operational monitoring implications