Random geometric blocks, circular wrapping and uniform restarts preserve conditional stationarity while estimating uncertainty in dependent time series.
Stationary bootstrap
The stationary bootstrap resamples a time series in consecutive runs whose lengths follow a geometric distribution, with random restarts and circular wrapping. It lets a quant estimate uncertainty while retaining stretches of the observed dependence, instead of treating every observation as an independent draw. Politis and Romano introduced the method for inference from weakly dependent stationary data. Its distinctive property is that the resampled process is stationary conditional on the observed sample. Politis and Romano (1994)
For a strategy's average return, an ordinary independent bootstrap breaks the order of observations. If the statistic is affected by autocorrelation or other temporal dependence, that changes the uncertainty being estimated. Consecutive runs preserve some local ordering. The block bootstrap article explains fixed-length moving blocks; the stationary bootstrap uses random run lengths instead.
Continue, wrap, or restart
Let the observed series be x0,…,xn−1. Choose a restart probability 0<p≤1 and construct an index sequence:
Draw the first index uniformly from 0,…,n−1.
At each following position, independently restart with probability p: draw a fresh uniform index from the whole sample.
Otherwise take the next original index, wrapping from n−1 to 0.
Stop after n output observations and recompute the statistic. Repeat independently to estimate its bootstrap distribution.
This is the sequential implementation described in Sullivan, Timmermann and White's Appendix C. A restart may land anywhere, including the same index or the next index; a restart cannot always be identified just by looking at the output values.
The number L of observations in a run, before truncating the final run to fit the output sample, has a geometric distribution:
P(L=ℓ)=p(1−p)ℓ−1,ℓ=1,2,…,E[L]=p1.
With p=0.25, the expected run length is four observations. Four is neither a fixed block size nor a maximum.
Take the fictional observations 10,20,30,40,50. One realised zero-based index path is 4,0,2,3,4: start at the final observation, continue through the circular boundary, restart at index 2, then continue twice. The output is 50,10,30,40,50, with mean
xˉ∗=550+10+30+40+50=36.
A fictional stationary-bootstrap path takes indices 4, 0, 2, 3, 4 from observations 10, 20, 30, 40, 50. It wraps from 50 to 10, restarts at 30, then continues to 40 and 50. The output mean is 36 and the expected block length is 4.
One realised resample. The first run contains two observations; three observations from the next run are retained before the sample stops. Wrapping continues a run; restarting begins a new one.
The original sample mean is 30. A replicate can have mean 36 without contradicting correct centring: the bootstrap mean averaged over all possible resamples is 30, conditional on these observations.
What “stationary” guarantees
The construction has an exact finite-sample property. Write It for the selected index. Conditional on the data, its transition probabilities are
Kij=(1−p)1{j=(i+1)modn}+np.
Every column sums to one. Consequently, a uniform index distribution remains uniform after a transition. Starting the time-homogeneous index chain uniformly makes its joint distributions invariant to shifts in output time. Applying the fixed values xi to those indices preserves that stationarity. This argument also explains why forcing the initial index to zero would lose the guarantee.
This is a property of the resampling mechanism. It does not establish that the historical return process was stationary, or that the bootstrap distribution is accurate for every statistic. The original paper separates construction from consistency results; its mean-variance analysis imposes dependence and moment conditions as well as tuning conditions. Politis and Romano, Section 3
Choosing how often to cut
Larger p creates shorter runs and more frequent cuts. At p=1, every observation is an independent uniform redraw. Smaller p retains longer stretches, but produces fewer fresh starts within a sample. In the limiting case of no restarts, an n-observation resample is just a circular rotation: its mean is always the original mean, giving a useless zero bootstrap standard error for that statistic.
A direct measure of this tuning choice is the probability of surviving k transitions without a restart, (1−p)k. With p=0.25, the probability of retaining a run across four transitions is 0.754≈31.64%. This measures uninterrupted copying, not the resampled autocorrelation itself.
For standard stationary-bootstrap asymptotics, the expected block length grows while remaining small relative to n: pn→0 and npn→∞. These are limiting conditions, not a prescription for a universal finite-sample setting. Results should be examined across defensible block-length choices, with the statistic's validity conditions in view. Politis's review of block resampling, Section 2.2, stresses that validity needs checking for the application.
Both kinds of boundary need scrutiny. A restart joins unrelated historical dates. A wrap treats the first observation as following the last, an adjacency absent from the original record. Longer runs reduce restart joins but can still cross that circular boundary. For path-dependent trading rules, distinguish resampling a previously computed return series from rebuilding signals, holdings and costs on a synthetic market path; those answer different questions.
A structural break is not repaired by generating stationary resamples from a mixture of regimes. Nor does resampling remove look-ahead bias already embedded in signals or returns. A stationary-looking bootstrap distribution cannot validate contaminated inputs.
For the underlying interval-estimation ideas, see SelfMadeQuant's Bootstrap Confidence Intervals lesson. Its public preview introduces the scope; the full teaching requires membership and starts from independent sampling, so it supplies foundations rather than this dependent-data construction.
Sources
Politis, D. N. and Romano, J. P. (1994), The Stationary Bootstrap, Journal of the American Statistical Association, 89(428), 1303–1313. Original method and conditional stationarity; Section 3 studies the mean and tuning conditions.
