Circular Spacings from Uniform Points

Why the Choice of Starting Point Matters

For points scattered around a circle, the gaps between consecutive points describe how evenly or unevenly they are distributed. Their laws are useful in problems about empty arcs, coverage, and the largest uncovered region. There is one important detail: a sequence of circular gaps needs a starting point. If that point is chosen as one of the sampled observations, the gaps have a particularly simple joint distribution.

Represent a circle of circumference one by \([0,1)\), with positions interpreted modulo one, and let \(X_1,\ldots,X_n\) be independent Uniform\((0,1)\) points. Anchor the traversal at \(X_1\) and proceed in the increasing direction around the circle. This convention—not starting at the smallest coordinate—will be used throughout.

The Joint Dirichlet Distribution

Conditional on \(X_1\), the offsets \(Y_2,\ldots,Y_n\) are independent Uniform\((0,1)\) random variables: subtracting a fixed point modulo one preserves the uniform distribution. Their joint conditional law does not depend on \(X_1\). Consequently the ordered offsets have the usual order-statistic density, which lets us derive the joint law of the gaps directly.

Proof. Write \(m=n-1\). The ordered offsets \(V_1,\ldots,V_m\) have joint density \(m!\) on \(0<v_1<\cdots<v_m<1\). Indeed, the \(m!\) possible orderings of independent continuous uniform observations have equal density one, and the ordered region collects all of them. The transformation \(d_1=v_1,\ d_2=v_2-v_1,\ldots,d_m=v_m-v_{m-1}\) has triangular Jacobian determinant one. The final gap is \(d_{m+1}=1-v_m\), so the ordered region becomes \(d_i>0\) and \(\sum_{i=1}^m d_i<1\). The density in these first \(m\) coordinates is therefore \(m!=(n-1)!\). This is precisely the density of \(\operatorname{Dirichlet}(1,\ldots,1)\), whose constant on the unit simplex is \(\Gamma(n)=(n-1)!\). \(\square\)

The density is constant in simplex coordinates, but the gaps are not independent: they must add to one. The law is exchangeable, so any one gap has the same marginal distribution, and any specified collection of \(k\) gaps has the same sum distribution as any other collection of \(k\).

Marginal and Aggregated Gap Probabilities

The constant simplex density gives exact distributions for one gap and for sums of gaps. For a single gap, the marginal is \(\operatorname{Beta}(1,n-1)\); its density and upper-tail probability are

More generally, let \(S_k\) be the sum of any specified \(k\) distinct gaps, where \(1\leq k\leq n-1\). Then \(S_k\sim\operatorname{Beta}(k,n-k)\), with density

To see this, group the gaps in the chosen set and let their total be \(s\). The \(k\) positive components within that group have a \((k-1)\)-dimensional simplex of scale \(s\), with volume factor \(s^{k-1}/(k-1)!\). The other \(n-k\) components have total \(1-s\), giving factor \((1-s)^{n-k-1}/(n-k-1)!\). Multiplying these factors by the joint density \((n-1)!\) yields the displayed density. This also verifies its normalization, since it is a beta density.

A related volume argument handles simultaneous lower bounds. For any fixed set of \(r\) distinct gaps and any \(a\geq0\),

When \(ra\leq1\), subtract \(a\) from each of the \(r\) constrained coordinates. The remaining coordinates are nonnegative and sum to \(1-ra\); scaling a simplex of dimension \(n-1\) by \(1-ra\) multiplies its volume by \((1-ra)^{n-1}\). If \(ra>1\), the constraints require more than the total available length, so the event is impossible.

Calculating Probabilities for the Largest Gap

The simultaneous-bound formula can be combined with inclusion-exclusion to study whether any gap exceeds a threshold. The method is especially transparent when two exceedances cannot occur together.

A Common Mistake: Starting at the Smallest Coordinate

A frequent error is to sort the original coordinates and assume that the cyclic gaps beginning at the minimum are also \(\operatorname{Dirichlet}(1,\ldots,1)\). The anchor convention matters. Starting at \(X_1\) chooses a sampled point without regard to its position relative to the coordinate cut at zero. Starting at the smallest coordinate selects a gap according to its relationship to that cut and generally gives a different law.

For example, with two points \(U_1,U_2\), let \(U_{(1)}<U_{(2)}\). The gap starting at the minimum is \(G=U_{(2)}-U_{(1)}\). Its density is \(2(1-g)\), \(0<g<1\), not the uniform density that a \(\operatorname{Dirichlet}(1,1)\) gap would have. In contrast, the gap from the anchored point \(X_1\) to \(X_2\), measured clockwise modulo one, is Uniform\((0,1)\). Therefore, before applying the Dirichlet law, specify where the circular sequence starts.