Uniform Spacings and the Simplex Constraint

From Ordered Observations to Gaps

A sample of points in the unit interval can be described either by the locations of its ordered observations or by the lengths of the intervals between them. The second description is useful because it turns the whole collection of observations into a vector of gaps. That vector is constrained: each gap is nonnegative, and together the gaps exactly fill the unit interval.

Let \(X_1,\ldots,X_n\) be independent random variables, each uniformly distributed on \((0,1)\), where \(n\geq 1\). Write their order statistics as \(X_{(1)}\leq X_{(2)}\leq\cdots\leq X_{(n)}\). To include the intervals at both ends of the unit interval, set \(X_{(0)}=0\) and \(X_{(n+1)}=1\).

There are \(n+1\) gaps, not \(n-1\): the intervals before the first observation and after the last observation count as spacings too. The word “uniform” identifies the sampling distribution; it does not mean that the realized gaps are equal in length.

The Simplex Constraint

The defining differences immediately impose a geometric restriction on the spacing vector. It is convenient to name the set of all vectors satisfying this restriction.

Although a spacing vector has \(n+1\) coordinates, its unit-sum condition means that only \(n\) coordinates can be chosen freely: the remaining coordinate must make the total equal to one. Thus the simplex is \(n\)-dimensional, lying inside \(\mathbb{R}^{n+1}\).

Proof. By the ordering and the endpoint definitions, \(X_{(i+1)}\geq X_{(i)}\) for \(i=0,\ldots,n\). Subtracting gives \(D_i=X_{(i+1)}-X_{(i)}\geq 0\) for every \(i\). For the total, substitute the differences and cancel consecutive terms:

The cancellation is telescoping: each interior order statistic appears once with a plus sign and once with a minus sign. Therefore the vector satisfies both defining conditions of \(\Delta_n\). \(\square\)

Worked Examples

Recovering the Ordered Sample

The gaps do not lose information about the ordered observations. Starting at zero, add the gaps from left to right to recover each observation. Conversely, taking successive differences of the ordered observations recovers the gaps.

Proof. First take any ordered sample and define its gaps by differences. For \(k=1,\ldots,n\), summing the first \(k\) gaps gives

So the cumulative-sum formula recovers every order statistic. To verify the reverse direction, begin with any \((d_0,\ldots,d_n)\in\Delta_n\), and define \(x_{(0)}=0\) and \(x_{(k)}=\sum_{j=0}^{k-1}d_j\) for \(k=1,\ldots,n\). Since each \(d_j\geq0\), these values are nondecreasing. Also, each partial sum is at most the full sum 1, so \(x_{(n)}\leq1\). The successive differences of these values are \(d_0,\ldots,d_{n-1}\), and the final difference is

where the last equality uses \(\sum_{j=0}^{n}d_j=1\). Thus every vector in \(\Delta_n\) comes from an ordered sample, and the cumulative sums uniquely determine that sample. This proves the bijection. \(\square\)

Strict Gaps With Probability One

The simplex condition allows zero coordinates. For example, a zero interior spacing means two consecutive ordered observations are equal; a zero endpoint spacing means an observation is at 0 or 1. These possibilities are included in the closed simplex as boundary cases. For an independent continuous uniform sample, however, they occur with probability zero.

Proof. A uniform random variable assigns probability zero to each individual point, so \(P(X_j=0)=P(X_j=1)=0\) for every \(j\). For any distinct \(j\) and \(k\), independence and the continuous uniform distribution give

The probability that any observation is an endpoint or that any pair of observations is equal is at most the sum of finitely many zero probabilities, hence is zero. Outside this event, all observations lie strictly between 0 and 1 and are distinct. Their order statistics then satisfy \(0<X_{(1)}<\cdots<X_{(n)}<1\), so every successive difference, including both endpoint gaps, is strictly positive. \(\square\)

This result distinguishes two useful statements: the spacing vector always satisfies the closed-simplex constraints, while a continuous uniform sample lands in the simplex’s relative interior with probability one. The first is a deterministic consequence of ordering; the second uses the sampling assumptions.

Common Mistake and Why the Constraint Matters

A frequent error is to count only the spaces between observations and omit the two endpoint gaps. For \(n\) ordered observations there are \(n-1\) interior intervals, but the partition of \([0,1]\) also includes the interval from 0 to the first point and the interval from the last point to 1. The total is therefore \((n-1)+2=n+1\) spacings.

Another mistake is to treat all \(n+1\) gaps as freely selectable. They must add to exactly one, so choosing \(n\) of them determines the last. A proposed vector with nonnegative entries but total \(0.9\), for instance, cannot be the complete set of gaps on \([0,1]\); it leaves \(0.1\) of the interval unaccounted for.

The simplex description is a geometric characterization, not by itself a probability distribution for the gap vector. It tells which vectors are possible under the ordering construction. Describing how probability is distributed across that simplex requires additional results. For now, the key structural fact is that ordered sample locations and their \(n+1\) gaps are equivalent descriptions, linked by differences and cumulative sums.