Why the Timing of an Allocation Jump Matters
A cumulative allocation path records how assignments have accumulated over trial time. If \(N_{n,k}(t)\) is the number of participants assigned to arm \(k\) by time \(t\), then its path is a step function: it changes at assignment times and remains constant between them. Depending on the scale used, a single assignment may create a small jump or a jump that remains visible in the limit.
The earlier material on functional central limit theorems motivates studying whole paths rather than just their values at a few fixed times. For a path with jumps, however, ordinary uniform comparison can be unnecessarily strict: two paths may have essentially the same jump pattern even if corresponding jumps occur at slightly different times. Skorokhod \(J_1\) convergence formalizes this allowance while still requiring the time discrepancy to vanish when the jumps themselves remain visible.
We work on the time interval \([0,1]\). A cumulative allocation vector can be represented by \(X_n(t)\in\mathbb{R}^K\), where the \(k\)th coordinate might be an allocation count, an allocation proportion, or a centered and scaled allocation. The relevant path space is \(D([0,1],\mathbb{R}^K)\), the space of functions that are right-continuous and have left limits. The paths are not required to be continuous.
The time change is not an arbitrary reordering of observations. It must be increasing, so the order of events is preserved, and it must approach the identity uniformly. It allows a jump in one path to line up with a nearby jump in another, but it cannot conceal a persistent shift in the timing of a jump whose size remains important.
Nearby Jumps and the \(J_1\) Criterion
The definition makes an important distinction. A jump can move in time while the path still converges, provided the displacement tends to zero and the jump can be aligned by admissible time changes. A jump of fixed nonzero size shifted by a fixed positive amount generally cannot be aligned by time changes that approach the identity. On the other hand, a jump whose size tends to zero may be shifted a fixed distance and still make a negligible contribution to the values of the path.
Worked Example: Aligning a Jump That Moves a Vanishing Distance
Consider the scalar paths \(x(t)=\mathbf{1}\{t\geq 0.50\}\) and \(x_n(t)=\mathbf{1}\{t\geq 0.50+1/n\}\), for \(n\) large enough that the shifted time is below 1. The jump height is 1 in both paths, but the jump in \(x_n\) occurs \(1/n\) later.
Define \(\lambda_n\) to be linear from \((0,0)\) to \((0.50,0.50+1/n)\), and linear from that point to \((1,1)\). It is continuous, strictly increasing, fixes both endpoints, and maps the jump time \(0.50\) to \(0.50+1/n\). On the first segment its slope is \(1+2/n\); on the second its slope is \(1-2/n\), which is positive for \(n>2\).
The greatest time displacement occurs at the joining point, so $$ \sup_{0\leq t\leq 1}|\lambda_n(t)-t|=\frac{1}{n}\longrightarrow 0. $$ Also, \(x_n(\lambda_n(t))=x(t)\) at every \(t\), including the jump time because both paths are right-continuous. Thus the value discrepancy is zero, and \(x_n\to x\) in \(J_1\).
In an allocation process, this situation can represent a visible cumulative change whose recorded time fluctuates by a vanishing amount as the trial size grows. The time change makes the paths agree without changing the order of assignments.
A useful justification follows directly from the definition: if a jump in the approximating path is aligned with a limit jump, the time change must move the jump time by their difference. If that difference goes to zero, this requirement is compatible with \(\sup_t|\lambda_n(t)-t|\to0\). If the difference stays bounded away from zero and the jump cannot be matched by other path changes, it is not.
Worked Example: A Fixed Shift of a Nonvanishing Jump
Let \(x(t)=\mathbf{1}\{t\geq 0.40\}\) and \(x_n(t)=\mathbf{1}\{t\geq 0.60\}\) for every \(n\). Both paths have a jump of size 1, but the jump times differ by 0.20.
For \(x_n(\lambda_n(t))\) to have its jump at the same time as \(x\), the time change must satisfy \(\lambda_n(0.40)=0.60\). Consequently, $$ \sup_{0\leq t\leq 1}|\lambda_n(t)-t| \geq |\lambda_n(0.40)-0.40|=0.20. $$ This lower bound cannot tend to zero. If a time change does not align the jumps, there is an interval between their jump locations on which one path is 0 and the other is 1, so the value discrepancy is 1. Either way, the two required \(J_1\) errors cannot both tend to zero. Therefore \(x_n\) does not converge to \(x\) in \(J_1\).
This conclusion concerns an isolated, nonvanishing jump with no compensating path changes. It is not a rule that every fixed timing displacement prevents convergence: the size of the shifted jump matters.
Worked Example: A Fixed Shift of a Vanishing Jump
Let the proposed limit be \(x(t)=0\) for all \(t\), and let \(x_n(t)=n^{-1}\mathbf{1}\{t\geq 0.75\}\). The jump occurs at 0.75 for every \(n\); its timing does not approach a time change of zero displacement relative to any moving jump in the limit. Nevertheless, choose the identity time change, \(\lambda_n(t)=t\). Then $$ \sup_t|\lambda_n(t)-t|=0,\qquad \sup_t|x_n(\lambda_n(t))-x(t)|=\frac{1}{n}\longrightarrow 0. $$ Thus \(x_n\to x\) in \(J_1\). The jump's height disappears, so its fixed location creates no lasting value discrepancy. A small increment in a suitably normalized cumulative allocation path can behave in this way.
Uniform Convergence and Continuous Limits
Uniform convergence requires the values of two paths to be close at the same time, with no time adjustment. It is therefore stronger than \(J_1\) convergence: if \(\sup_t\|x_n(t)-x(t)\|\to0\), choose the identity time change to obtain \(J_1\) convergence. The jump-alignment example shows why the converse need not hold when the limit has a jump.
If the limit \(x\) is continuous, however, \(J_1\) convergence does imply uniform convergence. To see this, suppose the time changes in the definition satisfy both \(J_1\) conditions. For any \(s\in[0,1]\), set \(t=\lambda_n^{-1}(s)\). Then $$ \|x_n(s)-x(s)\| \leq \|x_n(\lambda_n(t))-x(t)\|+\|x(t)-x(s)\|. $$ The first term is bounded by the \(J_1\) value error. Also, \(|t-s|\leq\sup_u|\lambda_n(u)-u|\). Since a continuous function on \([0,1]\) is uniformly continuous, the second term tends uniformly to zero as this time displacement tends to zero. Taking the supremum over \(s\) proves uniform convergence.
This result is useful when a centered or scaled allocation path has a continuous limiting trajectory: asymptotic timing adjustments do not produce a separate kind of discrepancy from uniform value error. When the proposed limit has jumps, timing alignment is substantive and the two notions should not be conflated.
What to Check in Allocation Paths
For a step-function allocation path, the size of a jump depends on how the process is scaled. For example, a single assignment changes the raw count \(N_{n,k}(t)\) by 1, the proportion \(N_{n,k}(t)/n\) by \(1/n\), and a centered proportion multiplied by \(\sqrt n\) by \(1/\sqrt n\). These are different path sequences, so the same assignment can be negligible in one representation and visible in another.
A disappearing largest jump is a useful diagnostic when a continuous limit is proposed: if the largest jump does not tend to zero, uniform convergence to a continuous path is impossible. But small jumps alone do not prove \(J_1\) convergence. The paths might still have incorrect levels, drift away over time, or accumulate discrepancies across many increments. Conversely, when the limit has jumps, checking only that the approximating paths have the right jump sizes is insufficient; the relevant jumps must also be alignable by time changes that approach the identity.
State whether the coordinates are counts, proportions, or centered and scaled quantities. Determine the jump size produced by one assignment on that scale.
Compare the locations and sizes of jumps in the approximating paths with those of the proposed limit. A jump whose size vanishes may have little effect even if its location differs substantially.
Ask whether an increasing time change can align the important jumps while its maximum displacement tends to zero. Check the value discrepancy after alignment as well.
Jump matching is not enough: verify that levels between jumps and the overall path values also approach the limit. For random paths, convergence in distribution requires control of the laws on the path space, not just a successful comparison of one realization.
How \(J_1\) Fits Among Other Comparisons
The main alternatives answer different questions. Uniform convergence compares values at identical times and rules out even small time shifts of a visible jump. \(J_1\) relaxes that requirement by allowing small, order-preserving time changes. Finite-dimensional convergence compares distributions at selected fixed times; by itself, it does not ensure that whole paths behave coherently between those times.
The \(M_1\) topology is another topology used for paths with jumps. It is designed to accommodate some limits formed by clusters of nearby jumps, including certain cases where a single \(J_1\) time change cannot match the jumps. This greater flexibility changes which path sequences count as convergent; it is not a reason to treat \(J_1\) timing requirements as optional. The topology should match the path limit and the question being studied.
| Comparison | What it permits | Key caution |
|---|---|---|
| Uniform | Only value differences at the same times, tending uniformly to zero | Can reject paths with matching jumps shifted by a vanishing amount |
| \(J_1\) | Value comparison after increasing time changes that approach the identity | Does not generally permit a persistent shift of an important jump |
| Finite-dimensional | Convergence of distributions at specified time points | Does not by itself control the path between those points |
| \(M_1\) | Some convergence involving clusters of nearby jumps that \(J_1\) cannot match | Uses a different, weaker path comparison; conclusions depend on choosing the appropriate topology |
Common Mistakes and Limitations
- Mistake: “Any fixed jump-time shift prevents \(J_1\) convergence.” This ignores jump size. A fixed shift of a jump whose height tends to zero can still give convergence, as the \(n^{-1}\) example shows. The obstruction arises when an important jump remains mismatched and the required time displacement does not vanish.
- Mistake: “The jump sizes match, so the paths converge.” Equal jump heights do not fix a persistent timing discrepancy. Check the time-change error as well as the value error.
- Mistake: “Every individual assignment has a small normalized jump, so the limit follows.” Small increments are only a local diagnostic. The accumulated path may still have the wrong shape or fail to approach the proposed limit.
- Limitation: \(J_1\) convergence is a statement about paths in a specified representation and scale. Changing from counts to proportions or to a centered and scaled process changes jump sizes and can change which discrepancies remain visible.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- For \(x(t)=\mathbf{1}\{t\geq 0.30\}\) and \(x_n(t)=\mathbf{1}\{t\geq 0.30+1/n\}\), what feature allows \(J_1\) convergence?
- Let \(x(t)=0\) and \(x_n(t)=n^{-1}\mathbf{1}\{t\geq 0.80\}\). Does the fixed jump time prevent \(J_1\) convergence? Give the relevant bound.
- Why does \(J_1\) convergence to a continuous limit imply uniform convergence?
- A single assignment changes a cumulative proportion by \(1/n\). What is its jump size if the centered proportion is multiplied by \(\sqrt n\)?
Solutions. (1) An increasing time change can map the limit jump time \(0.30\) to \(0.30+1/n\); its maximum displacement is \(1/n\), and after this change the paths match. (2) No. With the identity time change the value error is at most \(1/n\), which tends to zero, while the time error is zero. (3) The \(J_1\) value error controls the path after time change, and the remaining difference is bounded by the modulus of continuity of the limit over a time interval no larger than the maximum time displacement; continuity on \([0,1]\) makes that bound tend to zero. (4) The scaled jump is \(\sqrt n(1/n)=1/\sqrt n\), which tends to zero.