Why the Topology Matters
A risk function records a decision rule’s expected loss at each parameter value. When comparing rules, it is often useful to ask whether one risk function can be approximated by risks in a given class. But “approximated” has different meanings: closeness at each fixed parameter, uniform closeness over bounded parameter regions, or uniform closeness over the entire parameter space. Those choices can change which limiting risks count as members of the closure.
We compare the three notions for risk functions on \(\Theta=\mathbb{R}\). All functions in the examples are bounded and nonnegative. The examples can be realized as risks in a decision problem with a degenerate observation and an action \(a\in\mathbb{N}\): define the loss at parameter \(\theta\) for action \(a=n\) to be the \(n\)th function in the sequence. Thus each function in the sequence is the risk of a deterministic rule. A limit function need not itself be the risk of an available rule in that problem.
Three Notions of Closeness
They converge pointwise if, for every fixed \(\theta\in\mathbb{R}\), \(r_n(\theta)\to r(\theta)\).
They converge uniformly on compact sets (or compact-uniformly) if, for every compact \(K\subset\mathbb{R}\), \(\sup_{\theta\in K}|r_n(\theta)-r(\theta)|\to 0\).
They converge globally uniformly if \(\sup_{\theta\in\mathbb{R}}|r_n(\theta)-r(\theta)|\to 0\).
Each notion defines a topology on the relevant function space. For a set \(A\) of risk functions, its closure in that topology consists of the functions whose every open neighborhood intersects \(A\). We write these closures as \(\overline{A}^{\,p}\), \(\overline{A}^{\,c}\), and \(\overline{A}^{\,u}\), for pointwise, compact-uniform, and global-uniform closure, respectively.
A sequence converging in a topology has its limit in the closure of its range. The converse need not hold for every topology and every set: closure is defined by neighborhoods, not only by limits of sequences. In the examples below, we establish closure membership by convergence and nonmembership by exhibiting a neighborhood that misses the entire sequence.
How the Closures Are Ordered
Proof. Global uniform control implies compact-uniform control: if \(\sup_{\theta\in\mathbb{R}}|r(\theta)-s(\theta)|<\epsilon\), then for every compact \(K\), $$ \sup_{\theta\in K}|r(\theta)-s(\theta)|\leq \sup_{\theta\in\mathbb{R}}|r(\theta)-s(\theta)|<\epsilon. $$ Thus every compact-uniform neighborhood of a function contains a global-uniform neighborhood of that function. In other words, the global-uniform topology is finer than the compact-uniform topology.
Likewise, a compact-uniform neighborhood can require closeness on finitely many compact sets. Any finite set of parameter points is compact, so such a requirement controls closeness at each of those points. Hence the compact-uniform topology is finer than the pointwise topology. A finer topology has at least as many open neighborhoods; if a point belongs to the closure in the finer topology, every neighborhood in the coarser topology must also intersect \(A\). Applying this fact to both comparisons gives the stated inclusions. \(\square\)
The inclusions describe possible membership, not guaranteed strictness. A particular set of risks can have the same closure under all three topologies. The following examples show that the distinctions can matter.
Worked Examples of Different Risk Limits
Worked Example: Pointwise Closure Without Compact-Uniform Closure
For \(n\geq 1\), let \(r_n(\theta)=\exp(-n\theta^2)\), and set \(A=\{r_n:n\geq 1\}\). Determine whether \(r(\theta)=\mathbf{1}_{\{0\}}(\theta)\) belongs to each closure.
At \(\theta=0\), \(r_n(0)=1\) for every \(n\). At any fixed \(\theta\neq 0\), \(n\theta^2\to\infty\), so \(r_n(\theta)\to 0\). Therefore \(r_n(\theta)\to\mathbf{1}_{\{0\}}(\theta)\) at every fixed \(\theta\), and \(r\in\overline{A}^{\,p}\).
Now take the compact set \(K=[-1,1]\). At \(\theta=0\), the difference \(|r_n(\theta)-r(\theta)|\) is zero. For \(\theta\neq 0\), it equals \(\exp(-n\theta^2)\). As \(\theta\) approaches zero through nonzero values, this difference approaches \(1\). Its supremum on \(K\) is therefore \(1\), for every \(n\): $$ \sup_{\theta\in K}|r_n(\theta)-r(\theta)|=1. $$ The compact-uniform neighborhood of \(r\) requiring this supremum to be less than \(1/2\) contains no \(r_n\). Thus \(r\notin\overline{A}^{\,c}\). The global-uniform distance is also at least \(1\), so \(r\notin\overline{A}^{\,u}\).
The limiting risk is a pointwise limit, but the approximation fails uniformly even on a single compact interval. The narrow peak near zero becomes increasingly concentrated; checking each fixed parameter misses this behavior.
Worked Example: Compact-Uniform Closure Without Global-Uniform Closure
Let \(r_n(\theta)=\exp(-(\theta-n)^2)\), again with \(A=\{r_n:n\geq 1\}\). Consider the zero function \(r(\theta)=0\).
Fix any compact set \(K\subset\mathbb{R}\). It is contained in \([-M,M]\) for some finite \(M\). For \(n>M\), every \(\theta\in K\) satisfies \(|\theta-n|\geq n-M\), so $$ \sup_{\theta\in K}|r_n(\theta)| \leq \exp(-(n-M)^2)\longrightarrow 0. $$ Thus \(r_n\to 0\) compact-uniformly, and in particular pointwise. Hence \(0\in\overline{A}^{\,c}\) and \(0\in\overline{A}^{\,p}\).
For each \(n\), however, \(r_n(n)=1\), and \(0\leq r_n(\theta)\leq 1\) everywhere. Consequently, $$ \sup_{\theta\in\mathbb{R}}|r_n(\theta)-0|=1. $$ The global-uniform neighborhood of zero with radius \(1/2\) contains none of the functions in \(A\). Therefore \(0\notin\overline{A}^{\,u}\). The peak moves out along the parameter space: every fixed compact set eventually misses it, but the supremum over the whole space does not decrease.
Worked Example: A Limit in All Three Closures
Let \(r_n(\theta)=1/n\) for every \(\theta\in\mathbb{R}\), and let \(A=\{r_n:n\geq 1\}\). The candidate limit is again the zero function.
For every \(\theta\), \(|r_n(\theta)-0|=1/n\), so the sequence converges pointwise to zero. For every compact \(K\), its supremum difference is also \(1/n\), and the supremum over the entire parameter space is \(1/n\): $$ \sup_{\theta\in K}|r_n(\theta)|=\frac{1}{n}, \qquad \sup_{\theta\in\mathbb{R}}|r_n(\theta)|=\frac{1}{n}. $$ Both quantities tend to zero. The zero function therefore belongs to the pointwise, compact-uniform, and global-uniform closures of \(A\). The third example shows that the closure hierarchy can be strict for some sets but need not be strict for every set.
What These Closures Say About Risk Approximation
The topology determines the strength of the approximation claim. Pointwise closure supports approximation separately at each fixed parameter value. Compact-uniform closure provides one approximation bound that holds simultaneously across any chosen compact region. Global-uniform closure provides one bound simultaneously across the whole parameter space. In risk analysis, these are materially different guarantees: a small error at every fixed parameter does not rule out a large error somewhere in a region that changes with the approximating rule.
Compact-uniform approximation depends on which subsets of the parameter space are compact.
Say whether the risk error is controlled pointwise, on each compact set, or over the full parameter space.
A limit in a weaker topology does not establish membership in a stronger-topology closure.
A common mistake is to argue that pointwise convergence is uniform on a compact set because every individual parameter has a sufficiently large index. That reasoning allows the required index to depend on the parameter. Uniform convergence requires a single index threshold to work for every point in the compact set. In the first example, each nonzero parameter eventually has a small risk, but arbitrarily close to zero there are still points where the risk is nearly one. Compactness alone does not make pointwise convergence uniform; additional hypotheses would be needed.
A second mistake is to treat compact-uniform approximation as global-uniform approximation. The moving peak in the second example disappears from each fixed compact set, while remaining exactly height one somewhere on \(\mathbb{R}\). Consequently, conclusions about risk closure should name the topology explicitly rather than saying only that a risk is “approximable.”
Check Your Understanding
Use the three closure definitions to answer the questions.
- Why does global-uniform convergence imply compact-uniform convergence?
- For \(r_n(\theta)=\exp(-n\theta^2)\), what feature prevents compact-uniform convergence to \(\mathbf{1}_{\{0\}}(\theta)\)?
- For \(r_n(\theta)=\exp(-(\theta-n)^2)\), explain why the supremum on any fixed compact set tends to zero while the global supremum remains one.
- Can compact-uniform closure membership imply global-uniform closure membership? Give the example that supports your answer.
- Why is pointwise closure membership, by itself, insufficient to claim a uniform risk-approximation guarantee?