A sequential empirical process records every partial sample, not only the terminal empirical measure. Its extra time coordinate turns a Brownian bridge in the function index into a Brownian motion in time and supplies the natural limit theory for cumulative-sum and change-point statistics.
The Kiefer–Müller process
Definition. For \(0\leq t\leq1\) and \(f\in\mathcal F\), define \[ \alpha_n(t,f) = \frac1{\sqrt n}\sum_{i=1}^{\lfloor nt\rfloor} \{f(X_i)-Pf\}. \] A \(P\)-Kiefer–Müller process \(\mathbb K_P\) is centered Gaussian with covariance \[ \operatorname{Cov}\{\mathbb K_P(s,f),\mathbb K_P(t,g)\} =(s\wedge t)\{P(fg)-Pf\,Pg\}. \]
For fixed \(f\), \(t\mapsto\mathbb K_P(t,f)\) is Brownian motion with variance parameter \(\operatorname{Var}_P f\). At \(t=1\), \(f\mapsto\mathbb K_P(1,f)\) is the \(P\)-Brownian bridge. The term “bridge” here refers to centering functions by \(P\), not tying the time coordinate to zero at \(t=1\).
Theorem (sequential Donsker theorem). If \(\mathcal F\) is a measurable \(P\)-Donsker class with square-integrable envelope, then \[ \alpha_n\rightsquigarrow\mathbb K_P \qquad\text{in }\ell^\infty([0,1]\times\mathcal F). \]
Proof outline. Finite-dimensional convergence follows from the multivariate functional central limit theorem. For tightness, regard the process as indexed by product functions \[ (u,x)\mapsto1\{u\leq t\}\{f(x)-Pf\} \] under the product of Lebesgue measure on \([0,1]\) and \(P\). Half-line indicators in \(u\) form a VC class, and the Donsker entropy of \(\mathcal F\) combines with it through product-class bounds. The resulting asymptotic equicontinuity controls both \(|s-t|\) and \(\rho_P(f,g)\). \(\square\)
Tying down the time coordinate
Definition. The CUSUM or tied-down process is \[ \mathbb U_n(t,f) = \alpha_n(t,f)-\frac{\lfloor nt\rfloor}{n}\alpha_n(1,f). \]
Corollary. Under the sequential Donsker theorem, \[ \mathbb U_n\rightsquigarrow \mathbb U_P(t,f) =\mathbb K_P(t,f)-t\mathbb K_P(1,f), \] whose covariance is \[ \operatorname{Cov}\{\mathbb U_P(s,f),\mathbb U_P(t,g)\} =(s\wedge t-st)\{P(fg)-Pf\,Pg\}. \]
Proof. The map \(z\mapsto\{z(t,f)-tz(1,f)\}\) is continuous in the supremum norm, and \(\sup_t|\lfloor nt\rfloor/n-t|\leq1/n\). Apply the continuous-mapping theorem and calculate covariance bilinearly. \(\square\)
Example (mean change). For \(f(x)=x\), \(EX^2<\infty\), and variance \(\sigma^2\), \[ \frac1{\sigma}\sup_{0\leq t\leq1}|\mathbb U_n(t,f)| \rightsquig \sup_{0\leq t\leq1}|B(t)|, \] where \(B\) is a standard Brownian bridge. A large value indicates that an early partial-sample mean differs from the full-sample mean.
Local change-point alternatives
Proposition. Suppose observations before \(\tau n\) have law \(P\), while those after \(\tau n\) have laws \(P_n\) satisfying \[ \sqrt n(P_n-P)f\to\Delta(f). \] Then the tied-down process acquires the deterministic drift \[ d_\tau(t)\Delta(f), \qquad d_\tau(t)= \begin{cases} -t(1-\tau),&t\leq\tau,\\ -\tau(1-t),&t>\tau, \end{cases} \] up to the sign convention used for the post-change contrast.
Derivation. The expected partial sum contains no drift before \(\tau\), and \((t-\tau)\sqrt n(P_n-P)f\) after \(\tau\). The full-sample centering subtracts \(t(1-\tau)\sqrt n(P_n-P)f\). Combining the two pieces yields the triangular function above. Its absolute value is largest at \(t=\tau\), explaining why CUSUM localizes a single change point. \(\square\)
Sequential distribution functions
Example. With \(f_x=1_{(-\infty,x]}\), the process \[ \alpha_n(t,x) =\frac1{\sqrt n}\sum_{i\leq\lfloor nt\rfloor} \{1(X_i\leq x)-F(x)\} \] converges to a Gaussian process with covariance \[ (s\wedge t)\{F(x\wedge y)-F(x)F(y)\}. \] Suprema over both \(t\) and \(x\) test for an unknown-time change in the entire distribution rather than only its mean.
Dependence and monitoring
For time series, the covariance kernel changes to the long-run covariance \[ \Gamma(f,g)=\sum_{k\in\mathbb Z} \operatorname{Cov}\{f(X_0),g(X_k)\}, \] and independence-based symmetrization is no longer valid without modification. Mixing, martingale, physical-dependence, or block-bootstrap methods must establish the functional central limit theorem. Online monitoring also extends the time range beyond one and requires a boundary function chosen to control repeated false alarms.