A CLT for empirical processes involving time-dependent data

Abstract

For stochastic processes $\{X_t:t\in E\}$, we establish sufficient conditions for the empirical process based on $\{I_{X_t\le y}-\operatorname{Pr}(X_t\le y):t\in E,y\in\mathbb{R}\}$ to satisfy the CLT uniformly in $t\in E,y\in\mathbb{R}$. Corollaries of our main result include examples of classical processes where the CLT holds, and we also show that it fails for Brownian motion tied down at zero and $E=[0,1]$.Comment: Published in at http://dx.doi.org/10.1214/11-AOP711 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

A CLT for weighted time-dependent uniform empirical processes

“…For a uniform process {Xt : t ∈ E} (by which Xt is uniformly distributed on (0, 1) for t ∈ E) and a function w(x) > 0 on (0, 1), we give a sufficient condition for the weak convergence of the empirical process based on {w(x)(1 Xt≤x − x) : t ∈ E, x ∈ [0, 1]} in ℓ ∞ (E × [0, 1]). When specializing to w(x) ≡ 1 and assuming strict monotonicity on the marginal distribution functions of the input process, we recover a result of [9]. In the last section, we give an example of the main theorem.…”

“…Corollary 3.11. Under the WL-condition, the process G 0 (t, y) is sample bounded and uniformly continuous with respect to its L 2 distance; the same is true for a zero mean Gaussian process with covariance For the pre-Gaussian property of the empirical process considered in [9], we give a different proof rather than the constructive one in [9] using the generic chaining [15].…”

“…Weighted empirical processes consider suitable weight functions w(x) such that w(x)G n (x) converges weakly to the weighted Brownian bridge process w(x)B(x); in the literature, such a theorem is called Chibisov-O'Reilly theorem; see [2], [12], [3] etc. [9] considered a time dependent empirical process G n (t, y) := n −1/2 n 1 (1 Yi(t)≤y − P(Y i (t) ≤ y)), t ∈ E, y ∈ R, for independent and identically distributed (iid) stochastic processes Y 1 (t), Y 2 (t), · · · for t ∈ E. Under a condition the authors call the L-condition, this empirical process converges weakly in ℓ ∞ (E × R). In [5], the authors proved a CLT for weighted tail empirical processes under a small oscillation condition as the L-condition guarantees.…”

Bahadur–Kiefer representations for time dependent quantile processes

“…These include certain self-similar processes of which fractional Brownian motion is a special case. Their approach is based on an extension of a result of Vervaat [16] on the weak convergence of inverse processes in combination with results from their deep study with Kurtz [Kurtz, Kuelbs and Zinn [8]] of central limit theorems for time dependent empirical processes.…”

Estimation of treatment policies based on functional predictors

“…If X is a Gaussian process, then it suffices that Var( X ( t )) is bounded away from zero. Assumption (A4′) is used in Theorem 5 of Kuelbs et al (2013) to prove that a class of (unweighted) indicator functions is Donsker, but it is not clear whether their approach could be extended to our setting. …”