Limit Theorems for Quantile and Depth Regions for Stochastic Processes

Transcription

1 Limit Theorems for Quantile and Depth Regions for Stochastic Processes James Kuelbs and Joel Zinn Abstract. Since contours of multi-dimensional depth functions often characterize the distribution, it has become of interest to consider structural properties and limit theorems for the sample contours (see [ZS00b] and [KM12]). In particular, Kong and Mizera have shown that for finite dimensional data, directional quantile envelopes coincide with the level sets of half-space (Tukey) depth. We continue this line of study in the context of functional data, when considering analogues of Tukey s half-space depth [Tuk75]. This includes both a functional version of the equality of (directional) quantile envelopes and quantile regions as well as limit theorems for the sample quantile regions up to n asymptotics. Mathematics Subject Classification (2010). Primary 60F05; Secondary 60F17, 62E20. Keywords. quantile region, depth region. 1. Introduction The study of depth functions and resulting depth regions provide what is called a center-outward order for multidimensional data that allows one to gain insight into the underlying probability law. Many of the results obtained are for depths in R d, but more recently functional data, such as that obtained from streaming data, has received considerable attention. When trying to assess the viability of a particular statistical approach, it is important to analyze the method when applied to specific data, asking if the empirical version is consistent, and what rates of convergence one might have. In the case of functional data this often leads to considering the data functions at a discrete set of points - perhaps a large number of points in the domain of the functions. However, if the infinite dimensional process (or measure with infinite dimensional support) is unstable in some way with respect to the method one is using to order the typical functions being encountered, there Second author partially supported by NSF grants DMS and DMS

2 2 James Kuelbs and Joel Zinn is the question of what one might be modeling in these situations. In this paper, as in the development of general empirical processes, we are interested in some basic probabilistic properties of the statistics being proposed for halfspace depth and the related quantile processes, when no restrictions on the domain of the functions are imposed. If one has stability for the statistic being employed to provide the infinite dimensional ordering, this alleviates some of the concerns just mentioned. Papers that provided motivation, and some useful contrasts for what we do in this paper, include [MT94], [Mas02], [Nol99], and [KM12]. In addition, the papers [ZS00a] and [ZS00b] examine a number of unifying properties of a broad collection of such depths, and in [ZS00b] some convergence results for the related depth regions and their boundaries are established for R d -valued data. They also contain an extensive list of references. The paper [KM12] also provides results obtaining convergence of Tukey half-space depth regions for R d -valued data under conditions that are quite different from those in [ZS00b], but in each setting something close to a law of large numbers is an important assumption required for the proofs. An assumption of a similar nature appears in (3.14) and (3.15) of Theorem 3.9, and can be verified in many situations by applying the empirical quantile CLTs for stochastic processes obtained in [KZ13b] and [KZ14]. These CLTs, along with the approach in [KM12], were a primary motivation for the various limit theorems we establish here. In particular, the results in [KZ13b] and [KZ14] allow us to obtain n-asymptotics for the convergence of the half-space depth sets for many types of functional data. This includes data given by a broad collection of Gaussian processes, martingales, and independent increment processes. Furthermore, the limit theorems obtained have Gaussian limits uniformly over the parameter set of the data process and in the quantile levels α I for I a closed interval of (0, 1), and are established directly without first introducing a corresponding half-space depth. This is in contrast with the limit theorems for empirical medians in [Nol99] and [Mas02] based on the argmax of empirical Tukey depth processes, which have non-gaussian limits for data in R d when d 2. A first CLT of this type was obtained in [Swa07] for the empirical median process when the data was a sample continuous Brownian motion on [0, 1], and later in [Swa11] for the empirical α-quantile processes for each fixed α (0, 1). The proofs in these papers are quite different than those in [KZ13b] and [KZ14], which employ empirical process theory as developed for functional data in [KKZ13], a method of Vervaat from [Ver72], the CLT results in [AGOZ88], and the necessary and sufficient conditions for sample function continuity of a Gaussian process using generic chaining as in [Tal05]. A brief outline of the paper is as follows. In section 2 we introduce basic notation. Section 3 introduces additional notation and states the main results of this paper, namely Proposition 3.7 and Theorem 3.9, which indicate how half-space depth regions for stochastic processes based on evaluation maps are uniquely determined by related upper and lower quantile functions for

3 Quantile and Depth Regions 3 the process. In addition, under suitable conditions they show the empirical versions of these regions converge to the population versions with respect to a Hausdorff metric (also used for finite dimensional data in [KM12] and [ZS00b]), and include both consistency results and n-rates of convergence for these distances. As mentioned above, the main assumptions required in the proof of Theorem 3.9 can be verified by applying the empirical quantile CLTs in [KZ13b] and [KZ14] for many types of functional data, but we also obtain some consistency results for empirical quantile functions in Theorem 3.16 and Corollary They are of independent interest, and can be used in this setting to verify conditions (3.14) and (3.15) of Theorem 3.9. As is natural to expect, these consistency results are obtained under weaker conditions. However, they do not yield the n-rates of convergence given in Theorem 3.9, which follow when one can apply the CLT results. Theorem 3.9 applies quite generally, but the limiting regions can often be very small. This is pointed out in section 4, where we examine half-space depth regions based on data obtained by independently sampling continuous Brownian motions, showing that although Theorem 3.9 applies, the quantile regions and depth regions have probability zero. That these regions may have zero probability is a problem which holds for many other processes (see Remark 4.1), and hence motivates our Proposition 4.2. This result shows that if we suitably smooth the one dimensional distributions, then one can avoid this problem for many stochastic processes. Throughout the paper when we speak of smoothing a stochastic process we mean that we are applying the smoothing of Proposition 4.2. We postpone proofs to section 5, but a number of remarks are included in earlier sections to motivate and understand how the results fit together. 2. Basic Notation Throughout the paper E is a nonempty set, D(E) a collection of real-valued functions on E, D E is the minimal sigma-algebra making the evaluation maps θ t : D(E) R measurable, where θ t (z) = z(t), t E, z D(E), (2.1) and µ is a probability measure on (D(E), D E ). Of course, the (functional) data of interest are drawn from D(E) and µ is the population distribution or law on D E of the data. It will also be convenient to have i.i.d. stochastic processes X := {X(t) : t E}, X 1, X 2, on some probability space (Ω, F, P ) such that the common law they induce on (D(E), D E ) is µ. For each t E, we denote the distribution function of X(t) by F t (x) := F (t, x), x R. In addition, without loss of generality we assume that the sample paths of these processes are always in D(E), and for n 1 denote the empirical measures

4 4 James Kuelbs and Joel Zinn for µ on (D(E), D E ) by µ n (ω) = 1 n n δ Xj(ω), ω Ω. (2.2) j=1 At this stage the set E and the set of functions D(E) are purposely abstract. An important special case in [KZ15a] takes E to be the linear functions on R d of Euclidean norm one, and D(E) a subset (usually subspace) of the continuous functions on E containing {z : z(t) = t(x), t E, x R d }. In this setting the results of this paper apply to Tukey depth regions (and the related quantile regions), implying new empirical results for this important classical depth. The reason one does not proceed in a similar manner in the infinite dimensional case is that this choice of E is too large to always have the empirical quantile CLTs of [KZ13b] applicable in this setting, and also the resulting half-space depth may be zero with probability one. For example, Proposition 3.6 of [KZ13a] provides an explicit formula for halfspace depth for Gaussian measures on a Banach space which is zero except for points in a set of measure zero when all continuous linear functionals or, equivalently, all continuous linear functionals of norm one, are used to define E. The papers [KZ13a], [CC14], and [KZ15] contain other examples where zero depths appear, but [KZ15] also shows how to alleviate this problem using a smoothing (a random numerical shift) of the data. In particular, Theorems 1 and 2 of [KZ15] examine other aspects of this sort of problem, and establish limit theorems for half-region depth when the set E satisfies a compactness condition. It also provides some examples which show that the smoothing method we use avoids some non-intuitive properties that the modified halfregion depth proposed in other papers possesses. See examples 1-3 in [KZ15] for details. To describe the quantile and depth regions in our results we now recall the definition of left and right α-quantiles for real-valued random variables. Definition 2.1. Let ξ be a real-valued random variable with Borel probability law µ ξ, and for x R set F ξ (x) = P (ξ x). Then, for α (0, 1), the left and right α-quantiles of ξ (equivalently, of the distribution function F ξ or the probability law µ ξ ) are defined, respectively, as τ α,l (ξ) := τ α,l (F ξ ) := τ α,l (µ ξ ) := inf{x : F ξ (x) α} and (2.3) τ α,r (ξ) := τ α,r (F ξ ) := τ α,r (µ ξ ) := sup{x : F ξ (x ) α} = inf{x : F ξ (x) > α}. Next we turn to the definition of the left and right α-quantile functions determined by a measure ν on (D(E), D E ). In Remarks 2.3 and 2.4 that follow we indicate some simplifications of this notation that we employ for our fixed measure µ and its empirical measures µ n (ω). Definition 2.2. Let ν be a probability measure on (D(E), D E ), {θ t : t E} denote the evaluation maps in (2.1), and for each t E the distribution function of θ t with respect to ν is F θt. Then, for (α, t) (0, 1) E, the left

5 Quantile and Depth Regions 5 and right α-quantile functions determined by ν are τ α,l (t, ν) := τ α,l (θ t ) := τ α,l (F θt ) := inf{x: ν(f D(E): θ t (f) x) α} and (2.4) τ α,r (t, ν) := τ α,r (θ t ) := τ α,r (F θt ) := inf{x: ν(f D(E): θ t (f) x) > α}. (2.5) Remark 2.3. If the measure ν is our fixed measure, µ, we simplify to τ α,l (t) := τ α,l (t, µ) and τ α,r (t) := τ α,r (t, µ). (2.6) In case we also have τ α,l (t) = τ α,r (t) for all t E, then to denote their common value we simply write τ α (t), t E, (2.7) and note that τ α (t) is the unique function f(t) on E such that for each t E, f(t) is the left α-quantile of the random variable θ t ( ) on (D(E), D E, µ). In addition, note that if X := {X(t) : t E} is a stochastic process with sample paths in D(E) that induces law µ on (D(E), D E ), then the left and right α-quantile functions determined by µ can be defined by the distribution functions F t (x) := F (t, x) = P (X(t) x), since F t = F µ,θt, t E. Remark 2.4. If our measure is µ n (ω), we usually leave out the ω and write τ n α,l(t) := τ α,l (t, µ n ) and τ n α,r(t) := τ α,r (t, µ n ). (2.8) However, as in the proof of Theorem 1, there are times when including ω is helpful, and we then write τ n α,l(t, ω) := τ α,l (t, µ n (ω)) and τ n α,r(t, ω) := τ α,r (t, µ n (ω)). (2.9) Moreover, the empirical quantiles are such that for each ω Ω, t E τα,l(t, n ω) = inf{x: 1 n I Xj(t,ω) x) α} and (2.10) n j=1 τ n α,r(t, ω) = inf{x: 1 n n I Xj(t,ω) x) > α}. j=1 3. Equality of Quantile and Depth Regions, and Convergence Results In order to state our results on half-space quantile and depth regions for functional data (or stochastic processes), the following lemma recalls some basic facts about right and left quantiles of a real-valued random variable. Lemma 3.1. Let ξ be a real-valued random variable with distribution function F ξ. Then, for α (0, 1), τ 1 α,r (ξ) = τ α,l ( ξ), (3.1)

6 6 James Kuelbs and Joel Zinn and if F ξ (x) = α for some x > τ α,l (ξ), then F ξ (x) = α for all x [τ α,l (ξ), τ α,r (ξ)). In addition, if x [τ α,l (ξ), τ α,r (ξ)), then x is an α-quantile of ξ, i.e. F ξ (x) α and 1 F ξ (x ) 1 α. Next we turn to the definitions of α-quantile regions, half-space depths, and α-depth regions. Definition 3.2. Let ν be a probability measure on (D(E), D E ) with left and right α-quantile functions as in (2.4). If α (0, 1 2 ], then the α-quantile region (with respect to ν) is the subset of D(E) given by M α,ν := t E {z D(E) : τ α,l (t, ν) z(t) τ 1 α,r (t, ν)}. (3.2) Remark 3.3. If the measure ν is our fixed measure, µ, we simplify to M α := t E {z D(E) : τ α,l (t) z(t) τ 1 α,r (t)}, (3.3) where the left and right quantiles are (with respect to µ) as in (2.6), and again ignore its dependence on µ. Remark 3.4. If the measure ν is µ n (ω), we usually leave out the ω. That is, for α (0, 1 2 ] and n 1, the empirical α-quantile region (with respect to µ n) is denoted by M α,n := t E {z D(E) : τ n α,l(t) z(t) τ n 1 α,r(t)}, (3.4) where empirical α-quantile functions τ n α,l (t) and τ n 1 α,r(t) are as in (2.8). Of course, these functions and M α,n are defined for all ω Ω, and when that dependence needs special emphasis we will write τ n α,l (t, ω), τ n α,r(t, ω), and M α,n (ω), respectively. Otherwise the dependence on ω will be suppressed. Definition 3.5. Let ν be a probability measure on (D(E), D E ). For any realvalued function, h, on E, we define the half-space depth of h with respect to ν and the evaluation maps θ t, t E, by D(h, ν) := inf min{ν(z D(E) : z(t) h(t)), ν(z D(E) : z(t) h(t))}, t E (3.5) and the α-depth regions by N α,ν := {h D(E) : D(h, ν) α}. (3.6) Remark 3.6. If the measure ν is µ n (ω), we usually leave out the ω. That is, for n 1 the empirical half-space depths with respect to µ n are denoted by D(h, µ n ) and the empirical α-depth regions are N α,n := {h D(E) : D(h, µ n ) α}. (3.7) When the dependence on ω needs special emphasis we will write D(h, µ n )(ω) and N α,n (ω), respectively. Moreover, the depth D(h, ) not only depends on the measure, but is defined in terms of the evaluation maps indexed by E, so in what follows we may also refer to it as the E-depth with respect to

7 Quantile and Depth Regions 7 the measure involved, or simply as E-depth. In addition, when the stochastic process X induces the law µ on (D(E), D E ), we have D(h, µ) = inf min{p (X(t) h(t)), P (X(t) h(t))}. (3.8) t E We write Λ to denote the measurable cover function of a real-valued function Λ on Ω, see [Dud99], and for U, V subsets of D(E) we denote the Hausdorff distance between U and V (with respect to the sup-norm on D(E)) by d H (U, V ) = inf{ɛ > 0 : U V ɛ and V U ɛ }, (3.9) where U ɛ = {z D(E) : inf h U sup t E z(t) h(t) < ɛ}. If U or V is empty, but not both, then d H (U, V ) = Equality of depth and quantile regions. The following proposition shows certain quantile regions are equal to related depth regions. The proposition is quite general, and also applies to the empirical quantiles and the related empirical depths. Proposition 3.7. Assume the notation in section 2, and ( ). Then, for α (0, 1 2 ] and ν any probability measure on (D(E), D E) we have M α,ν = N α,ν. (3.10) In particular, the α-quantile regions and the α-depth regions with respect to µ and also the empirical measures µ n are such that and for n 1, ω Ω, M α = N α, (3.11) M α,n (ω) = N α,n (ω). (3.12) Remark 3.8. Although Proposition 3.7 holds quite generally, it is important to note that there are many examples where the sets in (3.11) are small. In fact, they may have µ probability zero for all α > 0. We present such an example in section 4, but many of the examples discussed in [KZ13a] also have similar properties. Nevertheless, it is important to keep in mind that if one wants to examine quantile regions of the type in Proposition 3.7, then some variety of half-space depth emerges. Hence, we will follow this example by showing the sets involved in (3.11) are much larger for smooth versions of the data, where before smoothing they possibly had positive half space depth with probability zero. Finally, with τ 0,l (t) = and τ 1,r (t) = + for all t E, which are their natural definitions, we easily see M 0 = D(E). Hence (3.11) also h olds for α = 0, since {h D(E) : D(h, µ) 0} = D(E). If D(E) is assumed to be a linear space, then the maps θ t, t E, are linear from D(E) into R, and hence M α is convex. If D(E) has a topology such that these maps are continuous, then M α is also closed. Of course, from (3.11) the sets N α then have similar properties.

8 8 James Kuelbs and Joel Zinn 3.2. Empirical Regions Converge In Proposition 3.7, D(E) is quite arbitrary, except that it supports the probability measure µ = L(X). However, for many standard stochastic processes X := {X(t) : t E} the set E is a compact interval of the real line or a compact subset of some metric space, and its sample paths may well be continuous, cadlag, or at least uniformly bounded on E. Hence, in such cases we can take D(E) to be the Banach space l (E) with sup-norm h = sup t E h(t), or some closed linear subspace of l (E) of smoother functions that reflect the regularity of the sample paths of X. The choice of D(E) = l (E) is convenient in that weak convergence results for empirical processes are readily available in this setting. Moreover, if D(E) = l (E) and the sample paths of X are in l (E), then stochastic boundedness of X and a fairly immediate argument, see the proof of Corollary 3.11, implies for given α (0, 1 2 ] and all ω Ω that τ α,l ( ), τ 1 α,r ( ), τα,l( ), n τ1 α,r( ) n D(E). (3.13) If D(E) is a closed subspace of l (E), then under suitable conditions we can still verify (3.13), but these arguments are more subtle. Hence in the next result (3.13) is an assumption, but following its statement we present some corollaries where (3.13) is verified directly. Theorem 3.9. Let D(E) be a closed linear subspace of l (E) with respect to the sup-norm such that for some α (0, 1 2 ] and all ω Ω (3.13) holds, and the measurable cover functions and τ n α,l( ) τ α,l ( ) (3.14) τ n 1 α,r( ) τ 1 α,r ( ) (3.15) converge in probability to zero with respect to P. Then, for the given α (0, 1 2 ] the sets M α, N α, M α,n, and N α,n are non-empty, and the measurable cover functions (of the Hausdorff distances) d H(M α,n, M α ) = d H(N α,n, N α ) (3.16) converge in probability to zero with respect to P. In addition, if 1 a n = O( n) converges to infinity, and the measurable cover functions a n τ n α,l( ) τ α,l ( ) and a n τ n 1 α,r( ) τ 1 α,r ( ) (3.17) are bounded in probability, then a n d H(M α,n, M α ) and a n d H(N α,n, N α ) (3.18) are bounded in probability with respect to P. Remark The assumptions (3.14) and (3.15) or (3.17) in Theorem 3.9 are non-trivial, but by applying the results in [KZ13b] and [KZ14] one can obtain a broad collection of stochastic processes for which they can be verified for all α (0, 1) with best possible a n, namely a n = n. Then, at least in some situations, one can apply the results in [KZ13a] to identify depth regions,

9 Quantile and Depth Regions 9 which when combined with Proposition 3.7 allow us to determine the related quantile regions for these processes. In section 4 we examine the combined effect of Proposition 3.7 and Theorem 3.9 when they are applied to sample continuous Brownian motion. Similar results also hold for a large number of other stochastic processes, some of which are mentioned in section 4. Corollary Let µ = L(X), D(E) = l (E), and assume α (0, 1 2 ] is such that there exists x α (0, ) with sup P ( X(t) x α ) < α. (3.19) t E Then, (3.14) and (3.15) imply (3.16), and (3.17) implies both (3.16) and (3.18) hold. The next two corollaries verify Theorem 3.9 for D(E) a closed linear subspace of l (E) consisting of smooth functions, provided it is assumed that the sample paths of X and both τ α,l, τ 1 α,r, satisfy the same smoothness conditions. Of course, the assumption that τ α,l, τ 1 α,r, are suitably smooth, can be verified by imposing suitable assumptions on the distribution functions F (t, x) = P (X t x). For example, in [KZ14] this follows immediately from the self-similarity property assumed on the stochastic process X, but it can also be verified for processes which are not self-similar by the following lemma. Lemma Let (E, d) be a metric space, for each t E assume that lim sup s t x R F (s, x) F (t, x) = 0, (3.20) and that for a given α (0, 1) there exists θ(α) > 0 such that inf f(t, x) c α > 0, (3.21) t E, x τ α,l (t) θ(α) where f(t, x) is the density of F (t, x) := P (X t x). In addition assume for the given α (0, 1) there exists y α (0, ) such that sup P ( X(t) y α ) < α (1 α), (3.22) t E and C(E) denotes the real-valued continuous functions on (E, d). Then, τ α,l ( ) = τ α,r ( ) := τ α ( ) C(E) l (E) for the given α (0, 1), and τ α ( ) y α. Remark If α (0, 1 2 ], then we can take x α = y α, where x α is defined as in (3.19) and y α is defined as in (3.22). The use of y α is relevant when α ( 1 2, 1). Furthermore, if {X(t) : t E} is sample path continuous, and (E, d) is a compact metric space, then X is a measurable random function which is finite with probability one and (3.22) holds. Moreover, X sample path continuous with probability one implies it is continuous in distribution as d(s, t) 0. Hence, since F (t, x) is continuous in x the convergence in distribution is uniform, which implies (3.20). Thus when (3.21) also holds we have τ α,l ( ) = τ α,r ( ) τ α ( ) C(E) for those α (0, 1). Similar conclusions

10 10 James Kuelbs and Joel Zinn also hold if E is a compact interval of R and {X(t) : t E} is continuous in distribution on E with cadlag sample paths. Corollary Let (E, d) be a compact metric space with D(E) = C(E), the space of real-valued continuous functions on (E, d), and assume for some α (0, 1 2 ], τ α,l, τ 1 α,r, are functions in C(E). Then, for the given α, (3.14) and (3.15) imply (3.16), and (3.17) implies both (3.16) and (3.18) hold with D(E) = C(E). Corollary Let E be a compact interval of the real line, and assume D(E) is the space of real-valued cadlag functions on E. If for some α (0, 1 2 ], τ α,l, τ 1 α,r, are cadlag functions on E, then for the given α (3.14) and (3.15) imply (3.16), and (3.17) implies both (3.16) and (3.18) hold with D(E) the cadlag functions on E. In Corollaries 3.14 and 3.15 we are assuming (3.14) and (3.15), or (3.17), and that the quantile functions τ α,l and τ 1 α,r are in D(E). Hence, to complete their proofs it suffices to show the empirical quantiles τ n α,l and τ n 1 α,r are also in the corresponding D(E). This follows since an argument from Lemma 3 in [KZ14] implies the left empirical quantile functions τ n α,l inherit the continuity or cadlag nature of the sample paths assumed for the process X. To obtain the same conclusion for the right empirical quantiles τ n 1 α,r, Lemma 3.1 combined with Lemma 3 in [KZ14] suffices Quantile Process Consistency Application of the results in [KZ13b] and [KZ14] to obtain (3.14) and (3.15), or (3.17), involve CLTs for α-quantile processes which hold uniformly in (t, α) E I, where I is a closed subinterval of (0, 1). Hence, of necessity, even when E is a single point, this requires the densities f(t, x) for X(t) to be strictly positive and continuous on J t := {x : 0 < F (t, x) < 1}. In the proofs of these CLTs we assumed J t := R, and for more general E our proofs also required that the densities {f(t, ) : t E} satisfying the uniform equicontinuity condition lim sup sup δ 0 t E u v δ f(t, u) f(t, v) = 0, (3.23) and for every closed interval I in (0, 1) there is an θ(i) > 0 satisfying inf f(t, x) c I,θ(I) > 0, (3.24) t E,α I, x τ α(t) θ(i) where τ α (t), α (0, 1), t E, is the unique α-quantile for the distribution F (t, ) when its density f(t, ) is strictly positive. Although these CLTs hold for a broad collection of stochastic processes, we were motivated to consider consistency results in hope of weakening these assumptions in that setting. This turns out to be the case as the condition (3.23) and that J t := R, t E, are no longer required for our consistency results. Furthermore, a local form of (3.24) holding for a fixed I rather than all I suffices (see (3.27). Of course, if I is a single α (0, 1) then (3.27)

11 Quantile and Depth Regions 11 already appeared in (3.21) of Lemma 3.12 to verify the continuity of the function τ α ( ) on E. Hence, the global strict positivity and continuity of the densities can be considerably weakened for consistency of α-empirical quantiles, making such results of independent interest. Here they provide sufficient conditions for (3.14) and (3.15), which combine to show that the measurable cover functions (of the Hausdorff distances) in (3.16) converge in probability to zero under these weakened conditions. For ω Ω, t E, and x R we denote the empirical distribution functions by F n (t, x) := 1 n I(X j (t, ω) x) = 1 n I Xj C n n t,x, n 1, C t,x C, (3.25) j=1 where C = {C t,x : t E, x R}, C t,x = {z D(E) : z(t) x}, and, as usual, we ignore writing that F n depends on ω Ω. Then, we have Theorem Let F (t, x) = P (X(t) x) have the density f(t, x) for t E, x R. In addition, assume P (lim sup n sup t E,x R j=1 F n (t, x) F (t, x) > 0) = 0, (3.26) and for I a closed interval of (0, 1) there exists θ(i) > 0 such that inf f(t, x) c I,θ(I) > 0. (3.27) t E,α I, x τ α,l (t) θ(i) Then, for all α I we have τ α,l ( ) = τ α,r ( ) τ α ( ), and there is a set Ω 0 such that P (Ω 0 ) = 1, and on Ω 0 lim sup n α I,t E τ n α,l(t) τ α (t) = 0. (3.28) Moreover, if D(E) is a linear subspace of l (E), is measurable on (D(E), D E ), and τ n α,l ( ), τ α( ) D(E), then τ n α,l( ) τ α ( ) (3.29) is measurable, and converges to zero with P probability one. Remark The condition (3.27) is used in two ways in the proof of Theorem The first is to show for all α I we have τ α,l ( ) = τ α,r ( ) τ α ( ), and the second is to verify (5.17) and (5.18). To prove the analogue of (3.28) and (3.29) for the processes τ n 1 α,r( ) τ 1 α ( ), n 1, (3.30) we define for ω Ω, t E, and x R the distribution functions H n (t, x) := 1 n I( X j (t, ω) x) and H(t, x) := P ( X(t) x), (3.31) n j=1 and, as usual, we ignore writing that H n depends on ω Ω. Then, we have

12 12 James Kuelbs and Joel Zinn Corollary Assume P (lim sup n sup t E,x R and for α (0, 1) there exists θ(α) > 0 such that H n (t, x) H(t, x) > 0) = 0, (3.32) inf f(t, x) c θ(α) > 0. (3.33) t E, x τ 1 α,r(t) θ(α) Then, τ 1 α,l ( ) = τ 1 α,r ( ) τ 1 α ( ), and there is a set Ω 0 such that P (Ω 0 ) = 1, and on Ω 0 lim sup τ1 α,r(t) n τ 1 α (t) = 0. (3.34) n t E Moreover, if D(E) is a linear subspace of l (E), is measurable on (D(E), D E ), and τ n 1 α,r( ), τ 1 α ( ) D(E), then τ n 1 α,r( ) τ 1 α ( ) (3.35) is measurable, and converges to zero with P probability one. Remark In order that Theorem 3.16 and Corollary 3.18 apply to obtain conclusions from Theorem 3.9, the nontrivial assumptions (3.26) and (3.32) must be verified. However, the assumptions that the sup-norm is measurable on (D(E), D E ), and the relevant quantile and empirical quantile functions are in a suitable D(E), often follow more directly. For example, Lemma 3.12 and Remark 3.13 provide some sufficient conditions for this when dealing with the quantile functions, and the comments following Corollaries 3.14 and 3.15 making reference to the argument of Lemma 3 in [KZ14] are useful for the empirical quantile functions. In particular, if {X(t) : t E} is sample path continuous, and (E, d) is a compact metric space, or E is a compact interval of R and {X(t) : t E} is continuous in distribution on E with cadlag sample paths, then the quantile functions and empirical quantile functions of Theorem 3.16 and Corollary 3.18 are in D(E), and the measurability of the sup-norm holds on D(E), provided D(E) is the continuous functions on E (respectively, the cadlag paths on E). 4. An Example, and How to Avoid Zero Half Space Depth As mentioned in Remark 3.8, Proposition 3.7 holds quite generally, but there are many examples where the sets in (3.11) have probability zero for all α > 0 with respect to the probability µ = L(X) on (D(E), D E ). Hence, we start with an example of this type. The example also satisfies the assumptions of Theorem 3.9, so even though the sets in (3.11) are small, the convergence results in Theorem 3.9 still hold. Following this we provide a smoothing result for functional data given by a stochastic process. This appears in Proposition 4.2, which shows the sets in (3.11) are much larger for smooth versions of the data, where before smoothing they possibly had positive half space depth with probability zero. The smoothing used in Proposition 4.2 is not a

13 Quantile and Depth Regions 13 smoothing of the sample paths of the functional data, but of each one dimensional distribution, and in the sense made explicit in Remark 4.3, the change in the data can be made arbitrarily small. Furthermore, the half-space depth and quantile regions of the resulting smoothed process can be shown to have positive probability in many situations. Following the statement of Proposition 4.2 we provide an example for which these regions can be explicitly obtained with fairly brief arguments. Let {Y (t) : t [0, 1]} be a centered sample continuous Brownian motion with variance parameter one and Y (0) = 0 with probability one. Set E = [0, 1] and D(E) = C[0, 1], the Banach space of continuous functions on [0, 1] in the sup-norm. We then have D E is the Borel subsets of C[0, 1], which we denote by B C[0,1]. Then, for E = [0, 1] and µ = L(Y ) on (C[0, 1], B C[0,1] ) µ(h C[0, 1] : D(h, µ) > 0) = 0, (4.1) where D(h, µ) is given by (3.5) with ν = µ and h D(E) = C[0, 1]. To verify (4.1) we first observe that D(h, µ) z(t) inf min{µ(z C[0, 1] : h(t) ), µ(z C[0, 1] : z(t) h(t) )} 0<t 1 t t t t (4.2) = inf min{1 Φ(h(t) ), Φ( h(t) )}, 0<t 1 t t where Φ is the standard normal distribution function. Hence, for all h C[0, 1] such that D(h, µ) lim inf min{1 Φ( h(t) ), Φ( h(t) )} = 0 (4.3) t 0 t t lim sup t 0 h(t) h(t) = or lim inf =. (4.4) t t 0 t Using the law of the iterated logarithm for Brownian motion at zero, we have that both terms in (4.4) hold for a set of functions h C[0, 1] with µ-probability one. Thus, (4.1) follows, and the sets in (3.11) are of µ-measure zero. Of course, these sets are clearly non-empty since they contain the zero function. Moreover, for standard Brownian motion on [0, 1] and α (0, 1), the left and right α-quantiles for each time t [0, 1] are equal, and τ α (t) = tτ α (1) = tφ 1 (α), and hence τ α (1) < 0 for 0 < α < 1 2, τ 1 (1) = 0, and τ α(1) > 0 for < α < 1. Therefore, for Brownian motion and α (0, 1 2 ] we have M α = {h C[0, 1] : tτ α (1) h(t) tτ 1 α (1)}, (4.5)

14 14 James Kuelbs and Joel Zinn and for α = 1 2 it consists only of the zero function. Since Proposition 3.7 holds, we also have for α (0, 1 2 ] that {h C[0, 1] : D(h, µ) α} = {h C[0, 1] : tτ α (1) h(t) tτ 1 α (1)}, (4.6) and by what we have shown above the sets in (4.5) and (4.6) have µ measure zero. Of course, one can also see this directly from the law of the iterated logarithm which implies both terms in (4.4) hold with µ measure one, but we thought it useful to proceed as we did because of Remark 4.1 below. In addition, applying Theorems 2 and 3 of [KZ14], we have assumptions (3.14), (3.15), and (3.17) holding for each α (0, 1). In particular, using Lemma 3 and Theorem 3 of [KZ14] one can avoid using measurable cover functions for these processes since we are assuming the input data has sample continuous or cadlag paths, and hence the conclusions of Theorem 3.9 in (3.16) and (3.18) hold, with limit sets as in (3.3) and (3.6), again without the use of measurable cover functions. Remark 4.1. Similar ideas imply (4.1) for sample continuous fractional Brownian motions which start at zero with probability one when t = 0 since the LIL at zero for these processes implies (4.4) with t replaced by t ρ, where ρ is the scaling parameter of the fractional Brownian motion. Also, if {Y (t) : t [0, 1]} is a symmetric stable process with stationary independent increments, sample paths in the space of cadlag functions on [0, 1], and Y (0) = 0 with probability one, then the same result holds by applying Theorem 5-iii, page 222, in [Ber96] provided the process is not identically zero. Now we turn to a proposition which indicates how we can smooth a broad collection of stochastic processes in order that the resulting smoothed process has strictly positive depth with probability one, and is always close to original process in a sense to be specified in Remark 4.3 below. The smoothing in Proposition 4.2 is also useful when applying the functional data results in [KKZ13], [KZ13b], and [KZ14], as many of these results depend on regularity assumptions for the distribution of X(t). Hence, by choosing the density f Z ( ) to be suitably smooth, we can be certain the necessary regularity conditions hold. For example, it is easy to see that the processes mentioned in Remark 4.1 can be smoothed using this result. Proposition 4.2. Let D(E) be a subset of l (E) which contains the constant functions and is closed under addition, assume Y := {Y (t) : t E} is a stochastic process with sample paths in D(E) and Z is a real-valued random variable independent of Y defined on the probability space (Ω, F, P ), and let X := {X(t) : t E}, where X(t) = Y (t) + Z for t E. Furthermore, assume Z has probability density f Z ( ) that is positive a.s. on R with respect to Lebesgue measure, the family of random variables {Y (t) : t E} is bounded in probability, and for h D(E) and µ = L(X) on (D(E), D E ), the depth

15 Quantile and Depth Regions 15 D(h, µ) is defined as in (3.5) with ν = µ. Then, for all h D(E) the E-half space depth is such that D(h, µ) > 0. (4.7) Remark 4.3. Since E( X Y p ) = E( Z p ) can be made small by choosing the density of Z to decrease rapidly at infinity and with probability near one in a small neighborhood of zero, the X process is close to the Y process in the L p -norm of their sup-norm difference. Therefore, at least in this sense the smoothing does not modify the data a great deal in order that the depth be positive with probability one. In addition, if {Y (t) : t [0, 1]} is standard Brownian motion and X(t) = Y (t) + Z, t [0, 1], where Z is a mean zero Gaussian random variable independent of {Y (t) : t [0, 1]} with variance σ 2 > 0, then we are able to see explicit changes in the quantile and depth regions given in (4.5) and (4.6) for {Y (t) : t [0, 1]} and those for the smoothed process {X(t) : t [0, 1]} obtained below. That is, since P (X(t) y) = Φ(y/ t + σ 2 ) for all y R, t [0, 1], where Φ is the distribution function of a mean zero, variance one Gaussian random variable, we have τ α (t) = t + σ 2 Φ 1 (α) for all α (0, 1) and t [0, 1]. Thus the α-quantile and depth regions for {X(t) : t [0, 1]} and α (0, 1 2 ] are given by M α,x = t [0,1] {z C[0, 1] : t + σ 2 Φ 1 (α) z(t) t + σ 2 Φ 1 (1 α)}. Since σ 2 > 0 and M α,x has non-empty interior with respect to the sup-norm topology on C[0, 1] for all α (0, 1 2 ), then the Cameron-Martin formula easily implies these regions have positive probability with respect to ν, the probability law of {X(t) : t [0, 1]} on C[0, 1]. Of course, since M α = {0} for α = 1 2 then ν(m 1 2,X ) = 0 as one should expect. Moreover, since (3.10) implies M α,x = N α,x, we also have ν(n α,x ) > 0 for all α (0, 1 2 ) and ν(n 1 ) = Proofs of the Results Proof of Proposition 3.7. If h M α, then τ α,l (t) h(t) τ 1 α,r (t) for all t E. Now h(t) τ α,l (t) implies µ(z : z(t) h(t)) α, and similarly h(t) τ 1 α,r (t) implies µ(z : z(t) h(t)) µ(z : z(t) τ 1 α,r (t)) α. Therefore, D(h, µ) min{α, α} = α. Conversely, assume D(h, µ) α and h D(E). Then, (3.5) with ν = µ implies for every t E that and µ(z D(E) : z(t) h(t)) α (5.1) µ(z D(E) : z(t) h(t)) α. (5.2)

16 16 James Kuelbs and Joel Zinn Thus, for every t E, we have from (5.1) that h(t) τ 1 α,r (t), and from (5.2) that h(t) τ α,l (t). Therefore, for every t E we have τ α,l (t) h(t) τ 1 α,r (t), which implies h M α, and the proposition is proved. Proof of Theorem 3.9. For α (0, 1 2 ], the set M α (respectively M α,n ) is nonempty since (3.13) implies τ α,l ( ) and τ 1 α,r ( ) are in M α (respectively τ n α,l ( ) and τ n 1 α,r( ) are in M α,n ), and Proposition 3.7 implies N α and N α,n are non-empty since M α = N α and M α,n = N α,n. Hence, we observe that and M ɛ α = {z D(E) : t E, τ α,l (t) ɛ < z(t) < τ 1 α,r (t) + ɛ} (5.3) M ɛ α,n = {z D(E) : t E, τ n α,l(t) ɛ < z(t) < τ n 1 α,r(t) + ɛ}. (5.4) Now, we ll use (3.13) to show that and {ω :M α,n (ω) M ɛ α} (5.5) = {ω : t E, τ n α,l(t, ω) > τ α,l (t) ɛ, τ n 1 α,r(t, ω) < τ 1 α,r (t) + ɛ} {ω :M α M α,n (ω) ɛ } (5.6) = {ω : t E, τ α,l (t) > τ n α,l(t, ω) ɛ, τ 1 α,r (t) < τ n 1 α,r(t, ω) + ɛ}. The proofs of (5.5) and (5.6) are similar, so we ll just check (5.5). Also, it is trivial that the right-hand of (5.5) is contained in the left-hand side. To prove the other inclusion fix ω such that M α,n (ω) M ɛ α. By (3.13) we have τ n α,l(, ω), τ n 1 α,r(, ω) M α,n (ω), and since we are assuming M α,n (ω) M ɛ α, for all t E τ α,l (t) ɛ < τ n α,l(t, ω) τ n 1 α,r(t, ω) < τ 1 α,r (t) + ɛ, which implies ω is in the right-hand side of (5.5). Therefore, both {ω : M α,n (ω) M ɛ α} and {ω : M α M α,n (ω) ɛ } contain the set A n, where A n = {ω : sup t E Now τα,l(t, n ω) τ α,l (t) < ɛ} {sup τ1 α,r(t, n ω) τ 1 α,r (t) < ɛ}. t E (5.7) {ω : d H (M α,n (ω), M α ) < ɛ} = {ω : M α,n (ω) M ɛ α} {ω : M α M α,n (ω) ɛ }, (5.8) and hence Therefore, {ω : d H (M α,n (ω), M α ) ɛ} (5.9) = {ω : M α,n (ω) M ɛ α} c {ω : M α M α,n (ω) ɛ } c A c n. P (d H (M α,n, M α ) ɛ) P (sup τα,l(t) n τ α,l (t) ɛ) (5.10) t E

17 Quantile and Depth Regions 17 +P (sup τ1 α,r(t) n τ 1 α,r (t) ɛ) t E where P denotes the outer probability given by P. Since P (Λ > c) = P (Λ > c) for all constants c, we thus have from (3.14) and (3.15) that the measurable cover function of the left term in (3.16) converges in probability to zero. Furthermore, applying (3.11) and (3.12) of Proposition 3.7, we also have the equality in (3.16), and hence the measurable cover function of the right term in (3.16) also converges in probability to zero. Therefore, the first part of the theorem holds. Since (5.10) holds for every ɛ > 0 we have with ɛ = L/a n > 0 that P (a n d H (M α,n, M α ) L) P (a n sup τα,l(t) n τ α,l (t) L) (5.11) t E +P (a n sup τ1 α,r(t) n τ 1 α,r (t) L), t E and hence (5.10) and the relationship between P -outer probability and measurable cover functions mentioned above implies a n d H (M α,n, M α ) is bounded in probability. Again, by applying (3.11) and (3.12) of Proposition 3.7, we also have a n d H (N α,n, N α ) is bounded in probability. Thus (3.18) holds, and the theorem follows. Proof of Corollary Since we are assuming (3.14),(3.15), and (3.17) to hold, it suffices to show the stochastic boundedness assumption in (3.19) implies (3.13) with D(E) = l (E). Since the sample paths of X are assumed to be in D(E) = l (E), we have for α (0, 1 2 ], n 1, t E that τα,l(t) n = inf{x : 1 n I(X j (t) x) α} β n and τ n 1 α,r(t) = inf{x : 1 n j=1 n I(X j (t) x) > 1 α} β, j=1 where β := sup 1 j n sup t E X j (t) = sup 1 j n X j <. If α (0, 1 2 ), then for every t E β τ n α,l(t) τ n α,r(t) τ n 1 α,l(t) τ n 1 α,r(t) β, and hence τα,l n ( ), τ 1 α,r( ) n l (E). Furthermore, if α = 1 2, then by deleting the middle two terms in the previous inequality we have τ n 1 2,l( ), τ n 1 2,r( ) l (E). Similarly, if α (0, 1 2 ] and (3.19) holds, then for all t E and since 0 < α α < 1, τ α,l (t) = inf{x : P (X(t) x) α} x α τ 1 α,r (t) = inf{x : P (X(t) x) > 1 α} x α. Since t E is arbitrary, arguing as above we also have for the given α (0, 1 2 ] that τ α,l( ), τ 1 α,r ( ) l (E). Therefore, (3.13) holds when D(E) = l (E).

18 18 James Kuelbs and Joel Zinn Proof of Lemma Since we are assuming (3.21), for each t E the distribution function F (t, x) is strictly increasing and continuous in x on the open interval (τ α,l (t) θ(α), τ α,l (t) + θ(α)) and hence for the given α we have τ α,l ( ) = τ α,r ( ) τ α ( ). Moreover, if α (0, 1 2 ]) and (3.22) holds, then as in the proof of Corollary 3.11 for the given α (0, 1 2 ]) we have sup t E τ α (t) y α <, and τ α ( ) l (E). Furthermore, if α ( 1 2, 1), then 1 α (0, 1 2 ) and arguing as before we have for t E that y α τ 1 α,l (t) τ 1 α,r (t) τ α,l (t) τ α,r (t) y α. Therefore, we again have sup t E τ α (t) y α <, and τ α ( ) l (E). Hence it suffices to show τ α ( ) C(E), so fix t E, α (0, 1), ɛ (0, θ(α)), and take δ > 0 such that d(s, t) δ and (3.20) implies sup F (s, x) F (t, x), (5.12) (s,x):d(s,t) δ,x R where < c α ɛ and c α is as in (3.21). Then, d(s, t) δ and (5.12) implies and F (t, τ α (s)) F (t, τ α (t)) + = α + < F (t, τ α (t) + ɛ) (5.13) F (t, τ α (s)) F (s, τ α (s)) = α > F (t, τ α (t) ɛ) (5.14) since (3.21) implies F (t, τ α (t) + ɛ) = α + τα(t)+ɛ τ α(t) f(t, x)dx α + c α ɛ > α + and τα(t) F (t, τ α (t) ɛ) = α f(t, x)dx α c α ɛ < α. τ α(t) ɛ Now (5.13) and (5.14) combine to imply τ α (t) ɛ τ α (s) τ α (t) + ɛ for d(s, t) δ. Since ɛ > 0 can be taken arbitrarily small, τ α ( ) C(E) and the lemma is verified. Proof of Theorem Since we are assuming (3.27), for each t E the distribution function F (t, x) is strictly increasing and continuous in x on the open interval (τ α,l (t) θ(i), τ α,l (t) + θ(i)), and hence for α I we have τ α,l ( ) = τ α,r ( ) τ α ( ). To establish (3.28) we use (3.26) to take Ω 0 such that P (Ω 0 ) = 1, and on Ω 0 lim sup n sup C t,x C F n (t, x) F (t, x) = 0. (5.15) Since we are also assuming (3.27), fix I (0, 1), 0 < ɛ θ(i), and ω Ω 0. Then, by (5.15) and our definition of Ω 0 there exists δ δ(ω, ɛ) > 0 such that for n n(δ(ω, ɛ)) we have sup F n (t, x) F (t, x) < δ, (5.16) C t,x C

19 Quantile and Depth Regions 19 Furthermore, if δ = δ(ω, ɛ) < ɛc I,θ(I), our choice of ɛ > 0 and (3.27) implies and That is, (5.17) holds since sup α I,t E sup F (t, τ α (t) ɛ) < α δ, (5.17) α I,t E inf F (t, τ α(t) + ɛ) > α + δ. (5.18) α I,t E F (t, τ α (t) ɛ) α inf α I,t E τα(t) τ α(t) ɛ f(t, x)dx, and (3.27) and our choice of δ = δ(ω, ɛ) < ɛc I,θ(I) then implies inf α I,t E Similarly, (5.18) holds since τα(t) τ α(t) ɛ inf F (t, τ α(t) + ɛ) α + α I,t E and (3.27) and our choice of δ implies inf α I,t E τα(t)+ɛ τ α(t) f(t, x)dx ɛc I,θ(I) > δ. inf α I,t E τα(t)+ɛ τ α(t) f(t, x)dx ɛc I,θ(I) > δ. f(t, x)dx, Thus for n n(δ(ω, ɛ)), (5.16), and by definition of τα,l n (t) that we have on Ω 0 that inf F n(t, τα,l(t)) n α, α I,t E inf F (t, τ α,l(t)) n inf F n(t, τα,l(t)) n δ α δ. α I,t E α I,t E Therefore, by (5.17) for all t E, all α I, and ω Ω 0 τ n α,l(t) τ α (t) ɛ, (5.19) provided n n(δ(ω, ɛ)). Similarly, for all x < τ n α,l (t) we have for all t E, all α I, and ω Ω 0 F (t, x) < F n (t, x) + δ(ω, ɛ) α + δ < F (t, τ α (t) + ɛ), provided n n(δ(ω, ɛ)), where the first inequality follows from (5.16), the second by definition of τ n α (t) and that x < τ n α (t), and the third by (5.18). Thus τ α (t) + ɛ > x for all x < τ n α (t), and combining this with (5.19) we have for all t E, all α I, and ω Ω 0 that τ α (t) ɛ τ n α (t) τ α (t) + ɛ, (5.20) provided n n(δ(ω, ɛ)). Since ɛ > 0 was arbitrary, letting n implies (3.28).

20 20 James Kuelbs and Joel Zinn Proof of Corollary Let ν = L( X), ν n = 1 n n j=1 δ X j on (D(E), D E ), and for α (0, 1) define the left α-quantiles τ α,l (t, ν) and τ α,l (t, ν n ) as in Definition 2.2. Then, since (3.32) holds, Theorem 3.16 implies lim sup τ α,l (t, ν n ) τ α,l (t, ν) = 0 (5.21) n t E on a subset E 0 of Ω such that P (E 0 ) = 1 provided for the given α there exists θ(α) > 0 such that inf h(t, x) c θ(α) > 0 (5.22) t E, x τ α,l (t,ν) θ(α) where h(t, x) is the density of X(t). Now for each t E we can take h(t, x) = f(t, x) for all x R, and hence Lemma 3.1 implies τ α,l (t, ν) = τ 1 α,r (t), t E, where as usual τ 1 α,r (t) is defined using L(X). Thus, (5.22) is equivalent to inf f(t, x) c θ(α) > 0, (5.23) t E, x+τ 1 α,r(t) θ(α) and setting u = x we have (5.23) equivalent to inf f(t, u) c θ(α) > 0, (5.24) t E, u+τ 1 α,r(t) θ(α) which follows from (3.33). Therefore, (5.21), holds, and since Lemma 3.1 also implies τ α,l (t, ν n ) = τ1 α,r(t), n t E, we thus have (3.34). Of course, (3.35) then follows immediately, and Corollary 3.18 is proved. Proof of Proposition 4.2. Since {Y (t) : t E} is bounded in probability there exists c (0, ) such that sup t E P ( Y (t) > c) < 1 2. Then, for t E P (X(t) h(t)) = P (Y (t) h(t) u)f Z (u)du, (5.25) and for u c + h we have R P (Y (t) h(t) u) P (Y (t) h (c + h )) P (Y (t) c). (5.26) Since our choice of c implies P (Y (t) c) P ( Y (t) c) > 1 2, (5.26) implies and combining (5.25) and (5.27) P (Y (t) h(t) u) > 1 2, (5.27) inf P (X(t) h(t)) 1 t E 2 P (Z c + h ) δ 1 (c, h ) > 0. (5.28) We also have for t E that P (X(t) h(t)) = P (Y (t) h(t) u)f Z (u)du, (5.29) and for t E and u (c + h ) that R P (Y (t) h(t) u) P (Y (t) h + (c + h )) P (Y (t) c). (5.30)

21 Quantile and Depth Regions 21 Our choice of c also implies P (Y (t) c) P ( Y (t) c) > 1 2, and hence (5.30) implies Combining (5.29) and (5.31) we thus have P (Y (t) h(t) u) > 1 2. (5.31) inf P (X(t) h(t)) 1 t E 2 P (Z (c + h )) δ 2 (c, h ) > 0. (5.32) Hence, (5.28) and (5.32) imply (4.7), and the proposition is proved. References [AGOZ88] Niels T. Andersen, Evarist Giné, Mina Ossiander, and Joel Zinn, The central limit theorem and the law of iterated logarithm for empirical processes under local conditions, Probab. Theory Related Fields 77 (1988), no. 2, [Ber96] Jean Bertoin, Lévy processes, Cambridge Tracts in Mathematics, vol. 121, Cambridge University Press, Cambridge, [CC14] Anirvan Chakraborty, and Probal Chaudhuri, On data depth in infinite dimensional spaces, Ann. Inst. Statist. Math. 66 (2014), no. 2, [Dud99] R. M. Dudley, Uniform central limit theorems, Cambridge Studies in Advanced Mathematics, vol. 63, Cambridge University Press, Cambridge, [KKZ13] James Kuelbs, Thomas Kurtz, and Joel Zinn, A CLT for empirical processes involving time-dependent data, Ann. Probab. 41 (2013), no. 2, [KM12] Linglong Kong and Ivan Mizera, Quantile tomography: using quantiles with multivariate data, Statist. Sinica 22 (2012), no. 4, [KZ13a] James Kuelbs and Joel Zinn, Concerns with functional depth, ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013), no. 2, [KZ13b], Empirical quantile clts for time-dependent data, High Dimensional Probabiity VI, The Banff Volume (Christian Houdre; David Mason; Jan Rosinski; Jon Wellner, ed.), Progress in Probability, Birkhauser, vol. 66, Springer Basel, 2013, pp (English). [KZ14], Empirical quantile central limit theorems for some self-similar processes, Journ. Theoret. Probab. 28 (2015), no. 1, [KZ15] James Kuelbs and Joel Zinn, Half-region depth for stochastic processes, Journ. of Multivariate Analysis 142 (2015), [KZ15a] James Kuelbs and Joel Zinn, Convergence of quantile and depth regions, (2015), to appear. [Mas02] J.C. Massé, Asymptotics for the Tukey median, J. Multivariate Anal. 81 (2002), [MT94] J.-C. Massé and R. Theodorescu, Halfplane trimming for bivariate distributions, J. Multivariate Anal. 48 (1994), no. 2, [Nol99] D. Nolan, On min-max majority and deepest points, Statist. Probab. Let. 43 (1999), [Swa07] Jason Swanson, Weak convergence of the scaled median of independent Brownian motions, Probab. Theory Related Fields 138 (2007), no. 1-2,

22 22 James Kuelbs and Joel Zinn [Swa11], Fluctuations of the empirical quantiles of independent Brownian motions, Stochastic Process. Appl. 121 (2011), no. 3, [Tal05] Michel Talagrand, The generic chaining, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005, Upper and lower bounds of stochastic processes. [Tuk75] John W. Tukey, Mathematics and the picturing of data, Proceedings of the International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 2, Canad. Math. Congress, Montreal, Que., 1975, pp [Ver72] Wim Vervaat, Functional central limit theorems for processes with positive drift and their inverses, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 23 (1972), [ZS00a] Yijun Zuo and Robert Serfling, General notions of statistical depth function, Ann. Statist. 28 (2000), no. 2, [ZS00b], Structural properties and convergence results for contours of sample statistical depth functions, Ann. Statist. 28 (2000), no. 2, James Kuelbs Department of Mathematics University of Wisconsin, Madison, WI Joel Zinn Department of Mathematics Texas A&M University, College Station, TX