Pointwise and uniform asymptotics of the Vervaat error process

Dedicated to the memory of Arthur Hing-Chiu Chan (1946–1999), PhD 1977, Carleton University

Endre Csáki^{1} Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127,

H-1364, Hungary. E-mail: csaki@renyi.hu

Miklós Csörgő^{2} School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, Canada K1S 5B6.

E-mail: mcsorgo@math.carleton.ca

Antónia Földes^{3} City University of New York, 2800 Victory BLVD., Staten Island, New York 10314, U.S.A.

E-mail: afoldes@gc.cuny.edu

Zhan Shi Laboratoire de Probabilités UMR 7599, Université Paris VI, 4 Place Jussieu, F-75252 Paris Cedex

05, France. E-mail: zhan@proba.jussieu.fr

Ričardas Zitikis^{4} Department of Statistics, University of Manitoba, 327 Machray Hall, Winnipeg, Manitoba, Canada

R3T 2N2. E-mail: Ricardas Zitikis@umanitoba.ca

Abstract. It is well known that, asymptotically, the appropriately normalized uniform Vervaat process, i.e., the integrated uniform Bahadur–Kiefer process properly normalized, behaves like the square of the uniform empirical process. We give a complete description of the strong and weak asymptotic behaviour in sup-norm of this representation of the Vervaat process and, likewise, we study also its pointwise asymptotic behaviour.

Keywords. Empirical process, quantile process, Bahadur–Kiefer process, Vervaat process, Vervaat error process, Kiefer process, Brownian bridge, Wiener process, strong approximation, law of the iterated logarithm, convergence in distribution.

2000 Mathematics Subject Classification. Primary 60F15; Secondary 60F05, 60F17.

1Research supported by the Hungarian National Foundation for Scientific Research, Grant No. T 029621.

2Research supported by an NSERC Canada Grant at Carleton University, Ottawa.

3Research supported by a PSC CUNY grant, No. 6-69413.

4Research supported by a grant of the University of Manitoba, Winnipeg, and by an NSERC Canada Grant at the Laboratory for Research in Statistics and Probability, Carleton University and University of Ottawa, Ottawa.

1 Introduction and preliminary results

Let U_{1}, U_{2}, … be independent copies of a random variable U uniformly distributed over the interval [0, 1]. Let

F_{n}(t) := \frac{1}{n} ∑_{k=1}^{n} 1\{U_{k} ≤ t\}, 0 ≤ t ≤ 1,

denote the empirical distribution function based on U_{1}, U_{2}, …, U_{n}, where 1 is the indicator function. Let F_{n}^{−1} be the left-continuous inverse of F_{n}. We denote the empirical and quantile processes over the interval [0, 1] by

α_{n}(t) := n^{1/2}(F_{n}(t) − t), 0 ≤ t ≤ 1,
β_{n}(t) := n^{1/2}(F_{n}^{−1}(t) − t), 0 ≤ t ≤ 1,

respectively. The sum

R_{n}(t) := α_{n}(t) + β_{n}(t), 0 ≤ t ≤ 1,

of the empirical and quantile processes is known in the literature as the Bahadur–Kiefer process (cf. Bahadur, 1966, Kiefer, 1967, 1970).

The Bahadur–Kiefer process enjoys some remarkable asymptotic properties, which are of interest in statistical quantile data analysis (cf., e.g., Csörgő, 1983, Shorack and Wellner, 1986). We summarize the most relevant results of Kiefer (1967, 1970) in this regard in the following theorem. Throughout the paper, we use the notation log x := log max(x, e) and log_{2} x := log log x.

Theorem A For every fixed t ∈ (0, 1), we have

n^{1/4}R_{n}(t) →_{d} (t(1 − t))^{1/4}N (|Ñ|)^{1/2}, n → ∞,  (1.1)
lim sup_{n→∞} \frac{n^{1/4}|R_{n}(t)|}{(log_{2} n)^{3/4}} = (t(1 − t))^{1/4} \frac{2^{5/4}}{3^{3/4}}  a.s.,  (1.2)

where N and Ñ are independent standard normal variables and →_{d} denotes convergence in distribution. Also,

lim_{n→∞} n^{1/4}(log n)^{−1/2} \frac{∥R_{n}∥}{(∥α_{n}∥)^{1/2}} = 1  a.s.,  (1.3)

where ∥f∥ := sup_{0≤t≤1} |f(t)| denotes the uniform sup-norm of f.

Theorem A is due to Kiefer (1967, 1970), except for (1.3), which he announced but proved only convergence in probability (cf. Theorem 1A, and the two sentences right after, in Kiefer, 1970). The upper bound for the almost sure convergence in (1.3) was proved by Shorack (1982), and the lower bound by Deheuvels and Mason (1990). For a review of these results and for further developments along these lines we refer to Deheuvels and Mason (1992), Einmahl (1996), Csörgő and Szyszkowicz (1998).

Via using the usual and the other laws of the iterated logarithm for α_{n}, (1.3) immediately implies

lim sup_{n→∞} n^{1/4}(log n)^{−1/2}(log_{2} n)^{−1/4}∥R_{n}∥ = 2^{−1/4}  a.s.,  (1.4)
lim inf_{n→∞} n^{1/4}(log n)^{−1/2}(log_{2} n)^{1/4}∥R_{n}∥ = \frac{π^{1/2}}{8^{1/4}}  a.s.,  (1.5)

while a direct application of (1.3) together with the weak convergence of α_{n} to a Brownian bridge B gives

n^{1/4}(log n)^{−1/2}∥R_{n}∥ →_{d} (∥B∥)^{1/2}, n → ∞.  (1.6)

Nevertheless, the following result, which one can immediately conclude also by combining (1.1) with (1.6), is true, and it was first formulated and proved directly by Vervaat (1972b).

Theorem B (Vervaat, 1972b) The statement

a_{n}R_{n} →_{d} Y, n → ∞,

cannot hold true in the space D[0, 1] (endowed with the Skorohod topology) for any sequence {a_{n}} of positive real numbers and any non-degenerate random element Y of D[0, 1].

In view of Theorems A and B now, it is of interest to see the asymptotic behaviour of the Bahadur–Kiefer process possibly in other norms as well. In this regard we quote

Theorem C (Csörgő and Shi, 1998) For any p ∈ [2, ∞), we have

lim_{n→∞} n^{1/4} \frac{∥R_{n}∥_{p}}{(∥α_{n}∥_{p/2})^{1/2}} = c_{0}(p)  a.s.,  (1.7)

where

c_{0}(p) := (E|N |^{p})^{1/p} = \sqrt{2} (\frac{Γ((p + 1)/2)}{\sqrt{π}})^{1/p} ,  (1.8)

N stands, as before, for a standard normal variable, and ∥f∥_{p} := (∫_{0}^{1} |f(t)|^{p} dt)^{1/p}, the L_{p} norm of f.

In particular, (1.7) yields L_{p} versions of the laws of the iterated logarithm (LIL’s) for the Bahadur–Kiefer process R_{n}. We also note that (1.7) combined with the weak convergence of α_{n} to a Brownian bridge B implies that, for p ∈ [2, ∞),

n^{1/4}∥R_{n}∥_{p} →_{d} c_{0}(p)(∥B∥_{q})^{1/2}, n → ∞.

This, in combination with (1.6), yields Theorem B again.

Vervaat’s (1972b) proof of Theorem B was based, in a most crucial and elegant way, on the following integrated Bahadur–Kiefer process

I_{n}(t) := ∫_{0}^{t} R_{n}(s)ds, 0 ≤ t ≤ 1.

Concerning the latter process, he established the weak convergence of

V_{n}(t) := 2n^{1/2}I_{n}(t)  (1.9)

to B^{2}, the square of a Brownian bridge, as well as a functional LIL for V_{n}, via proving the following theorem (for a discussion of related details we refer to Csörgő and Zitikis, 1999a, b).

Theorem D (Vervaat, 1972a, b) We have

lim_{n→∞} (log_{2} n)^{−1}∥V_{n} − α_{n}^{2}∥ = 0  a.s.  (1.10)
lim_{n→∞} ∥V_{n} − α_{n}^{2}∥ = 0  in probability.  (1.11)

In particular, in the space C[0, 1],

V_{n} →_{d} B^{2}, n → ∞.  (1.12)

For use of terminology, we call the process V_{n} of (1.9) the uniform Vervaat process. For further references and elaboration on this terminology we refer to Section 1 of Zitikis (1998).

As a consequence of (1.12), Vervaat (1972b) concluded Theorem B. We also note that (1.10) yields the LIL for V_{n} in sup-norm (cf. Corollary 1.1 (Vervaat 1972a, b) in Csörgő and Zitikis, 1999a, b).

2 The Vervaat error process, main results

Bahadur (1966) introduced R_{n} as the remainder term in the representation

β_{n} = −α_{n} + R_{n}

of the quantile process β_{n} in terms of the empirical process α_{n}. As we have seen in Theorems A and C, the remainder term R_{n}, i.e., the Bahadur–Kiefer process, is asymptotically smaller than the main term α_{n}, i.e., the empirical process, in both the L_{p} and sup-norm topologies.

In a similar vein, we can think of the process

Q_{n}(t) := V_{n}(t) − α_{n}^{2}(t), 0 ≤ t ≤ 1,  (2.1)

that appears in both statements (1.10) and (1.11) of Theorem D as the remainder term Q_{n} in the following representation

V_{n} = α_{n}^{2} + Q_{n}  (2.2)

of the uniform Vervaat process V_{n} in terms of the square of the empirical process. It is well-known (cf. Zitikis, 1998, for details and references) that the remainder term Q_{n} in (2.2) is asymptotically smaller than the main term α_{n}^{2}. Thus, just like in the case of R_{n}, one may like to know how small the remainder term Q_{n} is.

In view of Theorems A and C, one suspects that there should be substantial differences between the asymptotic pointwise, sup- and L_{p}-norms behaviour of the process Q_{n}. Indeed, Csörgő and Zitikis (1999a) established the following strong convergence result for ∥Q_{n}∥_{p}.

Theorem E (Csörgő and Zitikis, 1999a) For any p ∈ [1, ∞), we have

lim_{n→∞} n^{1/4} \frac{∥Q_{n}∥_{p}}{(∥α_{n}∥_{3p/2})^{3/2}} = \frac{1}{\sqrt{3}}c_{0}(p)  a.s.,  (2.3)

where c_{0}(p) is defined in (1.8).

For a comparison of this result to that of Theorem C, as well as for that of their consequences, we refer to Csörgő and Zitikis (1999a), who have also conjectured that in sup-norm the analogue statement of (2.3) should be of the following form:

lim_{n→∞} b_{n}n^{1/4} \frac{∥Q_{n}∥}{∥α_{n}∥^{3/2}} = c  a.s.,  (2.4)

where b_{n} is a slowly varying function converging to 0 and c is a positive constant.

One of the aims of this exposition is to prove that this conjecture is true with b_{n} = (log n)^{−1/2}. In addition, we also study the pointwise behaviour of the Vervaat error process Q_{n}. We summarize our results in the following theorem, which parallels Theorem A of Kiefer (1967, 1970) concerning the process R_{n}.

Theorem 2.1 For every fixed t ∈ (0, 1), we have

n^{1/4}Q_{n}(t) →_{d} (4/3)^{1/2}(t(1 − t))^{3/4}N (|Ñ|)^{3/2}, n → ∞,  (2.5)
lim sup_{n→∞} \frac{n^{1/4}|Q_{n}(t)|}{(log_{2} n)^{5/4}} = (t(1 − t))^{3/4} \frac{2^{11/4}3^{1/4}}{5^{5/4}}  a.s.,  (2.6)

where N and Ñ are independent standard normal variables. Also,

lim_{n→∞} n^{1/4}(log n)^{−1/2} \frac{∥Q_{n}∥}{(∥α_{n}∥)^{3/2}} = (4/3)^{1/2}  a.s.  (2.7)

As a consequence of this theorem, as well as that of Theorem E combined with (2.7), we have the following corollary, which confirms Conjecture 2.1 of Csörgő and Zitikis (1999a).

Corollary 2.1 The statement

a_{n}Q_{n} →_{d} Y, n → ∞,

cannot hold true in the space D[0, 1] for any sequence {a_{n}} of positive real numbers and for any non-degenerate random element Y of the space D[0, 1].

Another consequence via (2.7) is the following corollary.

Corollary 2.2 We have

lim sup_{n→∞} n^{1/4}(log n)^{−1/2}(log_{2} n)^{−3/4}∥Q_{n}∥ = \frac{2^{1/4}}{3^{1/2}}  a.s.,  (2.8)
lim inf_{n→∞} n^{1/4}(log n)^{−1/2}(log_{2} n)^{3/4}∥Q_{n}∥ = \frac{π^{3/2}}{3^{1/2}2^{5/4}}  a.s.,  (2.9)
n^{1/4}(log n)^{−1/2}∥Q_{n}∥ →_{d} (4/3)^{1/2}∥B∥^{3/2}, n → ∞,  (2.10)

where B is a standard Brownian bridge.

We note that (2.8) and (2.9) follow from (2.7) by means of the usual and the other LIL’s for α_{n}. As to (2.10), it results from a direct application of (2.7) together with the weak convergence of α_{n} to a Brownian bridge B.

Remark 2.1 In the literature we also find general forms of the Vervaat process that are based on random variables X_{1}, X_{2}, … replacing the uniform [0,1] random variables U_{1}, U_{2}, … In particular, such a general Vervaat process first appeared and was put to good use in Csörgő and Zitikis (1996). We refer to Zitikis (1998) for a detailed survey on this subject. For related though rather different limit theorems for the general Vervaat process, we refer to Csörgő and Zitikis (1998). It is obvious that the results of this paper can be generalized in such a way that they would cover general forms of the Vervaat process as well. However, a solution of this problem under reasonably optimal assumptions may constitute a rather challenging mathematical task which is definitely not within the scope of the present paper. For a recent review of Vervaat and Vervaat error processes we refer to Csörgő and Zitikis (1999b). □

3 The Vervaat error process in terms of a Kiefer process

We introduce some two-parameter Gaussian processes. Let {W (x, y), x ≥ 0, y ≥ 0} be a Wiener (Brownian) sheet, i.e., a two-parameter Gaussian process with EW (x, y) = 0 and covariance function

EW (x_{1}, y_{1}) W (x_{2}, y_{2}) = (x_{1} ∧ x_{2})(y_{1} ∧ y_{2}).

Next we define a Kiefer process {K(x, y), 0 ≤ x ≤ 1, y ≥ 0} by

K(x, y) := W (x, y) − xW (1, y),

where W (x, y) is a Wiener sheet. A Kiefer process K(x, y) can be characterized as a mean zero Gaussian process with covariance function

EK(x_{1}, y_{1}) K(x_{2}, y_{2}) = (x_{1} ∧ x_{2} − x_{1}x_{2})(y_{1} ∧ y_{2}).

Remark 3.1 In this paper we define, without loss of generality, all Kiefer processes K(x, y) and Brownian bridges B(x) to be equal to zero if x < 0 or x > 1. □

Concerning the uniform empirical process α_{n}, Komlós et al. (1975) established the following fundamental embedding theorem.

Theorem F (Komlós, Major and Tusnády, 1975) On a suitable probability space, the uniform empirical process {α_{k}(x), 0 ≤ x ≤ 1, k = 1, 2, … } and a Kiefer process {K(x, k), 0 ≤ x ≤ 1, k = 1, 2, … } can be so constructed that, as n → ∞,

max_{1≤k≤n} sup_{0≤x≤1} |α_{k}(x) − n^{−1/2}K(x, k)| = O (\frac{(log n)^{2}}{n^{1/2}} ) a.s.

Combining (1.4) with Theorem F, we arrive at (cf. Csörgő and Révész, 1975):

Proposition A On the probability space of Theorem F for the uniform quantile process {β_{k}(x), 0 ≤ x ≤ 1, k = 1, 2, … } with the Kiefer process {K(x, k), 0 ≤ x ≤ 1, k = 1, 2, … } of Theorem F, as n → ∞, we have

max_{1≤k≤n} sup_{0≤x≤1} |β_{k}(x) + n^{−1/2}K(x, k)| = O (\frac{(log n)^{1/2}(log_{2} n)^{1/4}}{n^{1/4}} )  a.s.  (3.1)

We note in passing (cf. Deheuvels, 1998 and references therein) that the almost sure rate of convergence in (3.1) cannot be improved even when approximating the uniform quantile process by any other Kiefer process.

For further use we quote the following two results of A.H.C. Chan (1977) (cf. Theorem 1.15.2, and Theorems S.1.14.2 and S.1.15.1, respectively, in Csörgő and Révész, 1981).

Theorem G (Chan, 1977) Let K(·, ·) be a Kiefer process, and let {h_{n}} be a non-increasing sequence of positive numbers such that lim_{n→∞}(log \frac{1}{h_{n}})/ log_{2} n = ∞. Then

lim_{n→∞} sup_{0≤t≤1−h_{n}} sup_{0≤s≤h_{n}} \frac{|K(t + s, n) − K(t, n)|}{(2nh_{n} log(1/h_{n}))^{1/2}} = 1 a.s.

Theorem H (Chan, 1977) Let 0 < ε_{T} ≤ \frac{1}{2}, 0 < a_{T} ≤ T be functions of T such that ε_{T} and a_{T} /T are non-increasing and a_{T} is non-decreasing. Define K ((x_{1}, x_{2}], t) = K(x_{2}, t) − K(x_{1}, t) (0 ≤ x_{1} < x_{2} ≤ 1). Then almost surely,

lim sup_{T →∞} sup_{0≤t≤T −a_{T}} sup_{0≤z≤a_{T}} sup_{0≤x≤1−ε_{T}} sup_{0≤s≤ε_{T}} β_{T} |K((x, x + s], t + z) − K((x, x + s], t)|

= lim sup_{T →∞} sup_{0≤t≤T −a_{T}} sup_{0≤x≤1−ε_{T}} β_{T} |K((x, x + ε_{T}], t + a_{T}) − K((x, x + ε_{T}], t)| = 1,

where

β_{T} = (2a_{T} ε_{T} (1 − ε_{T}) (log \frac{T}{ε_{T} a_{T}} + log_{2} T ))^{−1/2} .

If in addition, lim_{T →∞}(log \frac{T}{ε_{T} a_{T}})/ log_{2} T = ∞, then lim sup_{T →∞} can be replaced by lim_{T →∞}.

The main result of this section is the following strong approximation of the Vervaat error process Q_{n}(t) defined in (2.1) via a Kiefer process.

Theorem 3.1 On the probability space of Theorem F, using the there constructed Kiefer process K(·, ·), Q_{n}(·) can be approximated as follows. As n → ∞ we have

sup_{0<t<1} |Q_{n}(t) − Z_{n}(t)| = O(n^{−3/8}(log n)^{3/4}(log_{2} n)^{5/8})  a.s.,  (3.2)

where {Z_{n}(t), 0 < t < 1, n = 1, 2, … } is defined by

Z_{n}(t) := 2\frac{K(t, n)}{n} ∫_{0}^{1} (K (t − s\frac{K(t, n)}{n}, n) − K(t, n)) ds.  (3.3)

The proof of this theorem is based on the next two lemmas, which are of interest on their own.

Lemma 3.1 Let

A_{n}(t) := 2n^{1/2} ∫_{F_{n}^{−1}(t)}^{t}(α_{n}(u) − α_{n}(t)) du, 0 < t < 1, n = 1, 2, …  (3.4)

Then

Q_{n}(t) = A_{n}(t) − R_{n}^{2}(t),  (3.5)

and, consequently, as n → ∞,

sup_{0<t<1} |Q_{n}(t) − A_{n}(t)| = O(n^{−1/2}(log n)(log_{2} n)^{1/2})  a.s.  (3.6)

Proof. We have the following easy-to-check representation

V_{n}(t) = −α_{n}(t)β_{n}(t) + A_{n}(t) − (α_{n}(t) + β_{n}(t))β_{n}(t)

for all t ∈ (0, 1) and n = 1, 2, … (cf., Vervaat, 1972a, Shorack and Wellner, 1986, Zitikis, 1998). By (2.1), Q_{n}(t) = V_{n}(t) − α_{n}^{2}(t) = A_{n}(t) − R_{n}^{2}(t), yielding (3.5). The conclusion (3.6) follows from (3.5) and (1.4). □

Remark 3.2 While in this paper we are yet to study the stochastic process Q_{n}(t) in terms of A_{n}(t), it is interesting to note that, as a result of the representation (3.5), we already know the best possible stochastic behaviour of their difference A_{n}(t) − Q_{n}(t) = R_{n}^{2}(t). Moreover, on account of having (1.1)–(1.8), we can immediately write down exact analogues for the difference process A_{n}(t) − Q_{n}(t) via R_{n}^{2}(t). □

Lemma 3.2 On the probability space of Theorem F, using the there constructed Kiefer process K(·, ·), the stochastic process A_{n}(t) of Lemma 3.1 can be approximated such that when n → ∞,

sup_{0<t<1} |A_{n}(t) − Z_{n}(t)| = O(n^{−3/8}(log n)^{3/4}(log_{2} n)^{3/8}) a.s.,

where Z_{n}(t) is as in Theorem 3.1.

Proof. By a change of variables u = t − s(t − F_{n}^{−1}(t)) = t + sn^{−1/2}β_{n}(t) in (3.4), we have

A_{n}(t) = −2β_{n}(t) ∫_{0}^{1} (α_{n} (t + s\frac{β_{n}(t)}{n^{1/2}}) − α_{n}(t)) ds.

The usual LIL for β_{n} confirms that ∥β_{n}∥ = O((log_{2} n)^{1/2}) almost surely (when n → ∞). Therefore, by Theorem F, when n → ∞,

A_{n}(t) = −2n^{−1/2}β_{n}(t) ∫_{0}^{1} (K (t + s\frac{β_{n}(t)}{n^{1/2}}, n) − K(t, n)) ds
+ O (n^{−1/2}(log n)^{2}(log_{2} n)^{1/2}) ,  a.s.,  (3.7)

uniformly in t ∈ (0, 1).

For all t, s ∈ (0, 1), we have

|K (t + s\frac{β_{n}(t)}{n^{1/2}}, n) − K (t − s\frac{K(t, n)}{n}, n)|

:= |K(u + v, n) − K(u, n)|
≤ sup_{0≤u≤1} sup_{0≤v≤h_{n}} |K(u + v, n) − K(u, n)|,

where

u = t − s\frac{K(t, n)}{n}, v = s (\frac{β_{n}(t)}{n^{1/2}} + \frac{K(t, n)}{n})

and, on account of Proposition A, h_{n} = O(n^{−3/4}(log n)^{1/2}(log_{2} n)^{1/4}) (n → ∞), almost surely. Thus, according to Theorem G, almost surely, when n → ∞,

K (t + s\frac{β_{n}(t)}{n^{1/2}}, n) − K (t − s\frac{K(t, n)}{n}, n) = O (n^{1/8}(log n)^{3/4}(log_{2} n)^{1/8}) ,

uniformly in t, s ∈ (0, 1). Inserting this into (3.7) and using again the LIL for β_{n}, we arrive at:

A_{n}(t) = −2n^{−1/2}β_{n}(t) ∫_{0}^{1} (K (t − s\frac{K(t, n)}{n}, n) − K(t, n)) ds
+O(n^{−3/8}(log n)^{3/4}(log_{2} n)^{5/8})  a.s.  (3.8)

According to Proposition A,

−2n^{−1/2}β_{n}(t) = 2\frac{K(t, n)}{n} + O (\frac{(log n)^{1/2}(log_{2} n)^{1/4}}{n^{3/4}} )  (3.9)

almost surely and uniformly in t ∈ (0, 1). On the other hand, applying Theorem G to the integrand in (3.9) with h_{n} = O(n^{−1/2}(log_{2} n)^{1/2}), we obtain:

∫_{0}^{1} (K (t − s\frac{K(t, n)}{n}, n) − K(t, n)) ds = O (n^{1/4}(log n)^{1/2}(log_{2} n)^{1/4}) ,

almost surely and uniformly in t ∈ (0, 1). Plugging this and (3.9) into (3.8) yields that almost surely, when n → ∞,

A_{n}(t) = Z_{n}(t) + O (\frac{(log n)(log_{2} n)^{1/2}}{n^{1/2}} ) + O(n^{−3/8}(log n)^{3/4}(log_{2} n)^{5/8}),

uniformly in t ∈ (0, 1). This yields Lemma 3.2. □

Proof of Theorem 3.1. Follows from Lemmas 3.1 and 3.2. □

We mention that the process Z_{n}(t) was put to use in Csörgő and Zitikis (1999a) in their study of Q_{n}(t) in L_{p} norm. Moreover, they also remarked that their conjecture as stated in (2.4) is equivalent to stating it in terms of Z_{n} instead of Q_{n}. Likewise, in view of Theorem 3.1 now, the proof of Theorem 2.1 can, and will, be based on proving the statements (2.5)–(2.7) with the process Z_{n}(t) replacing Q_{n}(t) in all of them. The latter goal in turn will be achieved via using the next simple, though essential, observation that is borrowed from Einmahl (1996).

Proposition B Let {K(x, y), 0 ≤ x ≤ 1, y ≥ 0} be a Kiefer process. For any fixed 0 ≤ u < v ≤ 1, the process

\{ \frac{K(u + (v − u)x, y) − xK(v, y) − (1 − x)K(u, y)}{\sqrt{v − u}}, x ∈ [0, 1], y ≥ 0\}

is a Kiefer process that is independent of {K(s, y), s ∈ [0, u], y ≥ 0} and {K(s, y), s ∈ [v, 1], y ≥ 0}.

Based on Theorem 3.1 and this crucial observation, the respective proofs of the pointwise statements of (2.5)–(2.6) and the proof of the uniform property as in (2.7) of the process Q_{n} will take different routes. Hence, our next Section 4 is devoted to proving (2.5)–(2.6), while Section 5 will be on establishing the almost sure ratio statement of (2.7).

4 Pointwise behaviour of the Vervaat error process

This section is devoted to the proof of (2.5) and (2.6) of Theorem 2.1, concerning the pointwise asymptotics of Q_{n}(t). We first establish a strong approximation of Z_{n}(t) of (3.3) for any fixed t ∈ (0, 1) by a process in n, which is an integral of a Wiener sheet in its first parameter over a random interval that is independent of this Wiener sheet in hand.

Lemma 4.1 Given Z_{n}(t) as in (3.3), then for any fixed t ∈ (0, 1) one can define a Wiener sheet W^{∗}(·, ·) such that, as n → ∞, we have

Z_{n}(t) = 2 ∫_{0}^{|K(t,n)|/n} W^{∗}(y, n) dy + O (\frac{(log n)^{1/2} log_{2} n}{n^{1/2}})  a.s.,  (4.1)

where {W^{∗}(y, n), y ≥ 0, n = 1, 2, … } is independent of {K(t, n), n = 1, 2, … }.

Proof. Using Proposition B with u = 0, v = t, and writing 1 − x instead of x, we see that

K_{1}^{∗}(x, n) := \frac{K(t(1 − x), n) − (1 − x)K(t, n)}{t^{1/2}}, 0 ≤ x ≤ 1,  (4.2)

is a Kiefer process, independent of K(t, ·). Likewise, letting u = t, v = 1, we see that

K_{2}^{∗}(x, n) := \frac{K(t + (1 − t)x, n) − (1 − x)K(t, n)}{(1 − t)^{1/2}}, 0 ≤ x ≤ 1,

is also a Kiefer process, independent of K(t, ·). Moreover, K_{1}^{∗}(·, ·) and K_{2}^{∗}(·, ·) are independent. Consequently, the components of the vector

(K(t, ·), K_{1}^{∗}(·, ·), K_{2}^{∗}(·, ·))  (4.3)

are independent processes.

Defining x via

(1 − x)t = t − s\frac{K(t, n)}{n} if K(t, n) > 0,
t + (1 − t)x = t − s\frac{K(t, n)}{n} if K(t, n) < 0,

in the integral in the definition of Z_{n}(t) of (3.3), we arrive at the following representation of the latter process for each fixed t:

Z_{n}(t) = I_{1}(n)1{K(t, n) > 0} − I_{2}(n)1{K(t, n) < 0},  (4.4)

where

I_{1}(n) := 2t ∫_{0}^{K(t,n)/(nt)}(K(t(1 − x), n) − K(t, n)) dx,
I_{2}(n) := 2(1 − t) ∫_{0}^{−K(t,n)/(n(1−t))}(K(t + (1 − t)x, n) − K(t, n)) dx.

Considering I_{1}(n), and remembering that this is the case when K(t, n) > 0, using the definition (4.2) of K_{1}^{∗}(·, ·), we obtain: almost surely as n → ∞,

I_{1}(n) = 2t^{3/2} ∫_{0}^{K(t,n)/(nt)} K_{1}^{∗}(x, n) dx − 2tK(t, n) ∫_{0}^{K(t,n)/(nt)} x dx
= 2t^{3/2} ∫_{0}^{K(t,n)/(nt)} K_{1}^{∗}(x, n) dx − \frac{t(K(t, n))^{3}}{n^{2}t^{2}}
= 2t^{3/2} ∫_{0}^{K(t,n)/(nt)} K_{1}^{∗}(x, n) dx + O (\frac{(log_{2} n)^{3/2}}{n^{1/2}} ) ,  (4.5)

the last line following from the LIL for the Kiefer process with fixed t. Since K_{1}^{∗}(·, ·) is a Kiefer process independent of K(t, ·), we can use the representation

K_{1}^{∗}(x, n) = W_{1}^{∗}(x, n) − xW_{1}^{∗}(1, n),

where W_{1}^{∗}(·, ·) is a Wiener sheet, independent of K(t, ·). This independence property will be crucial in our use later on.

Observe that

∫_{0}^{K(t,n)/(nt)} K_{1}^{∗}(x, n) dx = ∫_{0}^{K(t,n)/(nt)} W_{1}^{∗}(x, n) dx − \frac{K^{2}(t, n) W_{1}^{∗}(1, n)}{2t^{2} n^{2}}.

By the LIL for the Kiefer process K and the Wiener process W_{1}^{∗}(1, ·), we have, when n → ∞, K^{2}(t, n) W_{1}^{∗}(1, n) = O(n^{3/2}(log_{2} n)^{3/2}) almost surely. Going back to (4.5), we obtain: as n → ∞, almost surely

I_{1}(n) = 2t^{3/2} ∫_{0}^{K(t,n)/(nt)} W_{1}^{∗}(x, n) dx + O (\frac{(log_{2} n)^{3/2}}{n^{1/2}} )

= 2 ∫_{0}^{K(t,n)/n} t^{1/2}W_{1}^{∗}(\frac{y}{t} , n) dy + O (\frac{(log_{2} n)^{3/2}}{n^{1/2}} ) .  (4.6)

Similarly, in the case K(t, n) < 0, we can show that as n → ∞, for I_{2}(n) of (4.4) we have, almost surely,

I_{2}(n) = 2 ∫_{0}^{−K(t,n)/n}(1 − t)^{1/2}W_{2}^{∗}(\frac{y}{1 − t}, n) dy + O (\frac{(log_{2} n)^{3/2}}{n^{1/2}} ) ,  (4.7)

where, just like W_{1}^{∗}(·, ·) of (4.6), the Wiener sheet W_{2}^{∗}(·, ·) of (4.7) is also independent of K(t, ·) (cf. (4.3)).

Combining (4.6) and (4.7) with (4.4) yields that, for each fixed t ∈ (0, 1), as n → ∞,

Z_{n}(t) = 2 ∫_{0}^{|K(t,n)|/n} W^{∗}(y, n) dy + O (\frac{(log_{2} n)(log n)^{1/2}}{n^{1/2}} ) a.s.,

where

W^{∗}(y, n) := t^{1/2}W_{1}^{∗}(\frac{y}{t} , n)1\{K(t, n) > 0\} − (1 − t)^{1/2}W_{2}^{∗}(\frac{y}{1 − t}, n)1\{K(t, n) < 0\}.

Since W^{∗}(·, ·) is a Wiener sheet, independent of K(t, ·), this yields Lemma 4.1. □

The rest of this section is devoted to the proof of (2.5) and (2.6) in Theorem 2.1. For the sake of clarity, they are proved separately.

Proof of (2.5). For each fixed n and T, \frac{3^{1/2}}{n^{1/2}T^{3/2}} ∫_{0}^{T} W^{∗}(y, n) dy is a standard normal variable. Thus, by conditioning, if T is a random variable independent of W^{∗}(·, n), then \frac{3^{1/2}}{n^{1/2}T^{3/2}} ∫_{0}^{T} W^{∗}(y, n) dy is a standard normal variable, independent of T. In view of the independence of K(t, ·) and W^{∗}(·, ·), we have, for each fixed n and t, (taking T := |K(t, n)|/n)

2 ∫_{0}^{|K(t,n)|/n} W^{∗}(y, n) dy
= (\frac{4}{3})^{1/2} \frac{(t(1 − t))^{3/4}}{n^{1/4}} (\frac{n^{1/2}|T |}{(t(1 − t))^{1/2}} )^{3/2} (\frac{3^{1/2}}{n^{1/2}T^{3/2}} ∫_{0}^{T} W^{∗}(y, n) dy)

=_{d} (\frac{4}{3})^{1/2} (t(1 − t))^{3/4}n^{−1/4}(|Ñ|)^{3/2}N ,

where “=_{d}” denotes identity in distribution, and N and Ñ are independent standard normal random variables. This, in light of (3.2) and (4.1), yields (2.5). □

Proof of (2.6). Fix again t ∈ (0, 1). Define

W_{t}(n) := \frac{K(t, n)}{(t(1 − t))^{1/2}} ,  n ≥ 1,
W_{t}^{∗}(x, y) := (t(1 − t))^{1/4}W^{∗} (\frac{x}{(t(1 − t))^{1/2}} , y) ,  x ≥ 0, y ≥ 0.

Clearly, W_{t} is a Wiener process (restricted on N^{∗}), and W_{t}^{∗}(·, ·) is a Wiener sheet, independent of {K(t, n), n = 1, 2, … }, thus of W_{t} as well. We also observe that

∫_{0}^{|K(t,n)|/n} W^{∗}(y, n) dy
= (t(1 − t))^{3/4} \frac{(2 log_{2} n)^{5/4}}{n^{1/4}} ∫_{0}^{|W_{t}(n)|/(2n log_{2} n)^{1/2}} \frac{W_{t}^{∗}(u(\frac{2 log_{2} n}{n})^{1/2}, n)}{n^{1/4}(2 log_{2} n)^{3/4}} du.

By Theorem 1.1 of Deheuvels and Mason (1992), as n → ∞, the set of limit points of

\{(\frac{W_{t}(n)}{(2n log_{2} n)^{1/2}} , \frac{W_{t}^{∗}(u(\frac{2 log_{2} n}{n})^{1/2}, n)}{n^{1/4}(2 log_{2} n)^{3/4}}) , u ∈ [0, 1]\}

is almost surely equal to

D := {(c, f) : f absolutely continuous with respect to the Lebesgue measure,

|c| ∈ (0, 1), f(0) = 0, c^{2} + ∫_{0}^{|c|}(f′(u))^{2} du ≤ 1}.  (4.8)

Consequently, with probability one,

lim sup_{n→∞} \frac{n^{1/4}}{(log_{2} n)^{5/4}} ∫_{0}^{|K(t,n)|/n} W^{∗}(y, n) dy = 2^{5/4}(t(1 − t))^{3/4} sup_{(c,f)∈D} ∫_{0}^{|c|} f(u) du.

We now determine the value of the “sup” expression on the right hand side. Integrating by parts, using the Cauchy–Schwarz inequality and (4.8), we get, for each (c, f) ∈ D,

∫_{0}^{|c|} f(u) du = ∫_{0}^{|c|}(|c| − u)f′(u) du
≤ (∫_{0}^{|c|}(|c| − u)^{2} du)^{1/2} (∫_{0}^{|c|}(f′(u))^{2} du)^{1/2} ≤ (\frac{|c|^{3}(1 − c^{2})}{3})^{1/2} .

Since |c|^{3}(1 − c^{2}) ≤ 2 × 3^{3/2}/5^{5/2} for any c ∈ [−1, 1], this yields

sup_{(c,f)∈D} ∫_{0}^{|c|} f(u) du ≤ \frac{2^{1/2}3^{1/4}}{5^{5/4}} .  (4.9)

Choosing

c = (3/5)^{1/2}, f(u) = \frac{2^{1/2}3^{1/4}}{5^{1/4}} u − \frac{5^{1/4}}{2^{1/2}3^{1/4}} u^{2},

it is seen that in (4.9) we have, in fact, an equality. Accordingly, with probability one,

lim sup_{n→∞} \frac{n^{1/4}}{(log_{2} n)^{5/4}} ∫_{0}^{|K(t,n)|/n} W^{∗}(y, n) dy = (t(1 − t))^{3/4} \frac{2^{7/4}3^{1/4}}{5^{5/4}} .

This yields (2.6) in view of (4.1) and (3.2). □

5 Uniform behaviour of the Vervaat error process

This section is devoted to establishing the uniform property (2.7) in Theorem 2.1. In view of Theorems 3.1 and F it suffices to show that

lim sup_{n→∞} \frac{n}{(log n)^{1/2}} \frac{∥Z_{n}∥}{∥K(·, n)∥^{3/2}} ≤ (\frac{4}{3})^{1/2}  a.s.,  (5.1)
lim inf_{n→∞} \frac{n}{(log n)^{1/2}} \frac{∥Z_{n}∥}{∥K(·, n)∥^{3/2}} ≥ (\frac{4}{3})^{1/2}  a.s.,  (5.2)

where ∥Z_{n}∥ := sup_{0≤t≤1} |Z_{n}(t)| and ∥K(·, n)∥ := sup_{0≤t≤1} |K(t, n)|.

Proof of (5.1). Recall that (cf. (3.3)), by definition,

Z_{n}(t) = 2\frac{K(t, n)}{n} ∫_{0}^{1} (K (t − s\frac{K(t, n)}{n}, n) − K(t, n)) ds
= 2 ∫_{0}^{K(t,n)/n}(K(t − z, n) − K(t, n)) dz.  (5.3)

Let

N = N(n) := ⌊n^{1/2}(log_{2} n)^{−4}⌋,  (5.4)

and let t_{i} = t_{i}(n) := i/N (for 0 ≤ i ≤ N). Define, for u ∈ [0, 1],

B_{i,n}^{∗}(u) := \frac{N^{1/2}}{n^{1/2}} (K (t_{i} + \frac{u}{N} , n) − uK(t_{i+1}, n) − (1 − u)K(t_{i}, n)) .

It follows from Proposition B that for each fixed n,

{B_{i,n}^{∗}(·), i = 0, 1, …, N − 1}

are independent Brownian bridges, and independent of {K(t_{i}, n), i = 0, 1, …, N − 1}. Let t ∈ [t_{i}, t_{i+1}]. Then

Z_{n}(t) = 2 (∫_{0}^{K(t_{i},n)/n} + ∫_{K(t_{i},n)/n}^{K(t,n)/n}) (K(t − z, n) − K(t, n)) dz.

By Theorem G, when n → ∞,

max_{0≤i≤N−1} sup_{t_{i}≤t≤t_{i+1}} \frac{|K(t, n) − K(t_{i}, n)|}{n} = O (\frac{(log n)^{1/2}}{N^{1/2}n^{1/2}} ) a.s.

On the other hand, by the LIL for the Kiefer process and Theorem G,

K(t − z, n) − K(t, n) = O (n^{1/4}(log n)^{1/2}(log_{2} n)^{1/4}) a.s.,

uniformly in 0 ≤ i ≤ N − 1, t ∈ [t_{i}, t_{i+1}] and z ∈ [K(t_{i}, n)/n, K(t, n)/n]. Therefore,

∫_{K(t_{i},n)/n}^{K(t,n)/n}(K(t − z, n) − K(t, n)) dz = O (\frac{(log n)(log_{2} n)^{1/4}}{N^{1/2}n^{1/4}} ) a.s.

As a consequence, for t ∈ [t_{i}, t_{i+1}], we have, almost surely when n → ∞,

Z_{n}(t) = 2 ∫_{0}^{K(t_{i},n)/n}(K(t − z, n) − K(t, n)) dz + O(\frac{(log n)(log_{2} n)^{1/4}}{N^{1/2}n^{1/4}} ) ,  (5.5)

where O is uniform in i = 0, 1, …, N − 1 and t ∈ [t_{i}, t_{i+1}].

Let i be such that K(t_{i}, n) ≤ 0. If t ∈ [t_{i}, t_{i+1}], then t = t_{i} + v/N for some v ∈ [0, 1]. Accordingly, by writing A_{i} := N|K(t_{i}, n)|/n,

∫_{0}^{K(t_{i},n)/n}(K(t − z, n) − K(t, n)) dz
= −\frac{n^{1/2}}{N^{3/2}} ∫_{0}^{A_{i}}(B_{i,n}^{∗}(v + y) − B_{i,n}^{∗}(v)) dy − \frac{K(t_{i+1}, n) − K(t_{i}, n)}{N} ∫_{0}^{A_{i}} y dy.

Since ∫_{0}^{A_{i}} y dy = N^{2}K^{2}(t_{i}, n)/2n^{2}, an application of the LIL for the Kiefer process and Theorem G yields that with probability one,

∫_{0}^{K(t_{i},n)/n}(K(t − z, n) − K(t, n)) dz
= −\frac{n^{1/2}}{N^{3/2}} ∫_{0}^{A_{i}}(B_{i,n}^{∗}(v + y) − B_{i,n}^{∗}(v)) dy + O (\frac{N^{1/2}(log n)^{1/2} log_{2} n}{n^{1/2}}) ,  (5.6)

uniformly in i = 0, 1, …, N − 1.

Note that B_{i,n}^{∗} can be represented as

B_{i,n}^{∗}(u) := −W_{i,n}^{∗}(u) + uW_{i,n}^{∗}(1),

where (for each fixed n) {W_{i,n}^{∗}, i = 0, 1, …, N − 1} are independent Wiener processes which are also independent of {K(t_{i}, n), i = 0, 1, …, N − 1}. Hence, almost surely,

∫_{0}^{A_{i}}(B_{i,n}^{∗}(v + y) − B_{i,n}^{∗}(v)) dy
= − ∫_{0}^{A_{i}}(W_{i,n}^{∗}(v + y) − W_{i,n}^{∗}(v)) dy + W_{i,n}^{∗}(1) ∫_{0}^{A_{i}} y dy
= − ∫_{0}^{A_{i}}(W_{i,n}^{∗}(v + y) − W_{i,n}^{∗}(v)) dy + O (\frac{N^{2}(log n)^{1/2} log_{2} n}{n}) ,  (5.7)

the last equality following from the LIL for the Kiefer process and the fact that almost surely, sup_{0≤i≤N−1} |W_{i,n}^{∗}(1)| = O((log n)^{1/2}). This last fact can be easily checked using the usual estimate for Gaussian tails and the Borel–Cantelli lemma, regardless of the dependency structure of the standard normal variables {W_{i,n}^{∗}(1), i = 0, 1, …, N − 1; n = 1, 2, … }.

Putting (5.5), (5.6) and (5.7) together, and taking into account the definition of N in (5.4), we obtain:

Z_{n}(t) = \frac{2n^{1/2}}{N^{3/2}} ∫_{0}^{A_{i}}(W_{i,n}^{∗}(v + y) − W_{i,n}^{∗}(v)) dy + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n}) ,  (5.8)

for t = t_{i} + v/N, 0 ≤ v ≤ 1, K(t_{i}, n) ≤ 0.

If K(t_{i}, n) > 0, then by introducing the Kiefer process K^{←}(t, n) := K(1 − t, n), we can write

2 ∫_{0}^{K(t_{i},n)/n}(K(t − z, n) − K(t, n)) dz
= 2 ∫_{0}^{K(t_{i},n)/n}(K^{←}(1 − t + z, n) − K^{←}(1 − t, n)) dz,

and, similarly to the argument leading to (5.8), we can find independent Wiener processes \{W\overleftarrow{i,n}, i = 0, 1, …, N − 1} which are also independent of {K(t_{i}, n), i = 0, 1, …, N − 1}, such that almost surely,

Z_{n}(t) = \frac{2n^{1/2}}{N^{3/2}} ∫_{0}^{A_{i}}(W\overleftarrow{i,n}(v + y) − W\overleftarrow{i,n}(v)) dy + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n}) ,  (5.9)

for t = t_{i} + v/N, 0 ≤ v ≤ 1, K(t_{i}, n) > 0.

Combining (5.8) and (5.9), we see that with probability one, for t = t_{i} + v/N, v ∈ [0, 1],

Z_{n}(t) = \frac{2n^{1/2}}{N^{3/2}} ∫_{0}^{A_{i}}(W_{i,n}(v + y) − W_{i,n}(v)) dy + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n}) ,  (5.10)

where

W_{i,n}(·) := W_{i,n}^{∗}(·)1\{K(t_{i}, n) ≤ 0\} + W\overleftarrow{i,n}(·)1\{K(t_{i}, n) > 0\},

i = 0, 1, …, N − 1, are Wiener processes, independent of {K(t_{i}, n), i = 0, 1, …, N − 1}. Note that we do not claim that the Wiener processes {W_{i,n}, i = 0, 1, …, N − 1} are independent between themselves.

For each n, we split {0, 1, …, N − 1} into two (random) parts:

J_{1} = J_{1}(n) := {i : |K(t_{i}, n)| ≤ n^{1/2}(log_{2} n)^{−1}, 0 ≤ i ≤ N − 1},
J_{2} = J_{2}(n) := {i : |K(t_{i}, n)| > n^{1/2}(log_{2} n)^{−1}, 0 ≤ i ≤ N − 1}.

If i ∈ J_{1}, then on applying the LIL for the Kiefer process and Theorem G we conclude

∫_{0}^{K(t_{i},n)/n}(K(t − z, n) − K(t, n)) dz = O (\frac{(log n)^{1/2}}{n^{1/4}(log_{2} n)^{3/2}} ) a.s.

uniformly in i ∈ J_{1}. In view of (5.5), we obtain: when → ∞,

max_{i∈J_{1}} sup_{t∈[t_{i},t_{i+1}]} |Z_{n}(t)| = O (\frac{(log n)^{1/2}}{n^{1/4}(log_{2} n)^{3/2}} )  a.s.  (5.11)

For i ∈ J_{2}, we consider the variables

X_{i,n} := \frac{3^{1/2}}{A_{i}^{3/2}} sup_{0≤v≤1} |∫_{0}^{A_{i}}(W_{i,n}(v + y) − W_{i,n}(v)) dy| .

According to (5.10), almost surely when n → ∞,

max_{i∈J_{2}} sup_{t∈[t_{i},t_{i+1}]} |Z_{n}(t)| ≤ \frac{2∥K(·, n)∥^{3/2}}{3^{1/2} n} max_{i∈J_{2}} X_{i,n} + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n}) ,

which, combined with (5.11), yields

∥Z_{n}∥ ≤ \frac{2∥K(·, n)∥^{3/2}}{3^{1/2} n} max_{i∈J_{2}} X_{i,n} + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n})  a.s.  (5.12)

We now show that the variables X_{i,n}, i ∈ J_{2} have Gaussian-like tails. To this end, we fix A > 0 and introduce the mean zero Gaussian process

X(v) := \frac{3^{1/2}}{A^{3/2}} ∫_{0}^{A}(W (v + y) − W (v)) dy, v ∈ [0, 1].

It is straightforward to compute its covariance:

EX(u)X(v) = \{ \begin{matrix}1 − \frac{3|v−u|}{2A} + \frac{|v−u|^{3}}{2A^{3}} & if |v − u| ≤ A, \\ 0 & if |v − u| > A, \end{matrix}  (5.13)

and therefore

E (X(v) − X(u))^{2} = \{ \begin{matrix}\frac{3|v−u|}{A} − \frac{|v−u|^{3}}{A^{3}} & if |v − u| ≤ A, \\ 2 & if |v − u| > A. \end{matrix}  (5.14)

According to a well-known inequality of Fernique (1964), for any x > 0,

P ( sup_{0≤v≤1} |X(v)| ≥ x (σ^{2} + 4 ∫_{1}^{∞} φ(p^{−s^{2}}) ds)) ≤ 4p^{2} ∫_{x}^{∞} e^{−s^{2}/2} ds,

where p ≥ 2 is arbitrary, and σ and φ are such that

E(X(v))^{2} ≤ σ^{2}, E (X(v) − X(u))^{2} ≤ φ^{2}(v − u).

In view of (5.13) and (5.14), we can choose σ = 1 and φ(h) = (3h/A)^{1/2}. Since

∫_{1}^{∞} φ(p^{−s^{2}}) ds ≤ \begin{matrix}√ & \overline{3} \\ √ & \overline{A} \end{matrix} ∫_{1}^{∞} e^{−(s log p)/2} ds = \frac{2\sqrt{3}}{\sqrt{pA} log p},

this leads to:

P ( sup_{0≤v≤1} |X(v)| ≥ x (1 + \frac{8\sqrt{3}}{\sqrt{pA} log p})) ≤ 4p^{2} ∫_{x}^{∞} e^{−s^{2}/2} ds.  (5.15)

Recall that W_{i,n} is independent of K(t_{i}, n). Therefore, applying (5.15) to W = W_{i,n} and A = A_{i} (for i ∈ J_{2}) we obtain:

P (X_{i,n} ≥ x (1 + \frac{8\sqrt{3} (log_{2} n)^{3}}{\sqrt{p} log p} )) ≤ 4p^{2} ∫_{x}^{∞} e^{−s^{2}/2} ds,

where, with N as in (5.4), we used the fact that A_{i} ≥ (log_{2} n)^{−6} for all i ∈ J_{2}.

Let ε ∈ (0, 1). We choose p = ε^{−2}(log_{2} n)^{6} and n_{0} = n_{0}(ε) such that for all n ≥ n_{0}, 8\sqrt{3} (log_{2} n)^{3}/\sqrt{p} log p ≤ ε. Thus for any x > 0 and n ≥ n_{0},

P(X_{i,n} ≥ (1 + ε)x) ≤ \frac{4(log_{2} n)^{12}}{ε^{4}} ∫_{x}^{∞} e^{−s^{2}/2} ds ≤ \frac{4(log_{2} n)^{12}}{ε^{4}} \frac{e^{−x^{2}/2}}{x} ,

which yields

P ( max_{i∈J_{2}} X_{i,n} ≥ (1 + ε)x) ≤ \frac{4N(log_{2} n)^{12}e^{−x^{2}/2}}{ε^{4}x} ≤ \frac{4n^{1/2}(log_{2} n)^{8}e^{−x^{2}/2}}{ε^{4}x} .

Taking x := (1 + ε)(log n)^{1/2}, we obtain:

P (max_{i∈J_{2}} X_{i,n} ≥ (1 + ε)^{2}(log n)^{1/2}) ≤ \frac{4(log_{2} n)^{8}}{ε^{4}(1 + ε)(log n)^{1/2}n^{ε}} .

Let n_{k} = ⌊k^{2/ε}⌋. By the Borel–Cantelli lemma and (5.12), almost surely when k → ∞,

∥Z_{n_{k}}∥ ≤ \frac{2(1 + ε)^{2}(log n_{k})^{1/2} ∥K(·, n_{k})∥^{3/2}}{3^{1/2} n_{k}} + O (\frac{(log n_{k})^{1/2}}{n_{k}^{1/4} log_{2} n_{k}} ) .  (5.16)

Let n_{k} ≤ n < n_{k+1}. Note that n_{k+1} − n_{k} = O(n_{k}^{1−ε/2}), k → ∞. By (5.3),

|Z_{n}(t) − Z_{n_{k}}(t)| ≤ 2 |∫_{0}^{K(t,n)/n} Δ_{k,n,t}(z) dz|

+2 |∫_{K(t,n_{k})/n_{k}}^{K(t,n)/n}(K(t − z, n_{k}) − K(t, n_{k})) dz| ,

where Δ_{k,n,t}(z) := K(t − z, n) − K(t − z, n_{k}) − K(t, n) + K(t, n_{k}). According to Theorem H and the LIL for the Kiefer process, when k → ∞,

Δ_{k,n,t}(z) = O (n_{k}^{(1−ε)/4}(log n_{k})^{1/2}(log_{2} n_{k})^{1/4}) a.s.,

uniformly in t ∈ (0, 1), z ∈ [0, K(t, n)/n] and n_{k} ≤ n < n_{k+1}. Thus, by the LIL for the Kiefer process,

∫_{0}^{K(t,n)/n} Δ_{k,n,t}(z) dz = O (n_{k}^{−(1+ε)/4}(log n_{k})^{1/2}(log_{2} n_{k})^{3/4}) a.s.,

uniformly in t ∈ (0, 1) and n_{k} ≤ n < n_{k+1}.

On the other hand, by the LIL for the Kiefer process and Theorem G,

K(t − z, n_{k}) − K(t, n_{k}) = O (n_{k}^{1/4}(log n_{k})^{1/2}(log_{2} n_{k})^{1/4}) a.s.,

uniformly in t ∈ (0, 1), z ∈ [K(t, n_{k})/n_{k}, K(t, n)/n] and n_{k} ≤ n < n_{k+1}. Since Corollary 1.12.4 of Csörgő and Révész (1981) implies that

sup_{0≤t≤1} |K(t, n) − K(t, n_{k})| = O (n_{k}^{1/2−ε/4}(log n_{k})^{1/2})  a.s.,  (5.17)

uniformly in n_{k} ≤ n < n_{k+1}, it follows that

max_{n_{k}≤n<n_{k+1}} sup_{t∈[0,1]} |\frac{K(t, n)}{n} − \frac{K(t, n_{k})}{n_{k}} | = O (n_{k}^{−1/2−ε/4}(log n_{k})^{1/2}) a.s.

Therefore, almost surely when k → ∞,

∫_{K(t,n_{k})/n_{k}}^{K(t,n)/n}(K(t − z, n_{k}) − K(t, n_{k})) dz = O (n_{k}^{−(1+ε)/4}(log n_{k})(log_{2} n_{k})^{1/4}) ,

uniformly in t ∈ [0, 1] and n_{k} ≤ n < n_{k+1}. We have therefore proved that

max_{n_{k}≤n<n_{k+1}} ∥Z_{n} − Z_{n_{k}}∥ = O (n_{k}^{−(1+ε)/4}(log n_{k})(log_{2} n_{k})^{1/4}) , a.s.

Going back to (5.16) and taking (5.17) into account, we obtain, when n → ∞,

∥Z_{n}∥ ≤ \frac{2(1 + ε)^{2} (log n)^{1/2}}{3^{1/2} n} ∥K(·, n)∥^{3/2} + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n})
= (1 + o(1))\frac{2(1 + ε)^{2} (log n)^{1/2}}{3^{1/2} n} ∥K(·, n)∥^{3/2}  a.s.,

where in the last line we used the other LIL for the Kiefer process (Kuelbs, 1979, Mogulskii, 1979):

(∥K(·, n)∥)^{−1} = O (\frac{(log_{2} n)^{1/2}}{n^{1/2}} )  a.s.  (5.18)

This also completes the proof of (5.1). □

Proof of (5.2). Let N and t_{i}, i = 0, 1, …, N be as before (cf. (5.4)). Combining Theorem G with

( sup_{0≤t≤1} K(t, n))^{−1} = O (\frac{log n}{n^{1/2}})  a.s.  (5.19)

(cf. Csáki and Shi, 1998), we conclude

lim_{n→∞} \frac{max_{0≤i≤N−1} K(t_{i}, n)}{sup_{0≤t≤1} K(t, n)} = 1  a.s.  (5.20)

Let t_{0}^{+} = t_{0}^{+}(ω, n) be such that K(t_{0}^{+}, n) = max_{0≤i≤N−1} K(t_{i}, n) and consider the set of indices I_{n} := {i : t_{0}^{+} − h_{n} ≤ t_{i} ≤ t_{0}^{+} + h_{n}}, where h_{n} = (log n)^{−4}. Fix ε ∈ (0, 1). For i ∈ I_{n}, we obtain from Theorem G, (5.19) and (5.20) that when n is sufficiently large,

K(t_{i}, n) ≥ K(t_{0}^{+}, n) − O ((nh_{n} log(1/h_{n}))^{1/2})
≥ (1 − ε)K(t_{0}^{+}, n) > 0.  (5.21)

Hence, for i ∈ I_{n}, we can apply (5.9) to t = t_{i} and v = 0 to see that when n → ∞,

Z_{n}(t_{i}) = \frac{2n^{1/2}}{N^{3/2}} ∫_{0}^{NK(t_{i},n)/n} W\overleftarrow{i,n}(y) dy + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n})

= \frac{2}{3^{1/2} n}(K(t_{i}, n))^{3/2}Y_{i,n} + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n}) ,  (5.22)

where, for each n, \{W\overleftarrow{i,n}, i = 0, 1, …, N − 1\} are independent Wiener processes which are also independent of {K(t_{i}, n), i = 0, 1, …, N − 1}, and

Y_{i,n} := \frac{3^{1/2}n^{3/2}}{N^{3/2}} \frac{1}{(K(t_{i}, n))^{3/2}} ∫_{0}^{NK(t_{i},n)/n} W\overleftarrow{i,n}(y) dy, i ∈ I_{n}.

For each n, {Y_{i,n}, i ∈ I_{n}} are independent standard normal variables. Since #(I_{n}) ≥ N/(log n)^{4}, we have, by the usual estimate for Gaussian tails,

P (max_{i∈I_{n}} |Y_{i,n}| < (1 − ε)^{1/2}(log n)^{1/2})

≤ (1 − \frac{2^{1/2} + o(1)}{(1 − ε)^{1/2}(π log n)^{1/2}} \frac{1}{n^{(1−ε)/2}} )^{N/(log n)^{4}}

≤ exp (−\frac{2^{1/2} + o(1)}{(1 − ε)^{1/2}(π log n)^{1/2}} \frac{N/(log n)^{4}}{n^{(1−ε)/2}} )

= exp (−\frac{2^{1/2} + o(1)}{(1 − ε)^{1/2}π^{1/2}} \frac{n^{ε/2}}{(log n)^{9/2}(log_{2} n)^{4}} ) ,

which is summable. By the Borel–Cantelli lemma, almost surely for all large n,

max_{i∈I_{n}} |Y_{i,n}| ≥ (1 − ε)^{1/2}(log n)^{1/2}.

Inserting this into (5.22) and (5.21) yields that, almost surely,

∥Z_{n}∥ ≥ max_{i∈I_{n}} Z_{n}(t_{i})

≥ \frac{2(1 − ε)^{3/2}(log n)^{1/2}}{3^{1/2} n} (K(t_{0}^{+}, n))^{3/2} + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n})

≥ \frac{2(1 − ε)^{2}(log n)^{1/2}}{3^{1/2} n} ( sup_{0≤t≤1} K(t, n))^{3/2} + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n}) ,

where in the last inequality we used (5.20). This also holds true for − inf_{0≤t≤1} K(t, n) in place of sup_{0≤t≤1} K(t, n), by means of a similar argument with K(t_{0}^{+}, n) replaced by −K(t_{0}^{−}, n) = − min_{0≤i≤N−1} K(t_{i}, n). Consequently,

∥Z_{n}∥ ≥ \frac{2(1 − ε)^{2}(log n)^{1/2}}{3^{1/2} n} ∥K(·, n)∥^{3/2} + O (\frac{(log n)^{1/2}}{n^{1/4} log_{2} n}) a.s.

In light of (5.18), this yields (5.2). □

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