Fields Institute Communications Volume 00, 0000

Our joint work with Miklós Csörgő

Endre Csáki

A. Rényi Institute of Mathematics, Hungarian Academy of Sciences P.O.B. 127, H-1364 Budapest, Hungary

                               csaki@renyi.hu

Antónia Földes

Department of Mathematics, City University of New York 2800 Victory Blvd., Staten Island, New York 10314, U.S.A.

                             afoldes@gc.cuny.edu

Zhan Shi

Laboratoire de Probabilités UMR 7599, Université Paris VI 4 Place Jussieu, F-75252 Paris Cedex 05, France

                            zhan@proba.jussieu.fr

Dedicated to Miklós Csörgő on the occasion of his 70-th birthday

Abstract. We discuss various topics. They include: strong approximation of local time and additive functionals; path properties of Cauchy principal value; iterated processes; empirical processes; Vervaat error process; Banach space valued processes.

1. Introduction

The work of Miklós Csörgő has had a great impact on modern probability and statistics. His books, papers and ideas are wells of informations for these fields’ generations of mathematicians. This survey attempts to give only a brief account of the papers, which he has co-authored with one or more of us during the twentyodd years that we have been his friends and collaborators. A number of these papers were written with Pál Révész. In the following, we summarize the contents of these papers which are focused on a few strongly related topics.

The basic object of these investigations is the standard Wiener process or Brownian motion. [The two names will be alternatively used in this paper for

2000 Mathematics Subject Classification. Primary 60J55; Secondary 60F15, 60F17, 60J15. Research of the first author supported by the Hungarian National Foundation for Scientific

Research, Grant Nos. T 029621, T 037886 and T043037.

Research of the second author supported by a PSC CUNY grant, No. 634680032. Cooperation between the authors supported by the joint French-Hungarian Intergovernmental

Grant ”Balaton”, No. F-39/2000.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society

the same process.] This is a mean zero Gaussian process {W (t), t ≥ 0} with covariance EW (t_{1})W (t_{2}) = min(t_{1}, t_{2}). In certain cases we need a Wiener process on the whole real line. Let {W_{1}(t), t ≥ 0} and {W_{2}(t), t ≥ 0} be two independent standard Wiener processes. Then a standard Wiener process on ℝ is defined as W (t) = W_{1}(t) for t ≥ 0 and W (t) = W_{2}(−t) for t < 0.

2. Local time and additive functionals

2.1 The increments of the local time. At the beginning of the 1980-s we were fascinated with the Brownian local time. The asymptotic behaviour of the increments of the Wiener process was well understood, as Csörgő and Révész [36], [37] proved their incredibly precise results in a couple of papers about how big and how small are these increments of the Wiener process. In our first joint paper our objective was to investigate the corresponding ”how big” question for the increments of the local time. Before quoting these results, we introduce a pair of conditions which will be frequently used in the sequel.

Condition A: 0 < a_{t} ≤ t is a nondecreasing function of t such that t/a_{t} is also nondecreasing.

Condition B:

lim_{t→∞} \frac{log(ta_{t}^{−1})}{log log t} = +∞.

Theorem A (Csörgő and Révész [36], [38]) Under Condition A we have

lim sup_{t→∞} β_{t}a_{t}^{−1/2}(W (t + a_{t}) − W (t))

= lim sup_{t→∞} β_{t}a_{t}^{−1/2} sup_{0≤s≤t−a_{t}}(W (s + a_{t}) − W (s))

= lim sup_{t→∞} β_{t}a_{t}^{−1/2} sup_{0<s<a_{t}}(W (t + s) − W (t)) = 1 a.s.,

where β_{t} = (2(log ta_{t}^{−1} + log log t))^{−1/2}. Supposing Condition B as well, we also have

lim_{t→∞} β_{t}a_{t}^{−1/2} sup_{0≤s≤t−a_{t}}(W (s + a_{t}) − W (s)) = 1 a.s.

As it turned out, the increments of the local time behave very similarly, though a slightly different normalization is needed. We start with a quick definition of the local time process. For any Borel set A on the real line let

H(A, t) := λ{s ≤ t : W (s) ∈ A}

be the occupation time of W, where λ is the Lebesgue measure. H(A, t) is a random measure which is absolutely continuous with respect to λ with probability 1, its Radon-Nikodym derivative is called the local time of W, and will be denoted by L(x, t), i.e.

H(A, t) = ∫_{A} L(x, t) dx.

The joint continuity of L(x, t) is a famous result of Trotter [79], who also investigated the modulus of continuity, separately for x and for t. The celebrated law of the iterated logarithm for the local time is due to Kesten [62]:

lim sup_{t→∞} \frac{L(0, t)}{(2t log log t)^{1/2}} = lim sup_{t→∞} \frac{sup_{−∞<x<∞} L(x, t)}{(2t log log t)^{1/2}} = 1  a.s.  (2.1)

Let us denote L(0, t) by L(t). Our main result in [14] was the following

Theorem 2.1 Under Condition A we have

lim sup_{t→∞} γ_{t}Y (t) = lim sup_{t→∞} γ_{t}a_{t}^{−1/2}(L(t) − L(t − a_{t})) = 1  a.s.,  (2.2)

where

Y (t) = Y (t, a_{t}) = a_{t}^{−1/2} sup_{0<s<t−a_{t}}(L(s + a_{t}) − L(s))

and

γ_{t} = (log ta_{t}^{−1} + 2 log log t)^{−1/2}.

Assuming Condition B as well, we also have

lim_{t→∞} γ_{t}Y (t) = 1 a.s.

2.2 Approximation by a Wiener sheet. Once we understood the asymptotic behaviour of the local time increments when t → ∞, we turned our attention to the whole two-variate process L(x, t) − L(0, t). The starting point of these investigations was a landmark paper of Dobrushin [48] formulated for random walk (instead of a Wiener process case) which we will quote later. This theorem tells us that the local time increments normalized appropriately has a distribution, which for large t is close to the distribution of the product of N_{1}\sqrt{|N_{2}|} where N_{1} and N_{2} are independent standard normal variables. This fact is even more intriguing combined with the following insightful result of Yor [83]:

Theorem B (Yor [83]) As λ → ∞,

(\frac{1}{λ}W (λ^{2}t), \frac{1}{λ}L(a, λ^{2}t), \frac{1}{2\sqrt{λ}} (L(a, λ^{2}t) − L(0, λ^{2}t)))

→^{w} (W (t), L(a, t), W^{∗}(a, L(0, t)))

where W^{∗}(a, u) is a Wiener sheet independent of W (t) and →^{w} denotes weak convergence.

A Wiener (Brownian) sheet {W^{∗}(a, u), a ≥ 0, u ≥ 0} is a mean-zero Gaussian process with covariance E W^{∗}(a_{1}, u_{1})W^{∗}(a_{2}, u_{2}) = min(a_{1}, a_{2}) min(u_{1}, u_{2}).

The above two results suggested that the local time difference L(x, t) − L(0, t) could be strongly approximated by σ_{x}W^{∗}(L^{∗∗}(0, t)) on such a way that

L(0, t) should be close to L^{∗∗}(0, t),

W^{∗}(t) and L^{∗∗}(0, t) should be independent,

and σ_{x} is a constant depending only on x. This conjecture was confirmed in [15] by

Theorem 2.2 There is a probability space with

• a standard Wiener process {W (t), t > 0} and its two-parameter local time process {L(a, t), a ∈ ℝ, t ≥ 0},

• a two-parameter Wiener process {B(a, u), a ≥ 0, u ≥ 0},

• a process {L^{1}(0, t), t ≥ 0}, with {L^{1}(0, t), t ≥ 0} =^{D} {L(0, t), t ≥ 0}

such that as t → ∞ • sup_{0≤a≤a^{∗}t^{δ/2}} |L(a, t) − L(0, t) − 2B(a, L^{1}(0, t))| = O(t^{((1+δ)/4−ϵ/2}) a.s.,

• |L^{1}(0, t) − L(0, t)| = O(t^{15/32} log^{2} t) a.s.,
• {L^{1}(0, t), t ≥ 0} and {B(a, u), a ≥ 0, u ≥ 0} are independent,

and for the constants above we have; a^{∗} > 0, 0 < δ < 7/100, 0 < ϵ < 1/72 − δ/7, =^{D} denotes equality in distribution.

The proof of this result was based on two major ingredients. The first of these two is an approximation theorem of Berkes and Philipp [4] for weakly dependent vectors. The second ingredient is a method we developed in this paper to achieve the stated independence of L^{1}(0, t) and B(a, u) in the theorem.

As a consequence of the above results, one can conclude various limit distributions and laws of the iterated logarithm, such as

\frac{L(a, t) − L(0, t)}{2\sqrt{aL(0, t)}} →^{D} N_{1},  t → ∞  for any a > 0,  (2.3)
\frac{L(a, t) − L(0, t)}{2a^{1/2}t^{1/4}} →^{D} N_{1}|N_{2}|^{1/2},  t → ∞  for any a > 0, (2.4)
lim sup_{t→∞} \frac{L(a, t) − L(0, t)}{2\sqrt{2aL(0, t) log log t}} = 1  a.s.  for any a > 0,  (2.5)

lim sup_{t→∞} \frac{L(a, t) − L(0, t)}{a^{1/2}t^{1/4}(log log t)^{3/4}} = \frac{2^{9/4}}{3^{3/4}}  a.s.  for any a > 0, (2.6)
lim sup_{t→∞} sup_{0<a<a^{∗}t^{δ}} \frac{L(a, t) − L(0, t)}{2\sqrt{2a^{∗}t^{δ}L(0, t)(log log t)}}
= lim sup_{t→∞} sup_{0<a<a^{∗}t^{δ}} \frac{3}{4}6^{−1/4} \frac{L(a, t) − L(0, t)}{(a^{∗}t^{δ})^{1/2}t^{1/4}(log log t)^{3/4}} = 1  a.s. (2.7)

for any a^{∗} > 0 and 0 ≤ δ < 7/200, →^{D} denoting convergence in distribution.

In fact, (2.3) and (2.4) also follow from Theorem B. However the rest of the above statements do not follow from any weak invariance principle. (2.5) and (2.6) were proved directly by Csáki and Földes [28]. An important step in attaining the above strong theorems was the following result which proved to be important in its own right; If W_{1}(·) is a standard Wiener processes and L_{2}(·, ·) is a Wiener local time, independent of W_{1}, then

lim sup_{t→∞} \frac{W_{1}(L_{2}(0, t))}{t^{1/4}(log log t)^{3/4}} = \frac{2^{5/4}}{3^{3/4}}  a.s.  (2.8)

2.3 Additive functionals. Let us consider a sequence of i.i.d. random variables X_{i}, i = 1, 2... taking values on the integer lattice ℤ. Put S_{0} = 0, S_{n} = X_{1} + X_{2} + ... + X_{n}. Let us denote the local time of the random walk S_{n} by ξ(x, n) := #{k : 0 < k ≤ n, S_{k} = x}. Define the additive functional A_{n} as

A_{n} := ∑_{i=1}^{n} f(S_{i}) = ∑_{x=−∞}^{∞} f(x)ξ(x, n),  (2.9)

where f(x), x ∈ ℤ is a real valued function. Clearly in the special case f(a) = 1, f(0) = −1, and f(x) = 0 otherwise, A_{n} = ξ(a, n) − ξ(0, n). Let us denote

f̄:= ∑_{k=−∞}^{∞} f(k).

The so called first-order results on A_{n} are establishing the following observation: If f̄ ≠ 0 then the asymptotic behaviour of A_{n} with appropriate normalization is the same as the behaviour of f̄L(0, n). The interested reader should consult Kallianpur and Robbins [60], Darling and Kac [44], Skorokhod and Slobodenyuk [78] and Borodin [7] to see the history of these first order limit results. However, we were interested in the so-called second order limit theorems for A_{n} which are focused on the behaviour of A_{n} when f̄ = 0 (clearly this is the case which contains the increments of the local time). The history of this topic goes back to the above mentioned famous result of Dobrushin [48] that we now formulate as the following theorem:

Theorem C (Dobrushin [48]) Assume that P(X_{1} = +1) = P(X_{1} = −1) = 1/2 and define the additive functional as in (2.9). If f(x), x ∈ ℤ has finite support and f̄ = 0, then

lim_{n→∞} P (\frac{A_{n}}{dn^{1/4}} < x) = P(N_{1}\sqrt{|N_{2}|} < x),  (2.10)

where N_{1} and N_{2} are two independent standard normal variables, and

d^{2} = 4 ∑_{k=−∞}^{∞} kf^{2}(k) + 8 ∑_{−∞<i<j<∞}∑ if(i)f(j) − ∑_{k=−∞}^{∞} f^{2}(k).

This result has several generalizations. The corresponding functional version was given by Kasahara [61] and Borodin [6].

Similarly to the discrete case, one can consider the additive functional of a standard Wiener process. Let g(x) be an integrable function on the real line and consider

G(t) := ∫_{0}^{t} g(W (s)) ds = ∫_{−∞}^{∞} g(x)L(x, t) dx, t ≥ 0.

Results on the additive functional G(t) are parallel to the results on A_{n}. Let us quote the functional form of the limit theorem given by Papanicolaou et al. [69], Ikeda and Watanabe [58], Kasahara [61] and Borodin [6]. They proved (under somewhat different assumptions on g) that

λ^{−1/4} (G(λt) − ḡL(0, t)) →^{w} σW_{1}(L_{2}(t))  as λ → ∞,  (2.11)

where ḡ := ∫_{−∞}^{∞} g(x) dx, W_{1} is a standard Wiener process, L_{2} is a Wiener local time at zero, such that W_{1} and L_{2} are independent, and σ is an explicitly given constant.

Our goal was to prove the strong approximation version of (2.11) for the random walk and the Wiener case as well. In both cases the method developed in [15] proved to be the appropriate tool to achieve our results in [16]. To avoid being repetitious we only quote the result of [16] in the case of a Wiener process.

Theorem 2.3 Assume that f(x) is an integrable function on ℝ and

∫_{−∞}^{∞} |x|^{1+δ}|f(x)| dx < ∞  for some δ > 0.  (2.12)

Then on a suitable probability space one can define a standard Wiener process W (t) with two other standard Wiener processes W_{1}(t) and W_{2}(t) such that

• W_{1}(t) and W_{2}(t) are independent,

• | ∫_{0}^{t} f(W (s)) ds − f̄L(0, t) − σW_{1}(L_{2}(0, t))| = O(t^{τ/2}) a.s., t → ∞,
• |L(0, t) − L_{2}(0, t)| = O(t^{κ/4}) a.s., t → ∞,

where f̄:= ∫_{−∞}^{∞} f(x) dx and

σ^{2} := 4 ∫_{−∞}^{0} (∫_{−∞}^{x} f(y) dy)^{2} dx + 4 ∫_{0}^{∞} (∫_{x}^{∞} f(y) dy)^{2} dx,  (2.13)

L(x, t) and L_{2}(x, t) resp., are the local times of W (·) and W_{2}(·) resp., and κ, τ are any numbers satisfying κ < 4τ,

\frac{1}{4} + \frac{1}{2(2 + δ)} < τ < \frac{1}{2}, \frac{7}{4} + \frac{1}{2(2 + δ)} < κ < 2.  (2.14)

Remark It was shown in [22] that the condition (2.12) can be relaxed to

∫_{−∞}^{∞} |x|^{1/2+δ}|f(x)| dx < ∞  for some δ > 0.  (2.15)

As a consequence of the above theorem we get the following LIL type result for the additive functionals.

Under the conditions of the above theorem we have

lim sup_{t→∞} \frac{| ∫_{0}^{t} f(W (s)) ds − f̄L(0, t)|}{t^{1/4}(log log t)^{3/4}} = σ \frac{2^{5/4}}{3^{3/4}} a.s.

Both of the above two results and their random walk counterparts became the starting point of many further investigations in this direction. The method of proof was successfully used to generalize these results for the additive functionals of various processes. Extensions were given for Markov chains by Csáki and Csörgő [13], for diffusions by Csáki and Salminen [32], for Markov processes by Eisenbaum and Földes [50], for simple symmetric random walk on the plane by Csáki et al. [30]. In [29] additive functionals of more general random walks in one and two dimensions were strongly approximated under various conditions. As a consequence of these results one always gets both LIL-type and weak convergence results.

2.4 Principal value of Brownian local time. An important special type of additive functionals is the following

Y_{α}(t) := ∫_{0}^{t} \frac{ds}{W^{α}(s)} = ∫_{0}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx,  (2.16)

where the integral ∫_{0}^{t} ds/W^{α}(s) (notation: z^{α} = |z|^{α} sgn(z)) is in the sense of Cauchy’s principal value). Strictly speaking, the first integral is defined as Cauchy’s principal value for 1 ≤ α < 3/2 and as Riemann integral for α < 1. The investigation of the process Y_{1}(t) which is called the Cauchy principal value of the Brownian local time goes back at least to Itô and McKean [59] and has become very active since the late 70s, due to applications in various branches of stochastic analysis. For example, it is a natural example in Fukushima [55] theory for Dirichlet processes and zero-energy additive functionals. Also, the principal values of Brownian local times are the key ingredient in establishing Bertoin [5]’s excursion theory for Bessel processes of small dimensions. For a detailed account on these facts and general properties of principal values of local times, we refer to the collection of research papers in Yor [84] and to the survey paper by Yamada [82].

Hu and Shi [56] proved the following LILs for the local and global behaviour of the principal value:

Theorem D (Hu and Shi [56])

lim sup_{t→∞} \frac{Y_{1}(t)}{\sqrt{t log log t}} = 2\sqrt{2}  a.s.  (2.17)

and

lim sup_{h→0} \frac{Y_{1}(h)}{\sqrt{h log log(1/h)}} = 2\sqrt{2} a.s.

This result supports the common belief that the principal value process Y_{1}(t) is very similar in behaviour to the Brownian motion. To explore further this phenomenon we investigated some path properties of Y_{α}(·) and especially Y_{1}(·). We studied the modulus of continuity and large increment properties (including the LIL) of Y_{α}(·), as well as appropriate properties of a simple symmetric random walk along these lines. Due however to lack of precise distributional properties of Y_{α}(·), when α ≠ 1, we could not obtain the desirable exact constants, though the rates we established are optimal. In our first theorem [22] we proved the upper bounds for the LIL, large increments and modulus of continuity.

Theorem 2.4 Under Condition A for 0 < α < 3/2 we have

lim sup_{t→∞} \frac{sup_{0≤u≤t−a_{t}} sup_{0≤s≤a_{t}} |Y_{α}(u + s) − Y_{α}(u)|}{a_{t}^{1−α/2} (log(t/a_{t}) + log log t)^{α/2}} ≤ c_{1}(α) a.s. (2.18)

lim sup_{h→0} \frac{|Y_{α}(h)|}{h^{1−α/2}(log log(1/h))^{α/2}} ≤ c_{1}(α)  a.s.  (2.19)
lim sup_{h→0} \frac{sup_{0≤t≤1−h} sup_{0≤s≤h} |Y_{α}(t + s) − Y_{α}(t)|}{h^{1−α/2} (log(1/h))^{α/2}} ≤ c_{1}(α)  a.s. (2.20)

Here, the constant c_{1}(α) is given by

c_{1}(α) = \frac{3 · 2^{7α/6}}{α^{2α/3}(3 − 2α)^{1−α/3}(2 − α)^{α/3}} .  (2.21)

Remark In the particular case a_{t} = t we get

lim sup_{t→∞} \frac{|Y_{α}(t)|}{t^{1−α/2}(log log t)^{α/2}} ≤ c_{1}(α)  a.s.  (2.22)

Concerning the constant in LIL, we have the following result [22].

Theorem 2.5 For 0 < α < 3/2, there exists a finite positive constant c_{2}(α) such that

lim sup_{t→∞} \frac{|Y_{α}(t)|}{t^{1−α/2}(log log t)^{α/2}} = c_{2}(α) ∈ [ 2^{3α/2} Γ(3 − α), c_{1}(α) ]  a.s. (2.23)

The LIL holds true also for random walks via the following invariance principle [22]. Let S_{i}, i = 1, 2, … be a simple symmetric random walk on the line, starting from 0, and let ξ(x, n) be its local time. Define

G_{α}(n) := ∑_{k=1}^{n} \frac{1_{[S_{k}≠0]}}{S_{k}^{α}} = ∑_{i=1}^{∞} \frac{ξ(i, n) − ξ(−i, n)}{i^{α}}.  (2.24)

Theorem 2.6 On a suitable probability space one can define a Wiener process {W (t), t ≥ 0} and a simple symmetric random walk {S_{n}, n = 1, 2, … } such that for any 0 < α < 3/2 and sufficiently small ε > 0 we have

|Y_{α}([t]) − G_{α}([t])| = o(t^{1−α/2−ε})  a.s.,  (2.25)

as t → ∞.

As a consequence of our Theorem 2.6, the LILs in (2.17), (2.22) and (2.23) remain true if Y_{α} is replaced by G_{α}.

As it is easily seen, Y_{α} is not defined for α ≥ 3/2. In this case, we considered instead the process

Z_{α}(t) := ∫_{0}^{t} \frac{1_{[|W (s)|≥1]}}{W^{α}(s)} ds = ∫_{1}^{∞} \frac{L(x, t) − L(−x, t)}{x^{α}} dx.  (2.26)

This is a ”nice” additive functional, for which Theorem 2.3 can be applied. The limit process associated with such functionals is V (t) = W_{1}(L_{2}(t)), where W_{1}(·) is a standard Wiener process and L_{2}(·) is a Wiener local time at zero, independent of W_{1}.

Considering the special case of Y_{1}, in [21] we characterized the modulus of continuity as follows.

Theorem 2.7 With probability one,

lim_{h→0} sup_{0≤t≤1} sup_{0≤s≤h} \frac{|Y_{1}(t + s) − Y_{1}(t)|}{\sqrt{h log(1/h)}} = 2.

Remark Y_{1}(t)/\sqrt{2} and W (t) have the same moduli of continuity (and the same remark applies to our next theorem below). We have already seen that Y_{1}(t)/2 and W (t) satisfy the same LIL. Heuristically speaking, that a factor \sqrt{2} is missing in the modulus of continuity, this comes from the fact that the Hausdorff dimension of the zero set of W is 1/2.

As to the large increments of Y_{1}(·), in [21] we proved

Theorem 2.8 Under Conditions A and B we have

lim_{t→∞} sup_{0≤u≤t−a_{t}} sup_{0≤s≤a_{t}} \frac{|Y_{1}(u + s) − Y_{1}(u)|}{\sqrt{a_{t} log(t/a_{t})}} = 2 a.s.

Remark Recently Csáki and Hu [31] was able to fill the gap in the above increment results by showing that Condition A is enough to get a limsup.

To look at the the corresponding two-dimensional question, let

{W(t) := (W_{1}(t), W_{2}(t)), t ≥ 0}

be a two-dimensional Wiener process, where W_{1}(t) and W_{2}(t) are two independent one-dimensional Wiener processes, with W_{1}(0) = W_{2}(0) = 0. Put

R(t) := ∥W(t)∥ = \sqrt{W_{1}^{2}(t) + W_{2}^{2}(t)} .

It is well-known that {R(t), t ≥ 0} is a two-dimensional Bessel process. In [23] we were interested in the additive functional

Z̃_{α}(t) := ∫_{0}^{t} \frac{ds}{R^{α}(s)},  (2.27)

the critical case being α = 2 (instead of 3/2). It can be seen that the integral in (2.27) converges for α < 2, but diverges for α ≥ 2 almost surely. In the latter case we defined the modified process

Z_{α}^{∗}(t) := ∫_{0}^{t} \frac{1}{R^{α}(s)}1_{\{R(s)≥1\}} ds.  (2.28)

Considering the random walk counterpart, let {S_{n}, n = 1, 2, … } be a simple symmetric random walk on the integer lattice ℤ^{2}, i.e. S_{n} = ∑_{k=1}^{n} X_{k}, where the random variables X_{i}, i = 1, 2, … are i.i.d., with

P(X_{1} = (0, 1)) = P(X_{1} = (0, −1)) = P(X_{1} = (1, 0)) = P(X_{1} = (−1, 0)) = \frac{1}{4}.

We also proposed to study the discrete process

U_{α}(n) := ∑_{k=1}^{n} \frac{1}{∥S_{k}∥^{α}} 1_{\{S_{k}≠0\}}.  (2.29)

Define

ξ(x, n) := #{k; 1 ≤ k ≤ n, S_{k} = x},

for any lattice point x ∈ ℤ^{2}. This is the local time process of {S_{n}, n = 1, 2, … }.

First we considered the case 0 < α < 2, for which we managed to show that the processes Z̃_{α}(·) and 2^{−α/2}U_{α}(·) are close enough to each other to share many of their properties. Based on some results of Revuz and Yor [72], Azencott [1] and Borovkov and Mogulskii [8], in [23] we proved the following lim sup and lim inf results for both of these processes.

Theorem 2.9 For 2/3 ≤ α < 2 we have

lim sup_{t→∞} \frac{Z̃_{α}(t)}{t^{1−α/2}(log log t)^{α/2}} = lim sup_{n→∞} \frac{2^{−α/2}U_{α}(n)}{n^{1−α/2}(log log n)^{α/2}} = K_{1}(α) a.s.

and

lim inf_{t→∞} t^{−(1−α/2)}(log log t)^{α/2}Z̃_{α}(t)

= lim inf_{n→∞} n^{−(1−α/2)}(log log n)^{α/2}2^{−α/2}U_{α}(n) = K_{2}(α) a.s.,

whith certain positive constants K_{1}(α) and K_{2}(α).

On the other hand, it turned out that when α > 2 then the two processes have to be investigated separately. However both processes, suitably centered, are close to certain iterated processes. We only quote the results from [23] in random walk case, and some of its consequences, parallel results are true for Z_{α}^{∗}(t).

Theorem 2.10 Let α > 2. There exists a probability space where one can define

• a two-dimensional simple symmetric random walk {S_{n}, n = 1, 2, … } with its local time {ξ(x, n), x ∈ ℤ^{2}, n = 1, 2, … }, and with the corresponding additive functional {U_{α}(n), n = 1, 2, … } as in (2.29);

• a process {ξ^{(1)}(0, n), n = 1, 2, … } =^{D} {ξ(0, n), n = 1, 2, … };

• a standard Wiener process {W (t), t ≥ 0}, independent of {ξ^{(1)}(0, n), n =

1, 2, … };

such that, for some ε > 0, as n → ∞,

• U_{α}(n) − f̄_{α}ξ(0, n) = σ_{α}W (ξ^{(1)}(0, n)) + O(log^{1/2−ϵ} n) a.s.,
• ξ(0, n) = ξ^{(1)}(0, n) + O(log^{1−ϵ} n) a.s.

where f̄_{α} := ∑_{x∈ℤ^{2}\\{0\}} \frac{1}{∥x∥^{α}} , σ_{α} := \sqrt{Var(U_{α}(ρ_{1}))} .

The above theorem has both weak and strong implications.

Theorem 2.11 For α > 2 we have

\frac{πU_{α}(n)}{f̄_{α} log n} →^{D} |E|, n → ∞,  (2.30)
\frac{U_{α}(n) − f̄_{α}ξ(0, n)}{σ_{α}\sqrt{log n}}\sqrt{2π} →^{D} E, n → ∞,  (2.31)
lim sup_{n→∞} \frac{U_{α}(n)}{log n log_{3} n} = \frac{f̄_{α}}{π}  a.s.,  (2.32)
lim sup_{n→∞} \frac{U_{α}(n) − f̄_{α}ξ(0, n)}{\sqrt{log n} log_{3} n} = \frac{σ_{α}}{\sqrt{2π}}  a.s.,  (2.33)

where E is a bilateral exponential random variable with density e^{−|x|}/2, x ∈ ℝ, and |E| is exponential with parameter 1.

2.5 Integral functionals. In [20] we studied the following two types of integral functionals of geometric stochastic processes which are of interest in financial modelling:

A(t) := ∫_{0}^{t} exp(X(u)) du,  B(t) := ∫_{0}^{∞} exp (Y (u) − \frac{u^{α}}{t} ) du,  (2.34)

for 0 ≤ t < ∞.

We managed to show, that under fairly general conditions on X(t) and Y (t) respectively, log A(t) and log B(t) behave like sup_{0≤u≤t} X(u) and sup_{0≤u<∞}(Y (u)− u^{α}/t). We only quote our first strong invariance theorem which deals with X(t).

Theorem 2.12 Let the stochastic process {X(t); 0 ≤ t < ∞} have almost surely continuous sample paths, P(X(0) = 0) = 1 and put

Z(t) := log A(t)  and  U(t) := sup_{0≤u≤t} X(u).

Assume that for the increment of X(t) we have

sup_{0≤s≤t−a_{t}} sup_{0≤v≤a_{t}} |X(s + v) − X(s)| = O(r(t, a_{t})) a.s.

as t → ∞, with some non-decreasing a_{t} (1 ≤ a_{t} ≤ t) and rate r(t, a_{t}). Then as t → ∞,

|Z(t) − U(t)| = O(r(t, a_{t}) + log t) a.s.

We applied these strong approximation theorems for a number of processes, such as Wiener process, fractional Brownian motion, Gaussian processes, and diffusion processes.

3. Iterated processes, and their local times

3.1 Iterated processes. K. Burdzy [9] proposed to investigate the process

Z(t) := {W_{1}(W_{2}(t)), 0 ≤ t < ∞},  (3.1)

where {W_{1}(t), t ∈ ℝ} and {W_{2}(t), t ≥ 0} are two independent standard Brownian motions. He called this process an iterated Brownian motion (IBM), and proved the following LIL:

Theorem E (Burdzy [9])

lim sup_{t→0} \frac{Z(t)}{t^{1/4}(log log(1/t))^{3/4}} = \frac{2^{5/4}}{3^{3/4}}  a.s.  (3.2)

Closely related processes to Z(t) are

H(t) := {W_{1}(|W_{2}(t)|), 0 ≤ t < ∞}  (3.3)

and

V (t) := {W_{1}(L_{2}(t) ), 0 ≤ t < ∞},

where L_{2} is a Wiener local time at 0, independent of W_{1}.

In 1993-94 many people got interested in these processes, one should consult [17] for proper references. In the above Theorem E we have an LIL for t → 0 and in (2.8) we have an LIL for the process V (t) as t → ∞ with the very same constant. The latter result combined with a famous result of Paul Lévy, mentioned earlier, implies that the same is true for the process Y (t) := W_{1}(max_{0≤s≤t} W_{2}(s)), and H(t), as well. It is easy to see that

\frac{V (t)}{t^{1/4}} =^{D} \frac{H(t)}{t^{1/4}} =^{D} \frac{Y (t)}{t^{1/4}} =^{D} N_{1}\sqrt{|N_{2}|},  (3.4)

where N_{1} and N_{2} are two independent standard normal variables. We have seen this distribution to appear in Dobrushin’s theorem (2.10) and in (2.4) as well. In all of these results we have in the above sense an iterated process created from a pair of independent processes. This gave us the idea that there must be a common way to investigate these three processes and started to study these iterated processes more closely. To introduce our first result in this direction, we recall the following definition: Let S be the Strassen class of functions, i.e., S ⊂ C[0, 1] is the class of absolutely continuous functions (with respect to the Lebesgue measure) on [0, 1] for which

f(0) = 0  and  ∫_{0}^{1}(f′(x))^{2} dx ≤ 1.  (3.5)

The set of ℝ^{2}-valued, absolutely continuous functions

{(g(y), h(x)), 0 ≤ y ≤ 1, 0 ≤ x ≤ 1}  (3.6)

for which g(0) = h(0) = 0 and

∫_{0}^{1}(g′(y))^{2} dy + ∫_{0}^{1}(h′(x))^{2} dx ≤ 1  (3.7)

will be called Strassen class S^{2}.

Now let C_{0}[0, 1] ⊂ C[0, 1] be the set of continuous functions f(·) on [0, 1] for which f(0) = 0. Let A be an operator on C_{0}[0, 1], satisfying (C.1) Acf = c^{ρ}Af (ρ ≥ 1, c > 0), (C.2) Af ≥ 0, (C.3) Af ∈ C_{0}[0, 1], (C.4) A is locally uniformly continuous on C_{0}[0, 1].

Some of our examples for Af(x) are the following: |f(x)|, max_{0<y≤x} f(y), and max_{0<y≤x} |f(y)|.

Theorem 3.1 Let W_{1}(·) and W_{2}(·) be two independent standard Wiener processes starting from zero, and let A be an operator satisfying conditions (C.1)–(C.4). Then for 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, the limit set of the vector

(\frac{W_{1}(yAW_{2}(xT))}{T^{ρ/4}(2 log log T)^{(ρ+2)/4}} , \frac{W_{2}(xT)}{(2T log log T)^{1/2}} )  (3.8)

is (g(yAh(x)), h(x)), where (g, h) ∈ S^{2}.

This theorem gives an easy way to show the above LILs, and it has many more consequences. Here we mention only one of them as an example.

Theorem 3.2 For 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, ρ ≥ 1 we have

lim sup_{T →∞} \frac{W_{1}(yAW_{2}(xT))}{T^{ρ/4}(2 log log T)^{(ρ+2)/4}} = 2^{1/2}λ_{x}^{1/2}y^{1/2}ρ^{ρ/4}(ρ + 2)^{−(ρ+2)/4}  a.s., (3.9)

where λ_{x} := sup_{f∈S} Af(x).

Using the invariance principle of Komlós et. al. [65], [66] it was also shown that if we construct an iterated random walk U(n) := S_{1}(|S_{2}(n)|) from two independent simple symmetric random walks S_{1} and S_{2}, then the iterated random walk is close to the iterated Brownian motion.

Theorem 3.3 On a rich enough probability space (Ω, F, P) one can construct an iterated Wiener process {H(t), t ≥ 0} and an iterated random walk {U(n), n = 1, 2, … } such that

max_{1≤k≤n} |U(k) − H(k)| = O(log n)  a.s., n → ∞  (3.10)

This theorem enables us to carry over many limit theorems from the IBM processes to the iterated random walk.

3.2 Local time and occupation time. In [18] we defined the local time L^{∗}(x, t) of H(t) = W_{1}(|W_{2}(t)|) as follows:

L^{∗}(x, t) := ∫_{0}^{∞} \overline{L}_{2}(s, t) d_{s}L_{1}(x, s),  x ∈ ℝ, t ≥ 0.  (3.11)

where \overline{L}_{2}(·, ·), L_{1}(·, ·) are the local time processes of |W_{2}(·)| and W_{1}(·), respectively. In particular, \overline{L}_{2}(x, t) := L_{2}(x, t) + L_{2}(−x, t), x ≥ 0, where L_{2}(·, ·) is the local time process of W_{2}(·). At about the same time Burdzy and Khoshnevisan [10] studied the local time of the process Z(t) = W_{1}(W_{2}(t)) and proved its Hölder continuity. Concerning L^{∗}, we established its joint continuity and studied its path behaviour aiming at the four classical Lévy classes of functions. However, these results are far from being optimal yet, and leave open many problems for further considerations, including even that of proving an LIL for L^{∗}(x, t) at x = 0. Indeed, a systematic study of the fine analytic properties of the process {L^{∗}(x, t), x ∈ ℝ, t ≥ 0} along the lines of those of the classical Brownian local time of P. Lévy seems to be a challenging problem. For further liminf type results we refer to [75].

We also considered the corresponding iterated random walk U(n) = S_{1}(|S_{2}(n)|) and defined its local time by

ξ^{∗}(x, n) := #{k : 0 < k ≤ n, U(k) = x}.

Then we established that on an appropriate probability space, as t → ∞,

sup_{x∈ℤ} |ξ^{∗}(x, t) − L^{∗}(x, t)| = O(t^{11/16+ε})  a.s.  (3.12)

which implies that all the above mentioned Lévy class type results are inherited by ξ^{∗}(x, t).

It is quite interesting to note that even though Z(t) = W_{1}(|W_{2}(t)|) and V (t) = W_{1}(L_{2}(t)) share many properties, the investigation of their respective local times reveals how different they really are. We started our investigation with studying the occupation time of V (t), and it turned out that we must confine our attention to it as V (t) has no local time. Actually because of the non-Markovian nature of V (t) it is more appropriate to talk about the non-existence of its occupation density. Another surprise was to realize that we were unable to establish a strong approximation result similar to (3.10), hence each result had to be established separately for V (t) and the corresponding iterated random walk. For simplicity, here we only explain how to define the occupation time of the iterated random walk. Let S_{1}(·) and S_{2}(·) be two independent simple symmetric random walks as above and denote the local time at zero of S_{2}(·) by ξ_{2}(n). In the spirit of V (t) = W_{1}(L_{2}(t)) we define R(n) := S_{1}(ξ_{2}(n − 1)) and the corresponding occupation time of R(n) is defined as

ξ̃(r, n) := #{k : 1 ≤ k ≤ n, R(k) = r}.  (3.13)

Then clearly

ξ̃(r, n) = ∑_{k=1}^{n} I{S_{1}(ξ_{2}(k − 1)) = r}

= ∑_{0≤s≤ ξ_{2}(n−1)}(ρ_{2}(s + 1) ∧ n − ρ_{2}(s))I{S_{1}(s) = r},  (3.14)

where I(·) is an indicator function and 0 = ρ_{2}(0) < ρ_{2}(1) < ... are the consecutive return epochs to zero of our second walk S_{2}(·). Thus we have

ξ̃(r, ρ_{2}(n)) = ∑_{s=1}^{n−1}(ρ_{2}(s + 1) − ρ_{2}(s))I{S_{1}(s) = r}.  (3.15)

Further studying (3.15) led us to the right way of interpreting ξ̃(r, n) and the occupation time of V (t) as well. It turned out that these occupation times has interesting limiting distributions. Here we only mention the following one.

Theorem 3.4 For any fixed integer r ≥ 0, as n → ∞, we have

\frac{ξ̃(r, n)}{n^{1/2}} →^{D} N_{1}^{2}|N_{2}|T_{1} =^{D} C^{2}|N_{2}|,  (3.16)

where N_{1} and N_{2} are independent standard normal random variables that are also independent of the stable (1/2) random variable T_{1}, and C is a standard Cauchy random variable independent of N_{2}.

As it was indicated above, the Lévy class type results for the occupation time of V (s) and R(s) were separately established. For further LIL-type results for ξ̃ we refer to Révész [71].

4. Empirical processes

4.1 Level crossings. Let U_{1}, U_{2}, … be a sequence of i.i.d. random variables uniformly distributed in (0, 1). The empirical process α_{n} based on the first n observations is defined as:

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

where F_{n}(t) := #{i : 1 ≤ i ≤ n, U_{i} ≤ t}/n, 0 ≤ t ≤ 1, is the empirical distribution function.

Consider the (normalized) level crossings of the empirical process:

𝕃_{n}(x) := n^{−1/2}# {t ∈ [0, 1] : α_{n}(t) = x} , x ∈ ℝ.

Let us also define the maximal level crossings:

𝕃_{n}^{∗} := sup_{x∈ℝ} 𝕃_{n}(x).

We recall the following results.

Theorem F (Révész [70]) Almost surely,

lim sup_{n→∞} \frac{𝕃_{n}(0)}{(2 log log n)^{1/2}} = 1.

Theorem G (Bass and Khoshnevisan [3]) We have,

lim sup_{n→∞} \frac{𝕃_{n}^{∗}}{(2 log log n)^{1/2}} = 1 a.s.
lim inf_{n→∞}(log log n)^{1/2} 𝕃_{n}^{∗} = 2^{1/2}π a.s.

In [41], we proved the following results.

Theorem 4.1 Almost surely, as n → ∞,

\{ (\frac{𝕃_{n}(0)}{(2 log log n)^{1/2}} , \frac{𝕃_{n}^{∗}}{(2 log log n)^{1/2}} ) \}

is relatively compact, with limit set equal to {(x, y) : 0 ≤ x ≤ y ≤ 1}.

Theorem 4.2 We have,

lim sup_{n→∞} \frac{1}{(log log n)^{1/2}} ∫_{−∞}^{∞}(𝕃_{n}(x))^{2}dx = (\frac{2}{3})^{1/2} a.s.,

lim inf_{n→∞}(log log n)^{1/2} ∫_{−∞}^{∞}(𝕃_{n}(x))^{2}dx = (\frac{2|a_{1}|}{3})^{3/2} a.s.,

where a_{1} < 0 is the largest real zero of the Airy function Ai(·).

Theorem 4.3 For any p ≥ 3, with probability one,

lim sup_{n→∞} \frac{1}{(log log n)^{(p−1)/2}} ∫_{−∞}^{∞}(𝕃_{n}(x))^{p}dx

= 2^{(p+1)/2}(p − 1)^{(p−1)/2}(p + 1)^{(p−3)/2} (B(\frac{1}{2}, \frac{1}{p − 1}))^{1−p} ,

where B(·, ·) stands for the usual beta function.

4.2 Vervaat error process. Let F_{n}(t) be as before the empirical distribution function from a uniform [0,1] sample. 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 [2], Kiefer [63], [64]). This process enjoys some remarkable asymptotic properties, which are of interest in statistical quantile data analysis (cf., e.g., Csörgő [33], Shorack and Wellner [77]). We summarize the most relevant results of Kiefer [63], [64], Shorack [76], Deheuvels and Mason [46] in the following theorem. For further developments one can consult Deheuvels and Mason [47], Einmahl [49], Csörgő and Szyszkowicz [42].

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

n^{1/4}R_{n}(t) →^{D} (t(1 − t))^{1/4}N_{1}(|N_{2}|)^{1/2}, n → ∞,  (4.1)
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.,  (4.2)

where N_{1} and N_{2} are independent standard normal variables. Also,

lim_{n→∞} n^{1/4}(log n)^{−1/2} \frac{∥R_{n}∥}{(∥α_{n}∥)^{1/2}} = 1  a.s.,  (4.3)

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

Via using the usual and the other laws of the iterated logarithm for α_{n}, (4.3) immediately implies

lim sup_{n→∞} n^{1/4}(log n)^{−1/2}(log log n)^{−1/4}∥R_{n}∥ = 2^{−1/4}  a.s.,  (4.4)
lim inf_{n→∞} n^{1/4}(log n)^{−1/2}(log log n)^{1/4}∥R_{n}∥ = \frac{π^{1/2}}{8^{1/4}}  a.s.,  (4.5)

while a direct application of (4.3) 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 → ∞.  (4.6)

Nevertheless, the following result, which one can immediately conclude also by combining (4.1) with (4.6), is true, and it was first formulated and proved directly by Vervaat [81].

Theorem I (Vervaat [81]) The statement

a_{n}R_{n} →^{w} Y, n → ∞

cannot hold true in the space D[0, 1] (endowed with the Skorokhod 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 H and I, it is of interest to see the asymptotic behaviour of the Bahadur–Kiefer process possibly in other norms as well. In this regard the following theorem was proved in [39], [40].

Theorem 4.4 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.,  (4.7)

where

c_{0}(p) := (E|N_{1}|^{p})^{1/p} = \sqrt{2} (\frac{Γ((p + 1)/2)}{\sqrt{π}})^{1/p} ,  (4.8)

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

Vervaat’s ([81]) elegant proof of Theorem I was based 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)  (4.9)

to B^{2}, the square of a Brownian bridge, as well as a functional LIL for V_{n}, via proving the following theorem.

Theorem J (Vervaat [80], [81]) We have

lim_{n→∞} (log log n)^{−1}∥V_{n} − α_{n}^{2}∥ = 0  a.s.  (4.10)
lim_{n→∞} ∥V_{n} − α_{n}^{2}∥ = 0  in probability.  (4.11)

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

V_{n} →^{D} B^{2}, n → ∞.  (4.12)

We call the process V_{n} of (4.9) the uniform Vervaat process.

Bahadur [2] 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 above, 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.

Similarly, one can consider the process

Q_{n}(t) := V_{n}(t) − α_{n}^{2}(t), 0 ≤ t ≤ 1,  (4.13)

that appears in both statements (4.10) and (4.11) of Theorem J. Then Q_{n} is the remainder term in the following representation

V_{n} = α_{n}^{2} + Q_{n}  (4.14)

of the uniform Vervaat process V_{n} in terms of the square of the empirical process. It is well-known (cf. Zitikis [85], for details and references) that the remainder term Q_{n} in (4.14) 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 H and 4.4, one suspects that there should be substantial differences between the asymptotic pointwise, sup- and L_{p}-norm behaviour of the process Q_{n}. Indeed, Csörgő and Zitikis [43] established the following strong convergence result for ∥Q_{n}∥_{p}.

Theorem K (Csörgő and Zitikis [43]) 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.,  (4.15)

where c_{0}(p) is defined in (4.8).

For a comparison of this result to that of Theorem 4.4, as well as for that of their consequences, we refer to Csörgő and Zitikis [43], who have also conjectured that in sup-norm the analogue statement of (4.15) should be of the following form:

lim_{n→∞} b_{n}n^{1/4} \frac{∥Q_{n}∥}{(∥α_{n}∥)^{3/2}} = c  a.s.,  (4.16)

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

One of our aims in [24] was to prove that this conjecture is true with b_{n} = (log n)^{−1/2}. In addition, we also studied the pointwise behaviour of the Vervaat error process Q_{n}. We summarize our results in the following theorem, which parallels Theorem H concerning the process R_{n}.

Theorem 4.5 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_{1}(|N_{2}|)^{3/2}, n → ∞,  (4.17)
lim sup_{n→∞} \frac{n^{1/4}|Q_{n}(t)|}{(log log n)^{5/4}} = (t(1 − t))^{3/4} \frac{2^{11/4}3^{1/4}}{5^{5/4}}  a.s.,  (4.18)

where N_{1} and N_{2} 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.  (4.19)

As a consequence of this theorem, as well as that of Theorem K combined with (4.19), we have the following corollary, which confirms the above conjecture.

Corollary 4.1 The statement

a_{n}Q_{n} →^{w} 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 of (4.19) is the following corollary.

Corollary 4.2 We have

lim sup_{n→∞} n^{1/4}(log n)^{−1/2}(log log n)^{−3/4}∥Q_{n}∥ = \frac{2^{1/4}}{3^{1/2}} a.s.,

lim inf_{n→∞} n^{1/4}(log n)^{−1/2}(log log n)^{3/4}∥Q_{n}∥ = \frac{π^{3/2}}{3^{1/2}2^{5/4}} a.s.,

n^{1/4}(log n)^{−1/2}∥Q_{n}∥ →^{D} (4/3)^{1/2}∥B∥^{3/2}, n → ∞,

where B is a standard Brownian bridge.

5. Banach space valued stochastic processes

Let {Y (t), t ∈ ℝ} := {X_{k}(t), t ∈ ℝ, k = 1, 2, … } be a sequence of independent Ornstein-Uhlenbeck processes with coefficients γ_{k} and λ_{k}, i.e. X_{k} is a stationary, mean zero Gaussian process with EX_{k}(s) X_{k}(t) = (γ_{k}/λ_{k}) exp(−λ_{k}|t − s|). This process was introduced by Dawson [45] and its path properties were studied by Csörgő and Lin [34], [35], Fernique [52], [53], [54], Iscoe et al. [57], Schmuland [73], [74]. For further development we refer to the books [67], [68] and the references therein. The basic ingredient in these investigations was the celebrated inequality of Fernique [51] and its various extensions.

In [25] we studied the infinite series

X(t) := ∑_{k=1}^{∞} X_{k}(t), −∞ < t < ∞

and established certain moduli of continuity and large increment results.

In subsequent papers [11], [12], [26], [27] investigations on moduli of continuity and large increments were extended to Banach space valued processes, ℓ^{2}- and ℓ^{p}-valued processes in particular. We quote two general theorems from [26].

Theorem 5.1 Let {Γ(t), −∞ < t < ∞} be a stochastic process with values in a separable Banach space B with norm ∥ · ∥. Let P be the probability measure generated by Γ(·). Assume that Γ(·) is P-almost surely continuous with respect to ∥ · ∥ and that for |t| ≤ t_{0}, 0 < x^{∗} ≤ x, and 0 < h ≤ h_{0} there exist non-negative non-decreasing functions σ_{1}(h) and σ_{2}(h) such that

P(∥Γ(t + h) − Γ(t)∥ ≥ xσ_{1}(h) + σ_{2}(h)) ≤ K exp(−γx^{β})

with some K, γ, β > 0. Then we have

P( sup_{0≤t≤T} sup_{0≤s≤a} ∥Γ(t + s) − Γ(t)∥ ≥ x(σ_{1}(a) + σ_{1}(a, k))

+σ_{1}^{∗}(a, k) + σ_{2}(a) + σ_{2}(a, k))

≤ 4 (\frac{T}{a} + 1) K 2^{2^{k+1}} exp(−γx^{β})

for any 0 ≤ T ≤ t_{0}, 0 < a ≤ h_{0}, x ≥ x^{∗} and any k ≥ 3, where

σ_{1}(a, k) = 2^{3+(1/β)} ∫_{2^{k−3}}^{∞} \frac{σ_{1}(ae^{−z})}{z} dz,

σ_{2}(a, k) = 6 ∫_{2^{k−3}}^{∞} \frac{σ_{2}(ae^{−z})}{z} dz,

σ_{1}^{∗}(a, k) = 4 (\frac{14}{γ})^{1/β} β ∫_{2^{(k−2)/β}}^{∞} σ_{1}(ae^{−z^{β}}) dz.

Before stating the next theorem, we give a definition: A function f(x) is called quasi-increasing on (a, b) if there exists a positive number c such that

f(x) ≤ cf(y) for all a < x < y < b.

Theorem 5.2 Assume the conditions of Theorem 5.1 with t_{0} = ∞ and that σ_{1}(h) and σ_{2}(h) are continuous functions such that σ_{1}(h)/h^{α} and σ_{2}(h)/h^{α} are quasi-increasing for some α > 0. Let a_{T} and b_{T} be continuous functions of T such that

\frac{b_{T}}{a_{T}} + σ_{1}(a_{T}) + \frac{1}{σ_{1}(a_{T})} → ∞, T → ∞

and

lim sup_{T →∞} a_{T} ≤ h_{0}.

Then we have

lim sup_{T →∞} sup_{0≤t≤b_{T}} sup_{0≤s≤a_{T}} β_{T} ∥Γ(t + s) − Γ(t)∥ ≤ 1 a.s.,

where

β_{T} = σ_{1}(a_{T})A_{T} + σ_{2}(a_{T}),

A_{T} = (\frac{1}{γ} (log (1 + \frac{b_{T}}{a_{T}} ) + log log (σ_{1}(a_{T}) + \frac{1}{σ_{1}(a_{T})})))^{1/β} .

Acknowledgement The authors are indebted to the referees for their helpful remarks.

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