Couplings and Strong Approximations to Time-Dependent Empirical Processes Based on I.I.D. Fractional Brownian Motions

Abstract

We define a time-dependent empirical process based on n i.i.d. fractional Brownian motions and establish Gaussian couplings and strong approximations to it by Gaussian processes. They lead to functional laws of the iterated logarithm for this process.

Appendix

1.1 A Gaussian Coupling Inequality

Einmahl and Mason [7] pointed out in their Fact 2.2 that the Strassen–Dudley theorem (see Theorem 11.6.2 in Dudley [6]) in combination with a special case of Theorem 1.1 of Zaitsev [28] (also see the discussion after its statement) yields the following Gaussian coupling. Here $\left| \cdot \right| _{N}$, $N\ge 1$, denotes the usual Euclidean norm on ${\mathbb {R}}^{N}$.

Coupling inequality

Let $Y_{1},\ldots ,Y_{n}$ be independent mean zero random vectors in ${\mathbb {R}}^{N}$, $N\ge 1$, such that for some $b>0$,

$$\begin{aligned} \left| Y_{i}\right| _{N}\le b,\quad i=1,\dots ,n. \end{aligned}$$

If $(\varOmega , {\mathcal {T}}, P)$ is rich enough, then for each $\delta >0$, one can define independent normally distributed mean zero random vectors $Z_{1},\ldots ,Z_{n}$ with $Z_{i}$ and $Y_{i}$ having the same covariance matrix for $i=1,\ldots ,n$, such that for universal constants $C_{1}>0$ and $C_{2}>0$,

$$\begin{aligned} P\left\{ \left| \sum _{i=1}^{n}\left( Y_{i}-Z_{i}\right) \right| _{N}>\delta \right\} \le C_{1}N^{2}\exp \left( -\frac{C_{2}\delta }{N^{2}b} \right) . \end{aligned}$$

(77)

Remark 7

Actually, Einmahl and Mason did not specify the $N^{2}$ in (77) and they applied a less precise result given Theorem 1.1 in [29] with $N^{2}$ replaced by $N^{5/2}$; however, their argument is equally valid when based upon Theorem 1.1 in [28]. Zaitsev [28] remarks that the assumptions of Theorem 1.1 of [29] imply those of Theorem 1.1 of [28]. See, in particular, the paragraph right above Remark 1.1 in [28]. Also see equation (18) in [30].

1.2 Pointwise Measurable Classes

Definition

A class ${\mathcal {G}}$ of measurable real-valued functions defined on a measurable space $\left( S,{\mathcal {S}} \right)$ is pointwise measurable if there exists a countable subclass ${\mathcal {G}}_{\infty }$ of ${\mathcal {G}}$ such that we can find for any function $f\in {\mathcal {G}}$ a sequence of functions $\{f_{m}\}$ in ${\mathcal {G}}_{\infty }$ for which $\lim _{m\rightarrow \infty }f_{m}(x)=f(x)$ for all $x\in S$. For more about pointwise measurability, see pages 109–110 and Example 2.3.4 of van der Vaart and Wellner [26], as well as Section 8.2 of Kosorok [11].

We shall show here that the classes of functions ${\mathcal {F}}\left( K,\gamma \right)$, $K\ge 1$, of the form (28), where $0\le \gamma <1<T<\infty$, are pointwise measurable. Let ${\mathbb {Q}}$ denote the set of rational numbers. For any $K\ge 1$, consider the countable class ${\mathcal {F}}_{\infty ,K}$ of functions of $g\in {\mathcal {C}}\left[ 0,T\right] \rightarrow \left\{ 0,1\right\}$ indexed by $u,v\in \left[ \gamma ,T\right] \cap \mathbb {Q\cup }\left\{ \gamma ,T\right\} , y\in {\mathbb {Q}}$ defined by

$$\begin{aligned} 1\left\{ g\left( v\right) -Kf_{H}(\left| v-u\right| )\le y,\ g\in {\mathcal {C}}\left( K\right) \right\} , \end{aligned}$$

where ${\mathcal {C}}\left( K\right)$ is as in (27). Clearly for each $\left( t,x\right) \in {\mathcal {T}}(\gamma )=\left[ \gamma ,T\right] \times {\mathbb {R}}$, we can choose sequences $s_{m}$ and $t_{m}\in \left[ \gamma ,T\right] \cap {\mathbb {Q\cup }}\left\{ \gamma ,T\right\}$ such that $t_{m}\searrow t$ and $s_{m}\nearrow t$. Also we can select a sequence $y_{m}\in {\mathbb {Q}}\searrow x$. We see that each

$$\begin{aligned} 1\left\{ g\left( t_{m}\right) -Kf_{H}(\left| t_{m}-s_{m}\right| )\le y_{m},g\in {\mathcal {C}}\left( K\right) \right\} \in {\mathcal {F}}_{\infty ,K}. \end{aligned}$$

Moreover, if $g\in {\mathcal {C}}\left( K\right)$, then $g\left( t_{m}\right) -Kf_{H}(\left| t_{m}-s_{m}\right| )\le g\left( t\right)$ and $g\left( t_{m}\right) -Kf_{H}(\left| t_{m}-s_{m}\right| )\rightarrow g\left( t\right)$. Thus if $g\left( t\right) \le x$ and $g\in {\mathcal {C}}\left( K\right)$, then

$$\begin{aligned} 1\left\{ g\left( t_{m}\right) -Kf_{H}(\left| t_{m}-s_{m}\right| )\le y_{m},g\in {\mathcal {C}}\left( K\right) \right\} =1\rightarrow 1=h_{t,x}^{\left( K\right) }\left( g\right) , \end{aligned}$$

whereas if $g\left( t\right) >x$, then for some $\delta >0$, $g\left( t\right) >x+\delta$ and all large enough m,

$$\begin{aligned} g\left( t_{m}\right) -Kf_{H}(\left| t_{m}-s_{m}\right| )>x+\delta /2 \text { and }x+\delta /4>y_{m}. \end{aligned}$$

This says that eventually $g\left( t_{m}\right) -Kf_{H}(\left| t_{m}-s_{m}\right| )>y_{m}$ and thus

$$\begin{aligned} 1\left\{ g\left( t_{m}\right) -Kf_{H}(\left| t_{m}-s_{m}\right| )\le y_{m},g\in {\mathcal {C}}\left( K\right) \right\} =0=h_{t,x}^{\left( K\right) }\left( g\right) . \end{aligned}$$

Hence ${\mathcal {F}}\left( K,\gamma \right)$ is pointwise measurable with countable subclass ${\mathcal {F}}_{\infty ,K}$.

For any $\kappa >0$ and $K\ge 1$, let ${\mathcal {G}}\left( \kappa ,K\right)$ denote the class of functions $g\in {\mathcal {C}}\left[ 0,T\right] \rightarrow \left[ 0,T^{\kappa }\right]$ indexed by $\left( t,x\right) \in {\mathcal {T}} \left( 0\right) =\left[ 0,T\right] \times {\mathbb {R}}$ defined by

$$\begin{aligned} t^{\kappa }h_{t,x}^{\left( K\right) }\left( g\right) =t^{\kappa }1\left\{ g\left( t\right) \le x,g\in {\mathcal {C}}\left( K\right) \right\} . \end{aligned}$$

(78)

Clearly by a slight modification of the above argument, ${\mathcal {G}}\left( \kappa ,K\right)$ is pointwise measurable.

1.3 Inequalities for Empirical Processes

In this subsection, ${\mathcal {G}}$ is a pointwise measurable class of measurable real-valued functions defined on a measurable space $\left( S,{\mathcal {S}}\right)$. For any $0<\sigma <1$, set

$$\begin{aligned} J\left( \sigma ,{\mathcal {G}}\right) =\int _{\left[ 0,\sigma \right] } \sqrt{1+\log N_{[\;]}(s,{\mathcal {G}},d_{P})}\,ds \end{aligned}$$

(79)

and

$$\begin{aligned} a\left( \sigma ,{\mathcal {G}}\right) =\sigma \left[ 1+\log N_{[\;]}(\sigma , {\mathcal {G}},d_{P})\right] ^{-1/2}. \end{aligned}$$

(80)

Lemma 19.34 in van der Vaart [25] gives the following moment bound. (Note the needed “$+1$” in the definition of $J(\sigma ,{\mathcal {G}})$ and $a\left( \sigma ,{\mathcal {G}} \right)$.)

Moment inequality

Let $\xi ,\xi _{1},\ldots ,\xi _{n}$ be i.i.d. and assume that ${\mathcal {G}}$ has a measurable envelope function G and $E\left( g^{2}\left( \xi \right) \right) <\sigma ^{2}<1$ for every $g\in {\mathcal {G}}$. We have, for a universal constant $A_{0}^{\prime }$,

$$\begin{aligned} \begin{array}{lll} &{}\displaystyle E \left\| \frac{1}{\sqrt{n}} \sum _{i=1}^{n} \left( g(\xi _{i})-Eg(\xi _{i}) \right) \right\| _{{\mathcal {G}}} \\ &{}\displaystyle \quad \le A_{0}^{\prime } \left[ J\left( \sigma ,{\mathcal {G}}\right) + \sqrt{n} \, E\left( G\left( \xi \right) 1\left\{ G\left( \xi \right) >\sqrt{n} a(\sigma , {\mathcal {G}}) \right\} \right) \right] . \end{array} \end{aligned}$$

(81)

Let $\epsilon$ be a Rademacher variable, i.e., $P\{\epsilon =1\}=P\{\epsilon =-1\}=1/2$, and consider independent Rademacher variables $\epsilon _{1},\ldots ,\epsilon _{n}$ independent of $\xi _{1}$, $\ldots ,\xi _{n}$. From a special case of a well-known symmetrization lemma, we have for any class of functions ${\mathcal {G}}$ in $L_{1}\left( P\right)$

$$\begin{aligned} \frac{1}{2} E\Big \Vert \sum _{i=1}^{n} \epsilon _{i} \big ( g(\xi _i) - Eg(\xi ) \big ) \Big \Vert _{{\mathcal {G}}} \le E \Big \Vert \sum _{i=1}^{n} \big ( g(\xi _{i})-Eg(\xi ) \big ) \Big \Vert _{{\mathcal {G}}} \le 2 E \Big \Vert \sum _{i=1}^{n} \epsilon _i g(\xi _{i}) \Big \Vert _{{\mathcal {G}}}. \end{aligned}$$

(See Lemma 6.3 of Ledoux and Talagrand [17].) In particular, we get

$$\begin{aligned} E\left\| \sum _{i=1}^{n}\epsilon _{i}g(\xi _{i})\right\| _{{\mathcal {G}} }\le & {} E\left\| \sum _{i=1}^{n}\epsilon _{i}\left( g(\xi _{i})-Eg\left( \xi \right) \right) \right\| _{{\mathcal {G}}}+E\left| \sum _{i=1}^{n} {\mathbb {\epsilon }}_{i}\right| \left\| Eg\left( \xi \right) \right\| _{{\mathcal {G}}} \nonumber \\\le & {} 2E\left\| \sum _{i=1}^{n}\left( g(\xi _{i})-Eg\left( \xi \right) \right) \right\| _{{\mathcal {G}}}+\sigma \sqrt{n}. \end{aligned}$$

Thus, we readily get from (81) with $A_{0}=2A_{0}^{\prime } +1$ and noting that the integrand of $J\left( \sigma ,{\mathcal {G}}\right)$ is greater than or equal to 1,

$$\begin{aligned} \begin{array}{lll} &{}\displaystyle E \left\| \frac{1}{\sqrt{n}} \sum _{i=1}^{n} \epsilon _{i} g(X_{i}) \right\| _{{\mathcal {G}}} \\ &{}\quad \displaystyle \le A_{0}\left[ J( \sigma ,{\mathcal {G}}) +\sqrt{n} \, E\left( G ( \xi ) 1\left\{ G\left( \xi \right) >\sqrt{n}\, a(\sigma ,{\mathcal {G}})\right\} \right) \right] . \end{array} \end{aligned}$$

(82)

We shall be using the moment bound (82) in conjunction with the following exponential inequality due to Talagrand [24]. This maximal version is pointed out by Einmahl and Mason [8, Inequality A.1 on p. 31].

Talagrand inequality

Let ${\mathcal {G}}$ be a pointwise measurable class of measurable real-valued functions defined on a measurable space $(S,{\mathcal {S}})$ satisfying $||g||_{\infty }\le M,\ g\in {\mathcal {G}}$, for some $0<M<\infty$. Let $X,X_{n}$, $n\ge 1$, be a sequence of i.i.d. random variables defined on a probability space $\left( \varOmega ,{\mathcal {A}},P\right)$ and taking values in S, then for all $t>0$ we have for suitable finite constants $A,A_{1}>0$,

$$\begin{aligned} \begin{array}{lll} &{}\displaystyle P\left\{ \max _{1\le m\le n}||\sqrt{m}\alpha _{m}||_{{\mathcal {G}}}\ge A\left( E \Big \Vert \sum _{i=1}^{n}\epsilon _{i}g(X_{i}) \Big \Vert _{{\mathcal {G}}}+t\right) \right\} \\ &{}\displaystyle \quad \le 2 \exp \left( -\frac{A_{1}t^{2}}{n\sigma _{{\mathcal {G}}}^{2}}\right) +2\exp \left( -\frac{A_{1}t}{M}\right) , \end{array} \end{aligned}$$

(83)

where $\sigma _{{\mathcal {G}}}^{2}=\sup _{g\in {\mathcal {G}}}\mathop {Var}(g(X))$.

1.4 Inequalities for Gaussian Processes

Let ${\mathbb {Z}}$ be a separable mean zero Gaussian process on a probability space $(\varOmega ,{\mathcal {A}},P)$ indexed by a set ${\mathbb {T}}$, equipped with a semimetric

$$\begin{aligned} \rho \left( s,t\right) =\sqrt{E \left( {\mathbb {Z}}\left( t\right) -{\mathbb {Z}} \left( s\right) \right) ^{2}}. \end{aligned}$$

(84)

For each $\varepsilon >0$, let $N\left( \varepsilon ,{\mathbb {T}},\rho \right)$ denote the minimal number of $\rho$ balls of radius $\varepsilon$ needed to cover ${\mathbb {T}}.$ Write $\left\| {\mathbb {Z}}\right\| _{{\mathbb {T}} }=\sup _{t\in {\mathbb {T}}}\left| {\mathbb {Z}}_{t}\right|$ and $\sigma _{ {\mathbb {T}}}^{2}\left( {\mathbb {Z}}\right) =\sup _{t\in {\mathbb {T}}}E\left( {\mathbb {Z}}_{t}^{2}\right)$.

According to Dudley [5], the entropy condition

$$\begin{aligned} \int _{\left[ 0,1\right] }\sqrt{\log N\left( \varepsilon ,{\mathbb {T}},\rho \right) }\,d\varepsilon <\infty \end{aligned}$$

(85)

ensures the existence of a separable, bounded, $\rho$ uniformly continuous modification of ${\mathbb {Z}}$. The following moment bound is a version of Corollary 2.2.8 in van der Vaart and Wellner [26]. (Also see their Problem 2.2.14.)

Gaussian moment inequality

For some universal constant $A_{4}>0$ and all $\sigma >0$, we have

$$\begin{aligned} E\left( \sup _{\rho \left( s,t\right) <\sigma }\left| {\mathbb {Z}}_{t}- {\mathbb {Z}}_{s}\right| \right) \le A_{4}\int _{\left[ 0,\sigma \right] } \sqrt{\log N\left( \varepsilon ,{\mathbb {T}},\rho \right) }\,d\varepsilon \end{aligned}$$

(86)

and for any $t_{0}\in {\mathbb {T}}$,

$$\begin{aligned} E\left( \left\| {\mathbb {Z}}\right\| _{{\mathbb {T}}}\right) \le E\left| {\mathbb {Z}}_{t_{0}}\right| +A_{4}\int _{\left[ 0,{\mathbb {D}} \right] }\sqrt{\log N\left( \varepsilon ,{\mathbb {T}},\rho \right) } \,d\varepsilon , \end{aligned}$$

(87)

with

$$\begin{aligned} {\mathbb {D}}=\sup _{s,t\in {\mathbb {T}}}\rho \left( s,t\right) \end{aligned}$$

(88)

denoting the diameter of ${\mathbb {T}}$.

Notice that if d is a semimetric on ${\mathbb {T}}$ such that for all $s,t\in T$, $d\left( s,t\right) \ge \rho \left( s,t\right)$, then

$$\begin{aligned} \sup _{\left\{ s:\rho \left( s,t\right) <\sigma \right\} }\left| \mathbb {Z }_{t}-{\mathbb {Z}}_{s}\right| \ge \sup _{\left\{ s:\text { }d\left( s,t\right) <\sigma \right\} }\left| {\mathbb {Z}}_{t}-{\mathbb {Z}} _{s}\right| \end{aligned}$$

and $N\left( \varepsilon ,{\mathbb {T}},d\right) \ge N\left( \varepsilon , {\mathbb {T}},\rho \right)$. Thus,

$$\begin{aligned} \int _{\left[ 0,1\right] }\sqrt{\log N\left( \varepsilon ,{\mathbb {T}},d\right) }\,d\varepsilon <\infty \end{aligned}$$

(89)

implies by the Dudley result the existence of a separable, bounded, d uniformly continuous modification of ${\mathbb {Z}}$. (Here note that $\rho$ uniformly continuous implies d uniformly continuous.) Moreover, the moment inequalities in (86) and (87) hold when $\rho$ is replaced by d and in the definition of ${\mathbb {D}}.$

In particular, these inequalities hold when ${\mathbb {Z}}={\mathbb {G}} _{(\gamma ,T)}$, the Gaussian process defined at the end of Subsect. 2.1, where ${\mathbb {T}}={\mathcal {F}}_{(\gamma ,T)}$ and $d=d_{P}$ is as defined in (11), and ${\mathbb {D}}=\sup \left\{ d_P \left( f,g\right) : f,g\in \mathcal { F}_{(\gamma ,T)}\right\}$ is the diameter ${\mathbb {D}}$ of ${\mathbb {T}}= {\mathcal {F}}_{(\gamma ,T)}$.

The following large deviation probability estimate for $\left\| {\mathbb {Z}} \right\| _{{\mathbb {T}}}$ is due to Borell [4]. (Also see Proposition A.2.1 in [26].) Let $M\left( X\right)$ denote the a median of $\left\| {\mathbb {Z}}\right\| _{{\mathbb {T}}}$, i.e., $P\left\{ \left\| {\mathbb {Z}}\right\| _{{\mathbb {T}}}\ge M\left( X\right) \right\} \ge 1/2$ and $P\left\{ \left\| {\mathbb {Z}}\right\| _{{\mathbb {T}} }\le M\left( X\right) \right\} \ge 1/2$. We shall assume that $M\left( X\right)$ is finite.

Borell’s inequality

For all $z>0$,

$$\begin{aligned} P\left\{ \left| \left\| {\mathbb {Z}}\right\| _{{\mathbb {T}}}-E\left( \left\| {\mathbb {Z}}\right\| _{{\mathbb {T}}}\right) \right| >z\right\} \le 2\exp \left( -\frac{z^{2}}{2\sigma _{{\mathbb {T}}}^{2}\left( {\mathbb {Z}} \right) }\right) . \end{aligned}$$

(90)

1.4.1 Application of the Landau–Shepp Theorem

We shall be using the following version of the Landau and Shepp [LS] [15] theorem (also see Sato [22], Theorem 2.5 of Marcus and Shepp [20] and Proposition A.2.3 in [26]):

Theorem [LS]

Let $X_{t},t\in T,$ be a real-valued separable Gaussian process such that w.p. 1, $\sup _{t\in T}\left| X_{t}\right| <\infty$, then for any $0<\beta <1/\left( 2\sigma ^{2}\right)$, where $\sigma ^{2}=\sup _{t\in T}\mathop {Var}\left( X_{t}\right)$, for all y sufficiently large

$$\begin{aligned} P\left\{ \sup _{t\in T}\left| X_{t}\right| >y\right\} <\exp \left( -\beta y^{2}\right) . \end{aligned}$$

(91)

Recall the definition of L in (4). Since L is finite, w.p. 1, we can apply the Landau and Shepp theorem to infer that for appropriate constants $C>0$ and $D>0$, for all $t>0,$

$$\begin{aligned} P\left\{ L>t\right\} \le C\exp \left( -Dt^{2}\right) . \end{aligned}$$

(92)

1.5 Four Maximal Inequalities

For the following inequalities, recall the mean zero Gaussian process G with covariance function defined in (7). Inequalities 1 and 2 are required for the proof of Proposition 2, and Inequalities 1A and 2A are needed in the proofs of Theorems 1 and 2.

Inequality 1

For all $0<\varrho <\infty$ and $\delta >0$, we have for some constant $\mu (\delta )$ and all $z>0$

$$\begin{aligned} P\left\{ \sup _{(t,x)\in [0,\varrho ]\times {\mathbb {R}}}t^{\delta }\left| G\left( t,x\right) \right| >\varrho ^{\delta }2^{\delta }\mu \left( \delta \right) +z\right\} \le 2\exp \left( -\frac{z^{2}\varrho ^{-2\delta }}{2^{2\delta +1}}\right) \end{aligned}$$

(93)

and for each $n\ge 1$ and for $t^{\delta }G^{\left( 1\right) }\left( t,x\right) ,\dots , t^{\delta }G^{\left( n\right) }\left( t,x\right)$ i.i.d. $t^{\delta } G\left( t,x\right)$

$$\begin{aligned} \begin{array}{ll} &{} P\left\{ \max _{1\le m\le n} \sup _{(t,x) \in [0,\varrho ]\times {\mathbb {R}}} \left| \frac{1}{\sqrt{n}}\sum _{i=1}^{m}t^{\delta }G^{\left( i\right) } \left( t,x\right) \right| > \varrho ^{\delta } 2^{\delta }\mu \left( \delta \right) +z\right\} \\ &{}\quad \le 4\exp \left( -\frac{z^{2}\varrho ^{-2\delta }}{2^{2\delta +1}}\right) . \end{array} \end{aligned}$$

(94)

Proof

Define for any integer $k\ge 0$,

$$\begin{aligned} {\mathcal {T}}_{k}=\left[ 2^{-k},2^{-k+1}\right] \times {\mathbb {R}}. \end{aligned}$$

Theorem 5 in [12] implies that, w.p. 1, for each integer k,

$$\begin{aligned} \sup \left\{ \left| G\left( t,x\right) \right| :\left( t,x\right) \in {\mathcal {T}}_{k}\right\} <\infty . \end{aligned}$$

(95)

Notice that for any $k\ge 0$

$$\begin{aligned} \sup \left\{ \left| G\left( t,x\right) \right| :\left( t,x\right) \in {\mathcal {T}}_{k}\right\} \overset{\mathrm {D}}{=}\sup \left\{ \left| G\left( t,x\right) \right| :\left( t,x\right) \in {\mathcal {T}} _{0}\right\} . \end{aligned}$$

Furthermore, (95) and separability of $G\left( t,x\right)$ permit us to apply the Landau–Shepp theorem (see (91)) to get

$$\begin{aligned} \mu _{0}:=E\left( \sup \left\{ \left| G\left( t,x\right) \right| :\left( t,x\right) \in {\mathcal {T}}_{0}\right\} \right) <\infty . \end{aligned}$$

Thus for any integer K

$$\begin{aligned}&E\left( \sup \left\{ t^{\delta }\left| G\left( t,x\right) \right| :\left( t,x\right) \in \left[ 0,2^{-K}\right] \times {\mathbb {R}}\right\} \right) \\&\quad \le \mu _{0}\sum _{k=K}^{\infty }2^{-\delta k}=2^{-\delta K}\mu _{0}/\left( 1-2^{-\delta }\right) =:2^{-\delta K}\mu \left( \delta \right) . \end{aligned}$$

This implies that, w.p. 1,

$$\begin{aligned} \sup \left\{ t^{\delta }\left| G\left( t,x\right) \right| :\left( t,x\right) \in \left[ 0,2^{-K}\right] \times {\mathbb {R}}\right\} <\infty . \end{aligned}$$

Also

$$\begin{aligned} \sup \left\{ \mathop {Var}\left( t^{\delta }G\left( t,x\right) \right) :\left( t,x\right) \in \left[ 0,2^{-K}\right] \times {\mathbb {R}}\right\} \le 2^{-2\delta K}. \end{aligned}$$

Applying Borell’s inequality (90) with ${\mathbb {Z}}\left( t,x\right) =t^{\delta }G\left( t,x\right)$, ${\mathbb {T}}=\left[ 0,2^{-K}\right] \times {\mathbb {R}}$, $E\left( \left\| {\mathbb {Z}}\right\| _{{\mathbb {T}}}\right) \le 2^{-\delta K}\mu \left( \delta \right)$ and $\sigma _{{\mathbb {T}} }^{2}\left( {\mathbb {Z}}\right) \le 2^{-2\delta K}$, we get for all $z>0$ and integers K

$$\begin{aligned} P\left\{ \sup _{(t,x)\in [0,2^{-K}]\times {\mathbb {R}}}t^{\delta }\left| G\left( t,x\right) \right| >2^{-\delta K}\mu \left( \delta \right) +z\right\} \le 2\exp \left( -\frac{z^{2}2^{2\delta K}}{2}\right) . \end{aligned}$$

Choose any $0<\varrho <\infty$ and integer K such that $2^{-K}\ge \varrho >2^{-K-1}.$ We see that

$$\begin{aligned} 2^{K+1}>\varrho ^{-1}\ge 2^{K}\ge \varrho ^{-1}/2. \end{aligned}$$

Hence, $\left[ 0,\varrho \right] \times {\mathbb {R}}\subset \left[ 0,2^{-K} \right] \times {\mathbb {R}}$. Therefore,

$$\begin{aligned}&P\left\{ \sup \left\{ t^{\delta }\left| G\left( t,x\right) \right| :\left( t,x\right) \in \left[ 0,\varrho \right] \times {\mathbb {R}}\right\} >\varrho ^{\delta }2^{\delta }\mu \left( \delta \right) +z\right\} \\&\quad \le P\left\{ \sup \left\{ t^{\delta }\left| G\left( t,x\right) \right| :\left( t,x\right) \in \left[ 0,2^{-K}\right] \times {\mathbb {R}} \right\} >2^{-\delta K}\mu \left( \delta \right) +z\right\} \\&\quad \le 2\exp \left( -\frac{z^{2}2^{2\delta K}}{2}\right) \le 2\exp \left( - \frac{z^{2}\varrho ^{-2\delta }}{2^{2\delta +1}}\right) . \end{aligned}$$

Inequality (94) follows from Lévy’s inequality (see Proposition A.1.2 in van der Vaart and Wellner [26]) along with separability of the Gaussian process $t^{\delta }G\left( t,x\right)$. $\square$

Inequality 1A

For all $0<\gamma <1<T<\infty$, we have for some constant $\mu$ and all $z>0$

$$\begin{aligned} P\left\{ \sup _{(t,x)\in {\mathcal {T}}\left( \gamma \right) }\left| G\left( t,x\right) \right| >\mu +z\right\} \le 2\exp \left( -\frac{z^{2}}{2} \right) \end{aligned}$$

(96)

and for each $n\ge 1$ and $G^{\left( 1\right) }\left( t,x\right) ,\dots ,G^{\left( n\right) }\left( t,x\right)$ i.i.d. $G\left( t,x\right)$

$$\begin{aligned} P\left\{ \max _{1\le m\le n}\sup _{((t,x)\in {\mathcal {T}}\left( \gamma \right) }\left| \frac{1}{\sqrt{n}}\sum _{i=1}^{m}G^{\left( i\right) }\left( t,x\right) \right| >\mu +z\right\} \le 4\exp \left( -\frac{z^{2} }{2}\right) . \end{aligned}$$

(97)

Proof

Theorem 5 in [12] implies that, w.p. 1,

$$\begin{aligned} \sup \left\{ \left| G\left( t,x\right) \right| :\left( t,x\right) \in {\mathcal {T}}\left( \gamma \right) \right\} <\infty . \end{aligned}$$

(98)

Furthermore, (98) permits us to apply the Landau–Shepp theorem to get

$$\begin{aligned} \mu :=E\left( \sup \left\{ \left| G\left( t,x\right) \right| :\left( t,x\right) \in {\mathcal {T}}\left( \gamma \right) \right\} \right) <\infty . \end{aligned}$$

Also

$$\begin{aligned} \sup \left\{ \mathop {Var}\left( G\left( t,x\right) \right) :(t,x)\in {\mathcal {T}}(\gamma ) \right\} \le 1. \end{aligned}$$

Applying Borell’s inequality (90) with ${\mathbb {Z}} (t,x) = G (t,x)$, ${\mathbb {T}}={\mathcal {T}}(\gamma )$, $E \left\| {\mathbb {Z}}\right\| _{{\mathbb {T}}} =\mu$ and $\sigma _{{\mathbb {T}}}^{2}\left( {\mathbb {Z}}\right) \le 1$, we get for all $z>0$

$$\begin{aligned} P\left\{ \sup _{(t,x)\in {\mathcal {T}}\left( \gamma \right) }\left| G\left( t,x\right) \right| >\mu +z\right\} \le 2\exp \left( -\frac{z^{2}}{2} \right) . \end{aligned}$$

Inequality (97) follows from Lévy’s inequality and separability of the Gaussian process $G\left( t,x\right)$. $\square$

In Inequalities 2 and 2A for $g \in c[0, T ]$,

$$\begin{aligned} h_{t,x} (g) = 1 \{g(t)\le x, g \in c_{\infty }\}, \end{aligned}$$

where $C_{\infty }$ is defined as in (10).

Inequality 2

For all $0<\varrho <T/2$ and $\delta >0$, we have for some $E(\delta )$ and for suitable finite positive constants $A, A_{1}>0$, for all $z>0$

$$\begin{aligned} {\begin{matrix} &{} P\left\{ \max _{1\le m\le n}\sup _{(t,x)\in [0,\varrho ]\times {\mathbb {R}}}|\sqrt{m}t^{\delta }\alpha _{m}\left( h_{t,x}\right) |>\sqrt{n} A\left( E(\delta )2^{\delta }\varrho ^{\delta }+z\right) \right\} \\ &{}\quad \le 2\left\{ \exp \left( -z^{2}A_{1}\left( 2\varrho \right) ^{-2\delta }\right) +\exp \left( -z\sqrt{n}A_{1}\left( 2\varrho \right) ^{-\delta }\right) \right\} . \end{matrix}} \end{aligned}$$

(99)

Note, in particular, Inequality 2 implies that for all $\lambda >1$ there is a $d>1$ such that

$$\begin{aligned} P\left\{ \sup \left\{ |t^{\delta }\alpha _{n}\left( h_{t,x}\right) |:\left( t,x\right) \in \left[ 0,\varrho \right] \times {\mathbb {R}}\right\} \ge d\varrho ^{\delta }\sqrt{\log n}\right\} <n^{-\lambda }. \end{aligned}$$

(100)

Proof

For any $k\ge 1$ and $g\in {\mathcal {C}}\left[ 0,T\right]$, let

$$\begin{aligned} g_{k}\left( t\right) =2^{kH}g\left( t2^{-k}\right) ,\ t\in \left[ 0,T \right] , \end{aligned}$$

and for any $k\ge 1$, $t\in \left[ 0,T \right]$, $x\in {\mathbb {R}}$ and $g\in {\mathcal {C}}\left[ 0,T \right]$ set

$$\begin{aligned} h_{t,x,k}\left( g\right) =h_{t,x}\left( g_{k}\right) =1\left\{ g_{k}\left( t\right) \le x, g_k \in C_{\infty }\right\} . \end{aligned}$$

Clearly w.p. 1

$$\begin{aligned} \sup _{(t,x)\in {\mathcal {T}}_{k}}\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i}) \right| =\sup _{(t,x)\in {\mathcal {T}}_{0}}\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x,k}(B_{i})\right| . \end{aligned}$$

Moreover, since

$$\begin{aligned} \left\{ B_{j}\right\} _{j\ge 1}\overset{\mathrm {D}}{=}\left\{ 2^{kH}B_{j}\left( \cdot /2^{k}\right) \right\} _{j\ge 1}, \end{aligned}$$

we see that

$$\begin{aligned} \sup _{(t,x)\in {\mathcal {T}}_{k}}\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x,k}(B_{i})\right| \overset{\mathrm {D}}{=}\sup _{(t,x)\in {\mathcal {T}}_{0}}\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i})\right| \end{aligned}$$

and thus

$$\begin{aligned} E\sup _{(t,x)\in {\mathcal {T}}_{k}}\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i})\right| =E\sup _{(t,x)\in {\mathcal {T}}_{0}}\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i})\right| . \end{aligned}$$

(101)

We readily see by inequality (82)

$$\begin{aligned} E\sup _{(t,x)\in {\mathcal {T}}_{0}}\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i})\right| \le 2\sqrt{n}E\left\| v_{n}\right\| _{ {\mathcal {T}}_{0}}+ \sqrt{n}, \end{aligned}$$

which by (9) is $\le 2\left( M\left( 1,2,H\right) +1\right) \sqrt{n} =:E_{0}\sqrt{n}.$ Thus

$$\begin{aligned} E\sup _{(t,x)\in {\mathcal {T}}_{0}}\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i})\right| \le E_{0}\sqrt{n}. \end{aligned}$$

(102)

Next, for all $\delta >0$ with K an integer such that $2^{-{K}}\ge \varrho >2^{-{K}-1}$

$$\begin{aligned} E\sup \left\{ \left| t^{\delta }\sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i})\right| :0\le t\le 2^{-K},x\in {\mathbb {R}}\right\} \end{aligned}$$

is by (101) and (102)

$$\begin{aligned} \le \sum _{k=K}^{\infty }2^{-k\delta }E\sup _{(t,x)\in {\mathcal {T}} _{k}}\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i})\right| \le E\left( \delta \right) 2^{-K\delta }\sqrt{n\,}, \end{aligned}$$

(103)

where $E\left( \delta \right) =E_{0}/\left( 1-2^{-\delta }\right)$.

Let

$$\begin{aligned} {\mathcal {H}}\left( \delta ,K\right) =\left\{ t^{\delta }h_{t,x}:\left( t,x\right) \in \left[ 0,2^{-K}\right] \times {\mathbb {R}}\right\} . \end{aligned}$$

From (103), we get

$$\begin{aligned} E\sup \left\{ \left| \sum _{i=1}^{n}\epsilon _{i}g(B_{i})\right| :g\in {\mathcal {H}}\left( \delta ,K\right) \right\} \le E\left( \delta \right) 2^{-K\delta }\sqrt{n\,}. \end{aligned}$$

(104)

Also observe that each $g\in {\mathcal {H}}\left( \delta ,K\right)$ satisfies $|g|\le 2^{-K\delta }$. Applying Talagrand’s inequality (83) with $M=2^{-K\delta }$, $\sigma _{{\mathcal {H}}\left( \delta ,K\right) }^{2} = 2^{-2K\delta }$ and the bound (104), we get that for any $\delta >0$ we have for suitable finite positive constants $A,A_{1}>0$, for all $z>0$

$$\begin{aligned} {\begin{matrix} &{} P\left\{ \max _{1\le m\le n}||\sqrt{m}\alpha _{m}||_{{\mathcal {H}}\left( \delta ,K\right) }\ge \sqrt{n}A(E\left( \delta \right) 2^{-K\delta }+z)\right\} \\ &{}\quad \le 2(\exp (-z^{2}A_{1}2^{2K\delta })+\exp (-z\sqrt{n}A_{1}2^{K\delta })). \end{matrix}} \end{aligned}$$

(105)

Inequality (99) follows from inequality (105). To see this, choose any $0<\varrho <T/2$ and integer K such that $2^{-K}\ge \varrho >2^{-K-1}.$ We see that

$$\begin{aligned} 2^{K+1}>\varrho ^{-1}\ge 2^{K}\ge \varrho ^{-1}/2. \end{aligned}$$

Hence $\left\{ t^{\delta }h_{t,x}:\left( t,x\right) \in \left[ 0,\varrho \right] \times {\mathbb {R}}\right\} \subset {\mathcal {H}}\left( \delta ,K\right)$, and

$$\begin{aligned}&P\left\{ \max _{1\le m\le n}\sup _{(t,x)\in [0,\varrho ]\times {\mathbb {R}}}|\sqrt{m}t^{\delta }\alpha _{m}\left( h_{t,x}\right) |\ge \sqrt{n }A(E\left( \delta \right) 2^{\delta }\varrho ^{\delta }+z)\right\} \\&\quad \le P\left\{ \max _{1\le m\le n}||\sqrt{m}\alpha _{m}||_{{\mathcal {H}} \left( \delta ,K\right) }\ge \sqrt{n}A(E\left( \delta \right) 2^{-K\delta }+z)\right\} \\&\quad \le 2(\exp (-z^{2}A_{1}2^{2K\delta })+\exp (-z\sqrt{n}A_{1}2^{K\delta })) \\&\quad \le 2\left\{ \exp \left( -z^{2}A_{1}(2\varrho )^{-2\delta }\right) +\exp \left( -z\sqrt{n}A_{1}(2\varrho )^{-\delta }\right) \right\} . \end{aligned}$$

$\square$

Inequality 2A

For all $0<\gamma <1<T<\infty$, we have for some $L(\gamma ,T)$ and all $z>0$ for suitable finite positive constants $A, A_{1}>0$, for all $z>0$

$$\begin{aligned} \begin{array}{lll} &{} P\left\{ \max _{1\le m\le n}\sup _{(t,x)\in {\mathcal {T}}\left( \gamma \right) }|\sqrt{m}\alpha _{m}\left( h_{t,x}\right) |\ge \sqrt{n}A\left( L(\gamma ,T)+z\right) \right\} \\ &{}\quad \le 2\left\{ \exp \left( -z^{2}A_{1}\right) +\exp \left( -z\sqrt{n} A_{1}\right) \right\} . \end{array} \end{aligned}$$

(106)

Proof

We see by inequality (82)

$$\begin{aligned} E\sup _{(t,x)\in {\mathcal {T}}\left( \gamma \right) }\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i})\right| \le 2\sqrt{n} E\left\| v_{n}\right\| _{{\mathcal {T}}\left( \gamma \right) }+ \sqrt{n}, \end{aligned}$$

which by (9) is $\le 2\left( M\left( \gamma ,T,H\right) +1\right) \sqrt{n}=:L(\gamma ,T)\sqrt{n}.$ Thus

$$\begin{aligned} E\sup _{(t,x)\in {\mathcal {T}}\left( \gamma \right) }\left| \sum _{i=1}^{n}\epsilon _{i}h_{t,x}(B_{i})\right| \le L(\gamma ,T)\sqrt{n }. \end{aligned}$$

(107)

Applying Talagrand’s inequality (83) with $M=1$, $\sigma _{{\mathcal {F}}_{\left( \gamma ,T\right) }}^{2}=1$ and the bound (107) gives (106). $\square$

Remark 8

Actually, to apply Talagrand’s inequality in the proofs of Inequalities 2 and 2A, as it is stated in (83), the classes of functions ${\mathcal {H}}\left( \delta ,K\right)$ and ${\mathcal {F}}_{\left( \gamma ,T\right) }$ should be pointwise measurable. Here we shall discuss how to take care of this detail in the proof of Inequality 2. A similar discussion works for the proof of Inequality 2A.

For any $k\ge 1$, let

$$\begin{aligned} {\mathcal {H}}\left( \delta ,K,k\right) =\left\{ g1\left\{ g\in {\mathcal {C}} \left( k\right) \right\} :g\in {\mathcal {H}}\left( \delta ,K\right) \right\} . \end{aligned}$$

where C(k) is defined as in (27).

The class ${\mathcal {H}}\left( \delta ,K,k\right)$ is pointwise measurable. Applying Talagrand’s inequality, we get with $M=2^{-K\delta }$ and $\sigma _{ {\mathcal {H}}\left( \delta ,K,k\right) }^{2}=2^{-2K\delta }$

$$\begin{aligned}&P\left\{ \max _{1\le m\le n}||\sqrt{m}\alpha _{m}||_{{\mathcal {H}}\left( \delta ,K,k\right) }\ge A\left( E\bigg \Vert \sum _{i=1}^{n}\epsilon _{i}g(B_{i})\bigg \Vert _{{\mathcal {H}}\left( \delta ,K,k\right) }+t\right) \right\} \\&\quad \le 2\exp \left( -\frac{2^{2K\delta }A_{1}t^{2}}{n}\right) +2\exp \left( -2^{K\delta }A_{1}t\right) . \end{aligned}$$

Obviously by the Wang [27] result (26), w.p. 1, $B \in \cup _{k=1}^{\infty }{\mathcal {C}}\left( k\right)$. Therefore, w.p. 1, for any $n\ge 1$, $B_{1},\dots ,B_{n}$, i.i.d. B there exists a $k\ge 1$ such that uniformly in $\left( t,x\right) \in \left[ 0,\varrho \right] \times {\mathbb {R}}$, $h_{t,x}^{\left( k\right) }\left( B_{i} \right) =h_{t,x}\left( B_{i} \right)$ and $t^{\delta }h_{t,x}^{\left( k\right) } \left( B_{i} \right) =t^{\delta }h_{t,x}\left( B_{i}\right)$, for $i=1,\dots ,n$. This says that, w.p. 1, for any $n\ge 1$, there exists a $k\ge 1,$ such that uniformly in $\left( t,x\right) \in \left[ 0,\varrho \right] \times {\mathbb {R}}$

$$\begin{aligned} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}t^{\delta }h_{t,x}\left( B_{i}\right) 1\left\{ B_{i}\notin {\mathcal {C}}\left( k\right) \right\} =0. \end{aligned}$$

Furthermore,

$$\begin{aligned} \sup _{\left( t,x\right) \in \left[ 0,\varrho \right] \times {\mathbb {R}}} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}t^{\delta } Eh_{t,x}\left( B_{i} \right) 1\left\{ B_{i}\notin {{\mathcal {C}}}\left( k\right) \right\} \le \sqrt{n}\varrho ^{\delta } P \left\{ B\notin {{\mathcal {C}}}\left( k\right) \right\} , \end{aligned}$$

which converges to zero for each fixed $n\ge 1$, as $k\rightarrow \infty$. By passing to the limit, as $k\rightarrow \infty$, we get for any $\delta >0$ and $t>0$

$$\begin{aligned} {\begin{matrix} &{} P\left\{ \max _{1\le m\le n}||\sqrt{m}\alpha _{m}||_{{{\mathcal {H}}}\left( \delta ,K\right) }\ge A\left( E\left\| \sum _{i=1}^{n}\epsilon _{i}g(B_{i})\right\| _{{{\mathcal {H}}}\left( \delta ,K\right) }+t\right) \right\} \\ &{}\quad \le 2\exp \left( -\frac{2^{2K\delta }A_{1}t^{2}}{n}\right) +2\exp \left( -2^{K\delta }A_{1}t\right) . \end{matrix}} \end{aligned}$$

Similarly, one can argue the validity of the Talagrand inequality using the index class ${{\mathcal {F}}}_{\left( \gamma ,T\right) }$.