On the Lower Classes of Some Mixed Fractional Gaussian Processes with Two Logarithmic Factors

Abstract

We introduce the fractional mixed fractional Brownian sheet and investigate the small ball behavior of its sup-norm statistic by establishing a general result on the small ball probability of the sum of two not necessarily independent joint Gaussian random vectors. Then, we state general conditions and characterize the sufficiency part of the lower classes of some statistics of the above process by an integral test. Finally, when we consider the sup-norm statistic, the necessity part is given by a second integral test.

1. Introduction

Let {𝐵_𝐻𝑖(𝑠),𝑠≥0} be a fractional Brownian motion (FBM) with index 0<𝐻_𝑖<1,𝑖∈ℕ^∗, and {𝐵_{𝐻𝑗,𝐻𝑘}(𝑠,𝑠^),𝑠≥0,𝑠^≥0} a fractional Brownian sheet (FBS) with index 0<𝐻_𝑗,𝐻_𝑘<1,𝑗∈ℕ^∗,𝑘∈ℕ^∗. We refer to [1] for further information about the FBM and the FBS. Denote by 𝜆₁ and 𝜆₂ two real numbers such that 𝜆₁𝜆₂≠0.

Define a fractional mixed fractional Gaussian process by a suitable combination of some appropriate fractional Gaussian processes. In the sequel, we consider the following three examples.

Example 1.1. The fractional mixed fractional Brownian motion (FMFBM) is defined by 𝑋𝑤₁,𝑤₂,𝑠=𝜆₁𝑠^𝐻2𝐵_𝐻1𝑤₁+𝜆₂𝑠^𝐻1𝐵_𝐻2𝑤₂,(1.1) where 𝐵_𝐻1 and 𝐵_𝐻2 are independent FBM with 𝐻₁≠𝐻₂.

Example 1.2. The fractional mixed fractional Brownian motion and fractional Brownian sheet (FMFBMFBS) are defined by 𝑋𝑤₁,𝑤₂,𝑤₃,𝑠=𝜆₁𝑠^𝐻2+𝐻3𝐵_𝐻1𝑤₁+𝜆₂𝑠^𝐻1𝐵_𝐻2,𝐻3𝑤₂,𝑤₃,(1.2) where the FBM 𝐵_𝐻1 and the FBS 𝐵_𝐻2,𝐻3 are independent.

Example 1.3. The fractional mixed fractional Brownian sheet (FMFBS) is defined by 𝑋𝑤₁,𝑤₂,𝑤₃,𝑤₄,𝑠=𝜆₁𝑠^𝐻3+𝐻4𝐵_𝐻1,𝐻2𝑤₁,𝑤₂+𝜆₂𝑠^𝐻1+𝐻2𝐵_𝐻3,𝐻4𝑤₃,𝑤₄,(1.3) where 𝐵_𝐻1,𝐻2 and 𝐵_𝐻3,𝐻4 are independent FBS with (𝐻₁,𝐻₂)≠(𝐻₃,𝐻₄).

The motivation supporting this paper is threefold.

(i)The first goal of the FMFBS deals with the potential applications. Since the FMFBM, the FMFBMFBS, and the FMFBS can be analyzed based on the large bodies of knowledge on FBM and FBS, it can be used in the same fields, that is, natural time series in economics, fluctuations in solid, hydrology, and, more recently, by new problems in mathematical finance, telecommunication networks, and the environment (see [2–4]).(ii)A second application deals with the small ball probability problem of the sum of two not necessarily joint centered Gaussian random vectors 𝑋 and 𝑌 in a separable Banach space 𝐸 with norm ‖⋅‖ (see [5]). The small ball behavior of the FMFBS under the uniform norm can be investigated as a special case of the small ball probability problem of the sum of two centered Gaussian random vectors, having a log-type small ball factor (see [6]).(iii)Last but not least, this article extends El-Nouty's results [6–9] and consequently answers some new questions. Recall first two definitions of the Lévy classes, stated in [10]. Let {𝑍(𝑡),𝑡≥0} be a stochastic process defined on the basic probability space (Ω,𝒜).

Deffinition 1.4. The function 𝑓(𝑡),𝑡≥0, belongs to the lower-lower class of the process 𝑍,(𝑓∈LLC(𝑍)), if, for almost all 𝜔∈Ω, there exists 𝑡₀=𝑡₀(𝜔) such that 𝑍(𝑡)≥𝑓(𝑡) for every 𝑡>𝑡₀.

Deffinition 1.5. The function 𝑓(𝑡),𝑡≥0, belongs to the lower-upper class of the process 𝑍,(𝑓∈LUC(𝑍)), if, for almost all 𝜔∈Ω, there exists a sequence 0<𝑡₁=𝑡₁(𝜔)<𝑡₂=𝑡₂(𝜔)<⋯ with 𝑡_𝑛→+∞, as 𝑛→+∞, such that 𝑍(𝑡_𝑛)≤𝑓(𝑡_𝑛),𝑛∈ℕ^∗.

In the spirit of [6–9, 11], the main aim of this paper is to characterize the lower classes of the uniform norm of the FMFBS for any 0<𝐻₁,𝐻₂,𝐻₃,𝐻₄<1. More precisely, we want to compare the influence of two FBSs and to measure the weight of a log-type small ball factor versus another one.

2. Main Results

Our first result is given in the following theorem.

Theorem 2.1. Let 𝑋 and 𝑌 be any two joint Gaussian random vectors in a separable Banach space with norm ‖⋅‖. Assume that there exist 𝐶_𝑋≥1 and 𝐶_𝑌≥1 such that one has, for any 𝜖>0 small enough, −𝐶_𝑋≤𝜖^1/𝛼log(1/𝜖)^𝛽logℙ‖𝑋‖≤𝜖≤−1𝐶_𝑋−𝐶_𝑌,≤𝜖^1/𝛼log(1/𝜖)^𝛽logℙ‖𝑌‖≤𝜖≤−1𝐶_𝑌,(2.1) with 0<𝛼,𝛼<+∞,0≤𝛽,𝛽<+∞ and (𝛼,𝛽)≠(𝛼,𝛽).
If (𝛼<𝛼) or (𝛼=𝛼 and 𝛽>𝛽), then there exists 𝐾_𝑋≥𝐶_𝑋 depending on 𝐶_𝑋 only such that one has, for any 𝜖>0 small enough, −𝐾_𝑋≤𝜖^1/𝛼log(1/𝜖)^𝛽logℙ‖𝑋+𝑌‖≤𝜖≤−1𝐾_𝑋.(2.2)

Since the study of the lower classes of the FMFBM (resp., FMFBMFBS) under the uniform norm was investigated in [8] (resp., [9]), we focus our attention to the FMFBS. Set 𝑌(𝑡)=sup_{0≤𝑠≤𝑡}sup_{0≤𝑤1,𝑤2,𝑤3,𝑤4≤𝑠}||𝑋𝑤₁,𝑤₂,𝑤₃,𝑤₄,𝑠||,𝑡≥0.(2.3) Note first that, by the scaling property, we have, for any 𝜖>0, ℙ𝑌(𝑡)≤𝜖𝑡^{𝐻1+𝐻2+𝐻3+𝐻4}==ℙℙ𝑌(s1u)p≤_0≤𝑠≤1𝜖∶s=up𝜙(𝜖),_{0≤𝑤1,𝑤2,𝑤3,𝑤4≤𝑠}||𝑋𝑤₁,𝑤₂,𝑤₃,𝑤₄,𝑠||≤𝜖(2.4) where 𝜙 is named the small ball function and 𝛾∶=𝐻₁+𝐻₂+𝐻₃+𝐻₄ the scaling factor.

Recall that the small ball behavior of the FBS under the uniform norm was studied in [12, 13].

Set 𝛼=min(𝐻₁,𝐻₂,𝐻₃,𝐻₄), which is in ]0,1[. We introduce the number 𝛽 taking its values in {0,1+1/𝛼}. As a direct consequence of Theorem 2.1, we have the following corollary.

Corollary 2.2. There is a constant 𝐾₀,0<𝐾₀≤1, depending on 𝐻₁,𝐻₂,𝐻₃,𝐻₄,𝜆₁ and 𝜆₂ only, such that one has, for any 𝜖>0 small enough, exp−log(1/𝜖)^𝛽𝐾₀𝜖^1/𝛼≤𝜙(𝜖)≤exp−𝐾⁰log(1/𝜖)^𝛽𝜖^1/𝛼.(2.5)

Recall that we suppose (𝐻₁,𝐻₂)≠(𝐻₃,𝐻₄). In the sequel, there is no loss of generality to assume also that 𝐻₁≤𝐻₂,𝐻₃≤𝐻₄.(2.6) Thus when (𝐻₁=𝐻₂,𝐻₃<𝐻₄ and 𝐻₁≤𝐻₃), (𝐻₁=𝐻₂,𝐻₃=𝐻₄ and 𝐻₁<𝐻₃), or (𝐻₁<𝐻₂,𝐻₃=𝐻₄ and 𝐻₃≤𝐻₁), we emphasize that 𝛽=1+1/𝛼, that is, we have a log-type small ball factor.

Note first that the minimum 𝛼 plays a key role. This is not really surprising. Indeed, this phenomenon was already observed in [8, 9].

It appears that the sufficiency part of the lower classes of 𝑌 can be stated in a general framework. Roughly speaking, we follow the same lines as those of [6, 7].

Let {𝑌₀(𝑡),𝑡≥0} be a real-valued statistic of the two independent FBSs, 𝐵_𝐻1,𝐻2 and 𝐵_𝐻3,𝐻4, such that 𝑌₀(𝑡) is a nondecreasing function of 𝑡≥0.

The following notation is needed. If 𝕂 is a Hausdorff compact space, we denote, by 𝐶(𝕂), the space of all continuous functions from 𝕂 to ℝ equipped with the classical sup-norm. Let 𝕏=𝐶([0,1]²)×𝐶([0,1]²) be the product space equipped with the product topology. Denote, by L (𝐵_𝐻1,𝐻2,𝐵_𝐻3,𝐻4), the Gaussian measure associated to 𝐵_𝐻1,𝐻2 and 𝐵_𝐻3,𝐻4 and defined on 𝔹, the Borel 𝜎-field of 𝕏.

We assume that 𝑌₀ satisfies the following three conditions:

(i)The scaling condition. There exists 𝛾₀>0 such that ℙ𝑌₀(𝑡)≤𝜖𝑡^𝛾0=ℙ𝑌₀(1)≤𝜖∶=𝜙(𝜖).(2.7)(ii)The convexity condition. There exists a convex and 𝔹-measurable function 𝑔: (𝕏,L(𝐵_𝐻1,𝐻2,𝐵_𝐻3,𝐻4))→ℝ such that, for any 𝑡≥0,𝑌₀(𝑡)=𝑔(𝐵_𝐻1,𝐻2(𝑠₁𝑡,𝑠₂𝑡), 𝐵_𝐻3,𝐻4(𝑠₃𝑡,𝑠₄𝑡);0≤𝑠₁,𝑠₂,𝑠₃,𝑠₄≤1), and 𝑌₀(𝑡)<+∞, with probability 1.(iii)The log-type small ball condition. There exist 𝛼₀∈]0,𝛾₀],𝛽₀∈ℝ and a constant 𝐾,0<𝐾≤1, depending on 𝐻₁,𝐻₂,𝐻₃,𝐻₄,𝛾₀,𝛼₀ and 𝛽₀ only such that we have, for any 𝜖>0 small enough, exp−log(1/𝜖)^𝛽0𝐾𝜖^1/𝛼0≤𝜙(𝜖)≤exp−𝐾lo𝜖g(1_1/𝛼0/𝜖)^𝛽0.(2.8)

Note that these conditions generalize those of [6, 7]. The small ball function still plays a key role. The convexity of the function 𝜓 defined by 𝜓(𝜖)=−log𝜙(𝜖),0<𝜖<1, is ensured by (ii) (see [14, 15]).

Our second result is given in the following theorem.

Theorem 2.3. Let 𝑓(𝑡) be a positive nondecreasing function of 𝑡≥0. Assume that there exists 𝑚>0 such that (𝑓(𝑡)/𝑡^{𝛾0−𝛼0})(log𝑡^𝛾0/𝑓(𝑡))^{−𝛽0𝛼0}≥𝑚.
If 𝑓(𝑡)𝑡^𝛾0isboundedand^0+∞𝑓(𝑡)^−1/𝛼0𝑡^{(𝛾0/𝛼0)−1}log𝑡^𝛾0𝑓(𝑡)^𝛽0𝜙𝑓(𝑡)𝑡^𝛾0𝑑𝑡<+∞,(2.9) then one has 𝑓∈LLC𝑌₀.(2.10)

The sup-norm statistic 𝑌 clearly satisfies the three above conditions with 𝛾₀=𝛾=𝐻₁+𝐻₂+𝐻₃+𝐻₄,𝛼₀=𝛼=min(𝐻₁,𝐻₂,𝐻₃,𝐻₄)=min(𝐻₁,𝐻₃),𝛽₀=𝛽∈{0,1+1/𝛼}, and 𝐾=𝐾₀. Now, we characterize the necessity part of the lower classes of the FMFBMFBS. Our main result is stated in the following theorem.

Theorem 2.4. Let 𝑓(𝑡) be a positive nondecreasing function of 𝑡≥0 such that 𝑓(𝑡)/𝑡^𝛾 is a nonincreasing function of 𝑡>0.
If 𝑓∈LLC(𝑌),(2.11) then we have lim_𝑡→+∞𝑓(𝑡)𝑡^𝛾=0,^0+∞𝑓(𝑡)^−1/𝛾𝜙𝑓(𝑡)𝑡^𝛾𝑑𝑡<+∞.(2.12)

First, we can notice that Theorem 2.3 involves 𝛾₀,𝛼₀ and 𝛽₀. If 𝛽₀=0, Theorem 2.3 looks like [7, Theorem 1] or else like [6, Theorem 1.1]. Theorem 2.4 has the same form as the necessity part of [11, Theorem 1.2] or as the theorems obtained in [6–9]. In his previous works on the study of the lower classes, the author showed that the methodology in [11] led to two integral tests, these tests are actually identical when 𝛾=𝛼 and 𝛽=0. Here, 𝛼=min(𝐻₁,𝐻₃)<𝛾=𝐻₁+𝐻₂+𝐻₃+𝐻₄. This is why the integral tests of Theorems 2.3 and 2.4 have different forms. Moreover, since 𝛼<𝛾, we must assume, as in [6–9], that 𝑓(𝑡)/𝑡^𝛾 is not only bounded, but also a nonincreasing function of 𝑡>0. This last assumption will play a key role in some proofs. Finally, although they are two different integral tests, Theorems 2.3 and 2.4 are sharp. Indeed, set, if 𝛽=0, 𝑓(𝑡)=𝜆𝑡^𝛾(loglog𝑡)^𝛼,𝑡≥3,𝜆>0,(2.13) or else (i.e., 𝛽=1+(1/𝛼)) 𝑓(𝑡)=𝑡^𝛾(𝜆logloglog𝑡)^1+𝛼(loglog𝑡)^𝛼,𝑡≥16,𝜆>0.(2.14) If 𝜆 is small enough, then Theorem 2.3 yields 𝑓∈LLC(𝑌), or else if 𝜆 is large enough, then 𝑓∈LUC(𝑌) by applying Theorem 2.4.

In Section 3, we prove Theorem 2.1. The proof of Theorem 2.4 is postponed to Sections 4 and 5. In the latter, we establish some key small ball estimates. Note also that these estimates can be of independent interest. The proofs, which are modifications of those of [6, 7], will be consequently omitted, in particular, the proof of Theorem 2.3.

3. Proof of Theorem 2.1

Recall, first, a Gaussian correlation inequality stated in [5].

Theorem A. Let 𝜇 be a centered Gaussian measure on a separable Banach space 𝐸. Then for any 0<𝜆<1, and any symmetric convex sets 𝐴 and 𝐵 in 𝐸, 𝜇(𝐴∩𝐵)𝜇𝜆²𝐴+1−𝜆²𝐵≥𝜇(𝜆𝐴)𝜇1−𝜆²^1/2𝐵.(3.1) In particular, 𝜇(𝐴∩𝐵)≥𝜇(𝜆𝐴)𝜇1−𝜆²^1/2𝐵.(3.2)

Rougly speaking, the proof follows the same lines as those in [15] and will be split into two parts: the lower bound and the upper one.

Part I. The Lower Bound
Theorem A implies, for any 0<𝛿<1 and 0<𝜆<1, ℙ‖𝑋+𝑌‖≤𝜖≥≥ℙℙ‖‖𝑋𝑋‖‖≤≤𝜆(1(1−−𝛿)𝛿𝜖),𝜖‖𝑌ℙ‖≤‖𝑌‖𝛿𝜖≤1−𝜆₂^1/2𝛿𝜖.(3.3)

Then we get, by using (2.1), logℙ‖𝑋+𝑌‖≤𝜖≥−𝐶_𝑋log1/𝜆(1−𝛿)𝜖^𝛽𝜆(1−𝛿)𝜖^1/𝛼−𝐶_𝑌log1/1−𝜆²^1/2𝛿𝜖^𝛽1−𝜆²^1/2𝛿𝜖^1/𝛼.(3.4) Hence, since (𝛼<𝛼) or (𝛼=𝛼 and 𝛽>𝛽), there exists 𝐶^𝑋≥𝐶_𝑋 depending on 𝐶_𝑋 only such that we have, for any 𝜖>0 small enough, 𝜖^1/𝛼log(1/𝜖)^𝛽logℙ‖𝑋+𝑌‖≤𝜖≥−𝐶^𝑋𝜆(1−𝛿)^1/𝛼,(3.5) and the lower bound follows by taking 𝛿→0 and 𝜆→1.

Part II. The Upper Bound
A new use of Theorem A implies, for any 0<𝛿<1 and 0<𝜆<1, ℙ‖𝑋‖≤𝜖(1−𝛿)𝜆≥≥ℙℙ‖‖𝑋𝑋++𝑌𝑌‖‖≤≤𝜖𝜆𝜖,ℙ‖𝑌‖‖𝑌≤‖(≤1−𝛿1𝜖𝛿−)𝜆𝜆₂^1/2𝛿𝜖(1−𝛿)𝜆.(3.6)

Then combining (2.1) with the fact that (𝛼<𝛼) or (𝛼=𝛼 and 𝛽>𝛽), there exists 𝐶^𝑋≥𝐶_𝑋 depending on 𝐶_𝑋 only such that we have, for any 𝜖>0 small enough, 𝜖^1/𝛼log(1/𝜖)^𝛽logℙ‖𝑋+𝑌‖≤𝜖≤−𝜆(1−𝛿)^1/𝛼𝐶^𝑋,(3.7) and the upper bound follows by taking 𝛿→0 and 𝜆→1.

By choosing 𝐾_𝑋=max(𝐶^𝑋,𝐶^𝑋), we complete the proof of Theorem 2.1.

Remark 3.1. When 𝑋 and 𝑌 are independent, there is a simple proof without using the correlation inequality in the spirit of [5]. A direct proof of Corollary 2.2 can also be done as in [9].

4. Proof of Theorem 2.4, Part I

To simplify the reading of our paper, we introduce the following notation. Set 𝑎_𝑡=𝑓(𝑡)/𝑡^𝛾 and 𝑏_𝑡=𝜙(𝑎_𝑡).

Suppose here that, with probability 1, 𝑓(𝑡)≤𝑌(𝑡) for all 𝑡 large enough. We want to prove that lim_𝑡→+∞𝑎_𝑡=0 and ∫^∞0𝑎^{𝑡−1/𝛾}𝑏_𝑡(𝑑𝑡/𝑡)<+∞.

In the sequel, there is no loss of generality to assume that 𝑓(𝑡) is a continuous function of 𝑡≥0.

Lemma 4.1. One has lim_𝑡→+∞𝑎_𝑡=0.(4.1)

To prove Theorem 2.4, we will show that 𝑓∈LUC(𝑌) when ∫^∞0𝑎^{𝑡−1/𝛾}𝑏_𝑡(𝑑𝑡/𝑡)=+∞ and lim_𝑡→+∞𝑎_𝑡=0.

Following the same lines as those in [11], our aim is to construct a suitable subset 𝐽 of ℕ such that we have the following property for an appropriate family of sets (𝐸_𝑖)_𝑖∈𝐽 in a basic probability space: given 𝜖>0, there exist a number 𝐾 and an integer 𝑝 such that ∀𝑛∈𝐽,𝑛≥𝑝⟹_{𝑚∈𝐽,𝑚>𝑛}ℙ𝐸_𝑛∩𝐸_𝑚≤ℙ𝐸_𝑛𝐾+(1+𝜖)_{𝑚∈𝐽,𝑚>𝑛}ℙ𝐸_𝑚.(4.2)

Lemma 4.2. When ∫^∞0𝑎^{𝑡−1/𝛾}𝑏_𝑡(𝑑𝑡/𝑡)=+∞ and lim_𝑡→+∞𝑎_𝑡=0, we can find a sequence {𝑡_𝑛,𝑛≥1} with the two following properties: 𝑡_𝑛+1≥𝑡_𝑛1+𝑎^𝑡1/𝛾_𝑛,^∞_𝑛=1𝑏_𝑡𝑛=+∞.(4.3)

Remark 4.3. The condition “𝑓(𝑡)/𝑡^𝛾 is a nonincreasing function of 𝑡>0” is essential to prove Lemma 4.2 (see [7, page 373]).

To continue the construction of the set 𝐽, we need the following definition and notation.

Definition 4.4. Consider the interval 𝐴_𝑘=[2^𝑘,2^𝑘+1[,𝑘∈ℕ. If 𝑎^{𝑡−1/𝛾}_𝑖∈𝐴_𝑘,𝑖∈ℕ^∗, then one notes 𝑢(𝑖)=𝑘.

Next, set 𝐼_𝑘={𝑖∈ℕ^∗,𝑢(𝑖)=𝑘∈ℕ} which is finite by Lemma 4.1 and 𝑁_𝑘=exp𝐾₀𝛾log2^𝛽𝑘^𝛽2^{𝛾(𝑘−1)/𝛼},(4.4) where 𝐾₀ was defined in Corollary 2.2.

Notation
(i)𝑈_𝑚,𝑘={𝑖∈ℕ^∗,𝑖∈𝐼_𝑘,𝑖<𝑚,card𝐼_𝑘∩[𝑖,𝑚[≤𝑁_𝑘},𝑚∈ℕ^∗,𝑘∈ℕ;(ii)𝑘₀=inf{𝑛∈ℕ,2^{𝛾𝑛/𝛼}≥2^𝛾/𝛼/(𝐾₀(2^𝛾/𝛼−1)(𝛾log2)^𝛽)+2^2𝛾/𝛼/𝐾²⁰(2^𝛾/𝛼−1)}, (𝑘₀ depends on 𝛾,𝛼,𝛽 and 𝐾₀ only);(iii)𝑉_𝑚=⋃_𝑘∈ℕ𝑈_𝑚,𝑘, where 𝑚 is fixed, 𝑢(𝑚)=𝑘₁, and 𝑘≥𝑘₁+𝑘₀;(iv)𝑊=⋃_𝑚≥1𝑉_𝑚.

Now, we can define the set 𝐽 as follows: 𝐽=ℕ^∗−𝑊.(4.5) Since it is assumed that 𝑓(𝑡)/𝑡^𝛾 is a nonincreasing function of 𝑡>0 (being a particular case of the condition “𝑓(𝑡)/𝑡^𝛾 is bounded”),𝑖<𝑚⟹𝑘=𝑢(𝑖)≤𝑢(𝑚)=𝑘₁.(4.6) Moreover, since 𝑘₀≥1, we get 𝑘<𝑘₁+𝑘₀. Hence 𝑉_𝑚 is always an empty set (by construction). Thus we obtain 𝐽=ℕ^∗.

We have, by Lemma 4.2, _𝑛∈𝐽𝑏_𝑡𝑛=+∞.(4.7)

Lemma 4.5. 𝑛∈𝐽,𝑚∈𝐽,𝑛<𝑚, such that card (𝐼_𝑢(𝑚)∩[𝑛,𝑚])>exp(𝐾₀2^{𝑢(𝑚)−1}), one has 𝑡_𝑚𝑡_𝑛≥expexp𝐾⁰42^{min(𝑢(𝑛),𝑢(𝑚))}.(4.8)

Proof. Set 𝑘=𝑢(𝑛),𝑘₁=𝑢(𝑚) and 𝐺=𝐼_𝑘1∩[𝑛,𝑚]={𝑖₁,𝑖₂,...,𝑖_𝑧}, where 𝑛≤𝑖₁<𝑖₂<⋯<𝑖_𝑧≤𝑚. We have 𝑡_𝑚𝑡_𝑛=𝑡^𝑚𝑡_𝑖𝑧𝑡_𝑖𝑧𝑡_𝑖𝑧−1⋯𝑡^𝑖1𝑡_𝑛.(4.9) Note that, when 𝑖∈𝐼_𝑘1, we have 𝑡_𝑖+1≥𝑡_𝑖(1+𝑎^𝑡1/𝛾_𝑖)≥𝑡_𝑖(1+2^−𝑘1−1). Moreover, since card(𝐺)>exp(𝐾₀2^𝑘1−1) by hypothesis, (4.9) implies 𝑡_𝑚𝑡_𝑛≥expexp𝐾₀2^𝑘1−1log1+2^−𝑘1−1≥expexp𝐾⁰42^𝑘1,(4.10) when 𝑛 hence 𝑘₁ are large enough.
Thus, since 𝑘≤𝑘₁, (4.10) implies (4.8).
The proof of Lemma 4.5 is now complete.

5. Proof of Theorem 2.4, Part II

Consider, now, the events 𝐸_𝑛={𝑌(𝑡_𝑛)<𝑓(𝑡_𝑛)}. We have directly ℙ(𝐸_𝑛)=𝑏_𝑡𝑛, and, by (4.7), ∑_𝑛∈𝐽𝑏_𝑡𝑛=+∞. To prove (4.2), we remark that, given 𝑛∈𝐽,𝐽 can be rewritten as follows: 𝐽=𝐽^∪(⋃_𝑘∈ℕ𝐽_𝑘)∪𝐽^, where 𝐽^={𝑚∈𝐽,𝑡_𝑛≤𝑡_𝑚≤2𝑡_𝑛},𝐽_𝑘={𝑚∈𝐽∩𝐼_𝑘,𝑡_𝑚>2𝑡_𝑛,card(𝐼_𝑘∩[𝑛,𝑚])≤exp(𝐾₀2^𝑘−1)}, and 𝐽^=𝐽−(𝐽^∪(⋃_𝑘∈ℕ𝐽_𝑘)).

Our first key small ball estimate is given in the following lemma.

Lemma 5.1. Consider 0<𝑡<𝑢, and 𝜃,𝜈>0. Then one has ℙ𝑌(𝑡)≤𝜃𝑡^𝛾∩𝑌(𝑢)≤𝜈≤exp𝐾₅ℙ𝑌(𝑡)≤𝜃𝑡^𝛾exp−𝐾⁵(𝑢−𝑡)𝜈^1/𝛾,(5.1) where 𝐾₅ depends on 𝐻₁,𝐻₂,𝐻₃,𝐻₄,𝜆₁, and 𝜆₂ only.

Proof. Set 𝐹₁={𝑌(𝑡)≤𝜃𝑡^𝛾} and 𝐹₂={𝑌(𝑢)≤𝜈}. We have ℙ𝐹₁∩𝐹₂=≤ℙℙ𝐹𝐹₁₁∩sup_{0≤𝑠≤𝑢}sup_{0≤𝑤1,𝑤2,𝑤3,𝑤4≤𝑠}||𝑋𝑤₁,𝑤₂,𝑤₃,𝑤₄,𝑠||≤𝜈≤ℙ𝐹₁∩sup_{𝑡≤𝑠≤𝑢}sup_{0≤𝑤1,𝑤2,𝑤3,𝑤4≤𝑠}||𝑋𝑤₁,𝑤₂,𝑤₃,𝑤₄,𝑠||≤𝜈∩sup_{𝑡≤𝑠≤𝑢}sup_{0≤𝑤1,𝑤2≤𝑠}||𝜆₁𝑠^𝐻3+𝐻4𝐵_𝐻1,𝐻2𝑤₁,𝑤₂||≤𝜈.(5.2) Denote, by [𝑥], the integer part of a real 𝑥. Let 𝛿>0. We consider the sequence 𝑡_𝑘,𝑘∈{0,…,𝑛}, where 𝑡₀=𝑡,𝑡_𝑘+1=𝑡_𝑘+𝛿, and 𝑛=[(𝑢−𝑡)/𝛿]. Consider also the rectangles 𝑅_𝑗=[𝑡_𝑗,𝑡_𝑗+1]×[𝑡_𝑗,𝑡_𝑗+1], where 𝑗∈{0,…,𝑛−1}. Their area is (𝑡_𝑗+1−𝑡_𝑗)²=𝛿². Let 𝐺_𝑗 be the event defined by 𝐺_𝑗=𝐹₁∩sup_{𝑡≤𝑠≤𝑡𝑗}sup_{0≤𝑤1,𝑤2≤𝑠}||𝜆₁𝑠^𝐻3+𝐻4𝐵_𝐻1,𝐻2𝑤₁,𝑤₂||≤𝜈.(5.3) We have 𝐹₁∩𝐹₂⊂𝐺_𝑗.
Moreover, we have also 𝐺_𝑗+1⊂𝐺_𝑗∩𝑍_𝑗≤4𝜈,(5.4) where 𝑍_𝑗=𝜆₁𝑡^𝐻3+𝐻4_𝑗+1𝐵_𝐻1,𝐻2𝑡_𝑗+1,𝑡_𝑗+1−𝐵_𝐻1,𝐻2𝑡_𝑗,𝑡_𝑗+1−𝐵_𝐻1,𝐻2𝑡_𝑗+1,𝑡_𝑗+𝐵_𝐻1,𝐻2𝑡_𝑗,𝑡_𝑗.(5.5) Before rewritting 𝑍_𝑗, we recall the integral representation of 𝐵_{𝐻𝑘,𝐻𝑙},𝑘∈ℕ^∗,𝑙∈ℕ^∗, given by 𝐵_{𝐻𝑘,𝐻𝑙}𝑠_𝑘,𝑠_𝑙=^𝑠𝑘_−∞^𝑠𝑙_−∞𝑔_𝐻𝑘𝑠_𝑘,𝑢_𝑘𝑔_𝐻𝑙𝑠_𝑙,𝑢_𝑙𝑊𝑑𝑢_𝑘,𝑢_𝑙,(5.6) where 𝑊(𝑢_𝑘,𝑢_𝑙),𝑢_𝑘∈ℝ,𝑢_𝑙∈ℝ, is a standard Brownian sheet, 𝑔_𝐻(𝑠,𝑢)=𝑘^−12𝐻(max(𝑠−𝑢,0)^𝐻−1/2−max(−𝑢,0)^𝐻−1/2), and 𝑘_2𝐻 is a normalizing constant.
Hence 𝑍_𝑗 can be rewritten by (5.6) as follows: 𝑍_𝑗=𝑍_𝑗,1+𝑍_𝑗,2, where 𝑍_𝑗,1=𝜆₁𝑘^−12𝐻₁𝑘^−12𝐻₂𝑡^𝐻3+𝐻4_𝑗+1^𝑡𝑗+1_𝑡𝑗^𝑡𝑗+1_𝑡𝑗𝑡_𝑗+1−𝑢₁^𝐻1−1/2𝑡_𝑗+1−𝑢₂^𝐻2−1/2𝑊𝑑𝑢₁,𝑢₂.(5.7) Note also that 𝑍_𝑗,1 and 𝑍_𝑗,2 are independent.
Since ℙ(|𝑍_𝑗,1+𝑥|≤4𝜈) is maximum at 𝑥=0 and 𝑍_𝑗,1 and 𝐺_𝑗 are independent, we have ℙ𝐺_𝑗+1≤ℙ𝐺_𝑗ℙ||𝑍_𝑗,1||≤4𝜈.(5.8) The integral representation of 𝑍_𝑗,1 implies that 𝔼(𝑍_𝑗,1)=0 and Var𝑍_𝑗,1=𝜆²¹𝑘^−22𝐻₁𝑘^−22𝐻₂4𝐻₁𝐻₂𝛿^{2(𝐻1+𝐻2)}𝑡^{2(𝐻3+𝐻4)𝑗+1}≥𝜆²¹𝑘^−22𝐻₁𝑘^−22𝐻₂4𝐻₁𝐻₂𝛿^2𝛾∶=𝐿²𝛿^2𝛾.(5.9) Denote, by Φ, the distribution function of the absolute value of a standard Gaussian random variable. Then we obtain ℙ𝑍_𝑗+1≤ℙ𝑍_𝑗Φ4𝜈𝐿𝛿^𝛾,(5.10) and therefore, ℙ(𝐹₁∩𝐹₂)≤ℙ(𝐹₁)Φ(4𝜈/𝐿𝛿^𝛾)^𝑛.
Choosing 𝛿=𝜈^1/𝛾, we get 𝐾₅=−logΦ(2/𝐿). Lemma 5.1 is proved.

Lemma 5.2. ∑_{𝑚∈𝐽′}ℙ(𝐸_𝑛∩𝐸_𝑚)≤𝐾^𝑏_𝑡𝑛 and ∑_{𝑚∈(∪𝑘𝐽𝑘)}ℙ(𝐸_𝑛∩𝐸_𝑚)≤𝐾^𝑏_𝑡𝑛, where 𝐾^ and 𝐾^ are numbers.

Proof. Setting 𝑢=𝑡_𝑚,𝑡=𝑡_𝑛,𝜃=𝑎_𝑡𝑛 and 𝜈=𝑓(𝑡_𝑚), Lemma 5.1 implies ℙ𝐸_𝑛∩𝐸_𝑚≤exp𝐾₅𝑏_𝑡𝑛exp−𝐾⁵𝑡_𝑚−𝑡_𝑛𝑓𝑡_𝑚^1/𝛾.(5.11) Consider, first, the case when 𝑚∈𝐽^.
Lemma 4.2 implies that, for all 𝑖≥𝑛, we have 𝑡_𝑖+1−𝑡_𝑖≥𝑡_𝑖𝑎^𝑡1/𝛾_𝑖=𝑓(𝑡_𝑖)^1/𝛾≥𝑓(𝑡_𝑛)^1/𝛾. Then we can establish 𝑡_𝑚−𝑡_𝑛≥(𝑚−𝑛)𝑓𝑡_𝑛^1/𝛾,𝑓𝑡_𝑚≤𝑓𝑡_𝑛𝑡^𝑚𝑡_𝑛^𝛾≤2^𝛾𝑓𝑡_𝑛.(5.12) Combining (5.11) with (5.12), we get ℙ𝐸_𝑛∩𝐸_𝑚≤exp𝐾₅𝑏_𝑡𝑛exp−𝐾⁵(𝑚−𝑛)2,(5.13) which is the first part of Lemma 5.2.
Consider, now, the case 𝑚∈𝐽_𝑘.
Combining (5.11) with the definition of 𝐽_𝑘, we have ℙ𝐸_𝑛∩𝐸_𝑚≤exp𝐾₅𝑏_𝑡𝑛exp−𝐾⁵2𝑎_𝑡𝑚^1/𝛾.(5.14) Since 𝑢(𝑚)=𝑘, we get ℙ𝐸_𝑛∩𝐸_𝑚≤exp𝐾₅𝑏_𝑡𝑛exp−𝐾₅2^𝑘−1,(5.15) and consequently, by noting that card𝐽_𝑘≤card(𝐼_𝑘∩[𝑛,𝑚])≤exp(𝐾₀2^𝑘−1) and by assuming 𝐾₀<𝐾₅, we have _𝑚_∈𝐽𝑘ℙ𝐸_𝑛∩𝐸_𝑚≤exp𝐾₅𝑏_𝑡𝑛exp𝐾₀−𝐾₅2^𝑘−1.(5.16) Lemma 5.2 is, therefore, proved.

To deal with the set 𝐽^, we first state a standard large deviation result and a technical lemma (see [6]).

Lemma A. Let 𝑋={𝑋(𝑠₁,𝑠₂),(𝑠₁,𝑠₂)∈[0,1]²} be a separable real-valued centered Gaussian process such that 𝑋(0,0)=0 with probability 1 and satisfying, for any [𝑠₁,𝑠₁+ℎ₁]×[𝑠₂,𝑠₂+ℎ₂]⊂[0,1]²,𝔼𝑋𝑠¹,𝑠₁+ℎ₁×𝑠₂,𝑠₂+ℎ₂²^1/2≤𝜅ℎ₁,ℎ₂≤𝑐_𝜅ℎ^𝛼1₁ℎ^𝛼2₂,𝛼₁>0,𝛼₂>0,(5.17) where 𝑋𝑠₁,𝑡₁×𝑠₂,𝑡₂=_{[𝑠1,𝑡1]×[𝑠2,𝑡2]}𝑋𝑑𝑢₁,𝑢₂.(5.18) Then one has, for 𝑐^𝜅−1𝛿>1,ℙsup_{(𝑠1,𝑠2)∈[0,1]2}||𝑋𝑠₁,𝑠₂||≥𝛿≤1𝐶exp−𝐶𝑐^𝜅−1𝛿²,(5.19) where 𝐶 is a positive constant independent of 𝑐_𝜅 and 𝛿.

Lemma B. One has, for 𝜖₁>𝜖/2, where 𝜖 is small enough, exp−𝐾₃|𝜖₁−𝜖|log1/𝜖^𝛽𝜖^1+1/𝛼≤𝜙𝜖¹𝜙(𝜖)≤exp𝐾³|𝜖₁−𝜖|log1/𝜖^𝛽𝜖^1+1/𝛼,(5.20) where 𝐾₃>0.

Building on Lemmas A and B, one can establish our last key small ball estimate in the following result.

Lemma 5.3. Let 𝜆 be a real number such that 1/2<𝜆<1. Set 𝑟=min1−max𝐻¹,𝐻₂,𝐻₃,𝐻₄3,(1−3𝜆)𝛼.(5.21) Then one has, for 𝑢≥2𝑡,ℙ𝑌(𝑡)≤≤𝜙𝜃𝑡(^𝛾𝜃,)𝑌𝜙((𝑢𝜈))≤ex𝜈p𝑢^𝛾2𝑡𝑢^𝑟𝐾₃lo𝜃g(1/𝜃^1+(1/𝛼))^𝛽+log(1/𝜈)^𝛽+31𝜈^1+(1/𝛼)𝐶_12,2exp−𝐶^12,24𝜆²¹𝐾^2𝐻_1,2𝑢𝑡^𝑟+𝐶1_34,2exp−𝐶^34,2+314𝜆²²𝐾^2𝐻_3,2𝑢𝑡^𝑟𝐶_12,1exp−𝐶^12,14𝜆²¹𝐾^2𝐻_1,1𝑢𝑡^𝑟+𝐶1_34,1exp−𝐶^34,14𝜆²²𝐾^2𝐻_3,1𝑢𝑡^𝑟,(5.22) where 𝐾_𝐻1,1,𝐾_𝐻1,2>0 depend on 𝐻₁(𝐻₁≤𝐻₂) only, 𝐾_𝐻3,1,𝐾_𝐻3,2>0 depend on 𝐻₃(𝐻₃≤𝐻₄) only, 𝐾₃>0 is defined as in Lemma B, and 𝐶_12,1,𝐶_12,2,𝐶_34,1,𝐶_34,2>0 are defined as in Lemma A.

Proof. Set 𝑄=ℙ(𝑌(𝑡)≤𝜃𝑡^𝛾,𝑌(𝑢)≤𝜈𝑢^𝛾).
Set 𝑣=√𝑢𝑡. If 𝑡=𝑜(𝑢), then 𝑡=𝑜(𝑣) and 𝑣=𝑜(𝑢).
Based on (5.6), 𝐵_𝐻1,𝐻2 and 𝐵_𝐻3,𝐻4 can be split as follows: 𝐵_𝐻1,𝐻2=𝐵_{𝐻1,𝐻2,1}+𝐵_{𝐻1,𝐻2,2},𝐵_𝐻3,𝐻4=𝐵_{𝐻3,𝐻4,1}+𝐵_{𝐻3,𝐻4,2},(5.23) where we have, for (𝑖,𝑗)∈{(1,2),(3,4)}, 𝐵_{𝐻𝑖,𝐻𝑗,1}𝑤_𝑖,𝑤_𝑗=_{|𝑥𝑖|≤𝑣}^𝑤𝑗_−∞𝑔_𝐻𝑖𝑤_𝑖,𝑥_𝑖𝑔_𝐻𝑗𝑤_𝑗,𝑥_𝑗𝑊𝑑𝑥_𝑖,𝑥_𝑗.(5.24) Note that 𝐵_{𝐻1,𝐻2,1} and 𝐵_{𝐻1,𝐻2,2} are independent as 𝐵_{𝐻3,𝐻4,1} and 𝐵_{𝐻3,𝐻4,2}.
Equation (5.23) implies that the FMFBS 𝑋 can be rewritten as follows: 𝑋=𝑋₁+𝑋₂, where 𝑋_𝑖𝑤₁,𝑤₂,𝑤₃,𝑤₄,𝑠=𝜆₁𝑠^𝐻3+𝐻4𝐵_{𝐻1,𝐻2,𝑖}𝑤₁,𝑤₂+𝜆₂𝑠^𝐻1+𝐻2𝐵_{𝐻3,𝐻4,𝑖}𝑤₃,𝑤₄.(5.25) Set 𝑌_𝑖(𝑡)=sup_{0≤𝑠≤𝑡}sup_{0≤𝑤1,𝑤2,𝑤3,𝑤4≤𝑠}||𝑋_𝑖𝑤₁,𝑤₂,𝑤₃,𝑤₄,𝑠||,𝑡≥0,𝑖∈{1,2}.(5.26) Then, given 𝛿>0, we have (see [11]) 𝑄≤𝜙(𝜃+2𝛿)𝜙(𝜈+2𝛿)+3ℙ𝑌₂(𝑡)>𝛿𝑡^𝛾+3ℙ𝑌₁(𝑢)>𝛿𝑢^𝛾.(5.27) Equation (5.20) implies 𝜙(𝜃+2𝛿)≤𝜙(𝜃)exp2𝛿𝐾₃lo𝜃g(1/𝜃^1+(1/𝛼))^𝛽,(5.28) and, consequently, 𝜙(𝜃+2𝛿)𝜙(𝜈+2𝛿)≤𝜙(𝜃)𝜙(𝜈)exp2𝛿𝐾₃lo𝜃g(1/𝜃^1+(1/𝛼))^𝛽+log(1/𝜈)^𝛽𝜈^1+(1/𝛼).(5.29) If we choose 𝛿=(𝑡/𝑢)^𝑟, then we get the first term of the RHS of Lemma 5.3.
Next, we want to obtain an upper bound of ℙ𝑌₂(=𝑡)ℙ>𝛿𝑡s^𝛾up_0≤𝑠≤1sup_{0≤𝑤1,𝑤2,𝑤3,𝑤4≤𝑠}||𝜆₁𝑠^𝐻3+𝐻4𝐿_{𝐻1,𝐻2,2}𝑤₁,𝑤₂+𝜆₂𝑠^𝐻1+𝐻2𝐿_{𝐻3,𝐻4,2}𝑤₃,𝑤₄||>𝛿,(5.30) where we have, for (𝑖,𝑗)∈{(1,2),(3,4)}, 𝐿_{𝐻𝑖,𝐻𝑗,2}𝑤_𝑖,𝑤_𝑗=_{|𝑥𝑖|≥𝑣/𝑡}^𝑤𝑗_−∞𝑔_𝐻𝑖𝑤_𝑖,𝑥_𝑖𝑔_𝐻𝑗𝑤_𝑗,𝑥_𝑗𝑊𝑑𝑥_𝑖,𝑥_𝑗.(5.31) We can show, by standard computations, that ℙ𝑌₂(𝑡)>𝛿𝑡^𝛾≤ℙsup_{0≤𝑤1,𝑤2≤1}||𝐿_{𝐻1,𝐻2,2}𝑤₁,𝑤₂||>𝛿+ℙsup_{0≤𝑤3,𝑤4≤1}2||𝜆₁||||𝐿_{𝐻3,𝐻4,2}𝑤₃,𝑤₄||>𝛿2||𝜆₂||.(5.32) Denote, by 𝜎_𝐻, the covariance function of a FBM 𝐵_𝐻. Set 𝜎_𝐻,2, the covariance function of the process {𝐵_𝐻,2(𝑤),0≤𝑤≤1}, defined by 𝐵_𝐻,2(𝑤)=_{|𝑥|≥𝑣/𝑡}𝑔_𝐻(𝑤,𝑥)𝑊(𝑑𝑥),(5.33) where 𝑊(𝑥),𝑥∈ℝ, is a Wiener process.
Since 𝔼𝐿_{𝐻1,𝐻2,2}𝑤₁,𝑤₂𝐿_{𝐻1,𝐻2,2}𝑤^1,𝑤^2=𝜎_𝐻1,2𝑤₁,𝑤^1×𝜎_𝐻2𝑤₂,𝑤^2,(5.34) we have, for any [𝑤₁,𝑤₁+ℎ₁]×[𝑤₂,𝑤₂+ℎ₂]⊂[0,1]²,𝔼𝐿=𝔼^𝐻₁^,𝐻₂^,2𝑤₁,𝑤₁+ℎ₁×𝑤_{2[𝑤1,𝑤1+ℎ1]×[𝑤2,𝑤2+ℎ2]},𝑤₂+ℎ₂²×𝐿_{𝐻1,𝐻2,2}𝑑𝑥₁,𝑥₂_{[𝑤1,𝑤1+ℎ1]×[𝑤2,𝑤2+ℎ2]}≤𝐿_{𝐻1,𝐻2,2}𝑑𝑥^1,𝑥^2_{𝑤1+ℎ1𝑤1}^𝑤1+ℎ1_𝑤1||𝜎_𝐻1,2𝑥₁,𝑥^1||𝑑𝑥₁𝑑𝑥^1×^𝑤2+ℎ2_𝑤2^𝑤2+ℎ2_𝑤2||𝜎_𝐻2𝑥₂,𝑥^2||𝑑𝑥₂𝑑𝑥^2∶=I×II.(5.35) Consider II first. We get, by the inequality of Cauchy-Schwarz, II≤^𝑤2+ℎ2_𝑤2^𝑤2+ℎ2_𝑤2𝑥^𝐻2₂𝑥^𝐻2₂𝑑𝑥₂𝑑𝑥^2≤ℎ²².(5.36) Consider I now.
A straight computation implies that there exists 𝐾_𝐻1,2>0 depending on 𝐻₁ such that 𝔼𝐵_𝐻1,2𝑥₁²≤𝐾^2𝐻_1,2𝑥²¹𝑣/𝑡^2𝐻1−2,(5.37) and, consequently, by the inequality of Cauchy-Schwarz, ||𝜎_𝐻1,2𝑥₁,𝑥^1||≤𝐾^2𝐻_1,2𝑥₁𝑥^1(𝑣/𝑡)^2𝐻1−2.(5.38) So we get 𝐼≤𝐾^2𝐻_1,2(𝑣/𝑡)^2𝐻1−2ℎ²¹.(5.39) Hence, combining (5.35) with (5.36) and (5.39), we have 𝔼𝐿_{𝐻1,𝐻2,2}𝑤₁,𝑤₁+ℎ₁×𝑤₂,𝑤₂+ℎ₂²≤𝐾^2𝐻₁^,2(𝑣/𝑡)^2𝐻1−2ℎ²¹ℎ²².(5.40) An application of Lemma A with 𝛼₁=𝛼₂=1, 𝑐_𝜅=𝐾_𝐻1,2(𝑣/𝑡)^𝐻1−1, and 𝑐^𝜅−1𝛿>1 implies that ℙsup_{0≤𝑤1,𝑤2≤1}||𝐿_{𝐻1,𝐻2,2}𝑤₁,𝑤₂||>𝛿2||𝜆₁||≤𝐶1_12,2exp−𝐶^12,2𝐾^2𝐻_1,2𝑣²/𝑡²^𝐻1−1𝛿²4𝜆²¹.(5.41) Similarly, we can establish ℙsup_{0≤𝑤3,𝑤4≤1}||𝐿_{𝐻3,𝐻4,2}𝑤₃,𝑤₄||>𝛿2||𝜆₂||≤𝐶1_34,2exp−𝐶^34,2𝐾^2𝐻_3,2𝑣²/𝑡²^𝐻3−1𝛿²4𝜆²².(5.42) Set 𝛿=(𝑡/𝑢)^𝑟. Recall that 𝑣²=𝑢𝑡 and 𝑟≤(1−max(𝐻₁,𝐻₂,𝐻₃,𝐻₄))/3. Combining (5.32) with (5.41) and (5.42), we get ℙ𝑌₂(𝑡)>𝛿𝑡^𝛾≤1𝐶_12,2exp−𝐶^12,24𝜆²¹𝐾^2𝐻_1,2𝑢𝑡^𝑟+𝐶1_34,2exp−𝐶^34,24𝜆²²𝐾^2𝐻_3,2𝑢𝑡^𝑟,(5.43) that is the second term of the RHS of Lemma 5.3.
Finally, we can establish a similar result for ℙ(𝑌₁(𝑢)>𝛿𝑢^𝛾), that is, ℙ𝑌₁(𝑢)>𝛿𝑢^𝛾≤1𝐶_12,1exp−𝐶^12,14𝜆²¹𝐾^2𝐻_1,1𝑢𝑡^𝑟+𝐶1_34,1exp−𝐶^34,14𝜆²²𝐾^2𝐻_3,1𝑢𝑡^𝑟,(5.44) which achieves the proof of Lemma 5.3.

Finally, we state the last technical lemma.

Lemma 5.4. There exists an integer 𝑝 such that if 𝑛>sup_𝑠≤𝑝(sup𝐼_𝑠), then for 𝑚∈𝐽^,𝑚>𝑛, given 𝜖>0, one has ℙ(𝐸_𝑛∩𝐸_𝑚)≤(1+𝜖)𝑏_𝑡𝑛𝑏_𝑡𝑚.

Proof. Let 𝑢(𝑛)=𝑘^ and 𝑢(𝑚)=𝑘₁. We have, by Lemma 4.5, 𝑡_𝑚𝑡_𝑛≥expexp𝐾⁰42^{min(𝑘′,𝑘1)}.(5.45) Let 𝑝∈ℕ. Then 𝑘^>𝑝 and 𝑘₁>𝑝. Thus we have min(𝑘^,𝑘₁)>𝑝.
Set 𝑡=𝑡_𝑛,𝑢=𝑡_𝑚,𝜃=𝑎_𝑡𝑛 and 𝜈=𝑎_𝑡𝑚. Note that log(1/𝜃)≤1/𝜃≤2^{(𝑘′+1)𝛼},log(1/𝜈)≤1/𝜈≤2^{(𝑘1+1)𝛼}, and 1/𝑏_𝑡𝑛𝑏_𝑡𝑚=exp(𝜓(𝜃)+𝜓(𝜈)).
By using Lemma 5.3 and letting 𝑝→+∞, we end the proof of Lemma 5.4.

Lemmas 5.2 and 5.4 yield that (4.2) holds. Combining Borel-Cantelli's second lemma with (4.2) and (4.7), we show that, given 𝜖>0, _𝑛∈𝐽ℙ𝐸_𝑛≥1+2𝐾𝜖⟹1+12𝜖≤ℙ_𝑛∈𝐽𝐸_𝑛=ℙ_𝑛∈𝐽𝑌𝑡_𝑛≤𝑓𝑡_𝑛,(5.46) and, consequently, 𝑓∈LUC(𝑌). The proof of Theorem 2.4 is now complete.

Acknowledgment

The author thanks the referee for the insightful comments.