Survival Exponents for Some Gaussian Processes

Abstract

The problem is a power-law asymptotics of the probability that a self-similar process does not exceed a fixed level during long time. The exponent in such asymptotics is estimated for some Gaussian processes, including the fractional Brownian motion (FBM) in (−𝑇₋,𝑇), 𝑇≥𝑇₋≫1 and the integrated FBM in (0,𝑇), 𝑇≫1.

1. The Problem

Let 𝑥(𝑡),𝑥(0)=0 be a real-valued stochastic process with the following asymptotics: 𝑃𝑥(𝑡)<1,𝑡∈Δ_𝑇=𝑇^{−𝜃𝑥+𝑜(1)},𝑇→∞, (1.1) where 𝜃_𝑥 is the so-called survival exponent of 𝑥(𝑡). Below we focus on estimating 𝜃_𝑥 for some self-similar Gaussian processes in extended intervals Δ_𝑇=(0,𝑇) and (−𝑇₋,𝑇), 𝑇≥𝑇₋≫1. Usually the estimation of the survival exponents is based on Slepian’s lemma. The estimation requires reference processes with explicit or almost explicit values of 𝜃. Unfortunately, the list of such processes is very short. This includes the fractional Brownian motion (FBM), 𝑤_𝐻(𝑡), of order 0<𝐻<1 both with one- and multidimensional time. According to Molchan ([1]) 𝜃_𝑤𝐻=1−𝐻forΔ_𝑇=(0,𝑇),𝜃_𝑤𝐻=𝑑forΔ_𝑇=(−𝑇,𝑇)^𝑑. (1.2) Another important example is the integrated Brownian motion 𝐼(𝑡)=∫^𝑡₀𝑤(𝑠)𝑑𝑠 with the exponent 𝜃_𝐼=14,Δ_𝑇=(0,𝑇), (1.3) (Sinai [2]).

The nature of this result is best understood in terms of a series of generalizations where the integrand is a random walk with discrete or continuous time (see, e.g., Isozaki and Watanabe [3]; Isozaki and Kotani [4]; Simon [5]; Vysotsky [6, 7]; Aurzada and Dereich [8]; Dembo et al. [9]; Denisov and Wachtel [10]. The extension of (1.3) to include the case of the integrated fractional Brownian motion, 𝐼_𝐻(𝑡)=∫^𝑡₀𝑤_𝐻(𝑠)𝑑𝑠, remains an important; but as yet unsolved problem.

Below we consider the survival exponents for the following Gaussian processes: 𝐼_𝐻(𝑡),𝑡∈(0,𝑇);𝜒_𝐻(𝑡)=sign(𝑡)𝑤_𝐻(𝑡),𝑡∈(−𝑇,𝑇); FBM in Δ_𝑇=(−𝑇^𝛼,𝑇), 0≤𝛼≤1; the Laplace transform of white noise with Δ_𝑇=(0,𝑇); the fractional Slepian’s stationary process whose correlation function is 𝐵_𝑆𝐻(𝑡)=(1−|𝑡|^2𝐻)₊, 0<𝐻≤1/2.

Our approach to the estimation of 𝜃 is more or less traditional. Namely, any self-similar process 𝑥(𝑡) in Δ_𝑇=(0,𝑇) generates a dual stationary process 𝑥(𝑠)=𝑒^−ℎ𝑠𝑥(𝑒^𝑠), 𝑠<ln𝑇∶=𝑇, where ℎ is the self-similarity index of 𝑥(𝑡). For a large class of Gaussian processes, relation (1.1) induces the dual asymptotics 𝑃𝑥(𝑠)≤0,0<𝑠<𝑇=exp−𝜃_𝑥𝑇(1+𝑜(1)),𝑇→∞, (1.4) with the same exponent 𝜃_𝑥=𝜃_𝑥, [1, 11]. More generally, the dual exponent is defined by the asymptotics 𝑃𝑥(𝑡)≤0,𝑡∈Δ_𝑇⧵(−1,1)=exp−𝜃_𝑥𝑇(1+𝑜(1)). (1.5) To formulate the simplest condition for the exponents to be equal, we define one more exponent 𝜃̆_𝑥 by means of the asymptotics 𝑃𝑡^∗_𝑇≤1=𝑇^{−̆𝜃𝑥+𝑜(1)}, (1.6) where 𝑡^∗_𝑇 is the position of the maximum of 𝑥(𝑡) in Δ_𝑇, that is, 𝑥(𝑡^∗_𝑇)=sup(𝑥(𝑡),𝑡∈Δ_𝑇).

Lemma 1.1 (see [1, 11]). Let 𝑥(𝑡),𝑥(0)=0 be a self-similar continuous Gaussian process in Δ_𝑇=(−𝑇₋,𝑇),𝑇₋≤𝑇 and (𝐻_𝑥(Δ_𝑇),‖‖_𝑇) be the reproducing kernel Hilbert space associated with 𝑥(𝑡). Suppose that there exists such an element 𝜑 of 𝐻_𝑥(Δ_𝑇) that 𝜑(𝑡)≥1,|𝑡|>1 and ‖𝜑‖²_𝑇=𝑜(ln𝑇). Then 𝜃_𝑥,𝜃_𝑥, and 𝜃̆_𝑥 can exist simultaneously only; moreover, the exponents are equal to each other.

The equality 𝜃=𝜃 reduces the original problem to the estimation of 𝜃. Nonnegativity of the correlation function of 𝑥(𝑠) guarantees the existence of the exponent 𝜃, [12]. In turn, the inequality of two correlation functions, 𝐵₁(𝑠)≤𝐵₂(𝑠), 𝐵_𝑖(0)=1, implies, by Slepian’s lemma, the inverted inequality for the corresponding exponents: 𝜃₁≥𝜃₂.

An essentially different approach is required to find the explicit value of 𝜃 for FBM in Δ_𝑇=(−𝑇^𝛼,𝑇) and to estimate 𝜃 in (1.4) for the fractional Slepian process with a small parameter 𝐻.

2. Examples

2.1. Integrated Fractional Brownian Motion

Consider the process 𝐼_𝐻(𝑡)=^𝑡₀𝑤_𝐻(𝑠)𝑑𝑠, (2.1) where 𝑤_𝐻(𝑡) is the fractional Brownian motion, that is, a Gaussian random process with the stationary increments: 𝐸|𝑤_𝐻(𝑡)−𝑤_𝐻(𝑠)|²=|𝑡−𝑠|^2𝐻, 𝑤_𝐻(0)=0. Molchan and Khokhlov [13, 14] analyzed the exponent 𝜃_𝐼𝐻 theoretically and numerically and formulated the following Hypothesis: 𝜃_𝐼𝐻=𝐻(1−𝐻) for Δ_𝑇=(0,𝑇)) and 𝜃_𝐼𝐻=1−𝐻 for Δ_𝑇=(−𝑇,𝑇).

The unexpected symmetry 𝜃_𝐼𝐻=𝜃_𝐼1−𝐻 for Δ_𝑇=(0,𝑇) caused some doubt as to the numerical results. To support the hypothesis, Molchan [11] derived the following estimates of 𝜃 for 𝐼_𝐻(𝑡): 𝜌𝐻(1−𝐻)≤𝜃⁺_𝐼𝐻≤𝜃_𝐼^−/+_𝐻≤(1−𝐻), (2.2) where 𝜌 is a small constant and (+) and (−/+) are indicators of the intervals Δ_𝑇=(0,𝑇) and Δ_𝑇=(−𝑇,𝑇), respectively. Note that, in the case of 𝐻<1/2 and Δ_𝑇=(−𝑇,𝑇), it is unknown whether the exponent exists. In such cases we have to operate with upper 𝜃 and lower 𝜃 exponents. Therefore, 𝜃_𝐼^−/+_𝐻 in (2.2) for 𝐻<1/2 is any number from the interval (𝜃,𝜃). The relation (2.2) can be improved as follows.

Proposition 2.1. For Δ_𝑇=(0,𝑇), one has(a) 𝜃_𝐼𝐻≥𝜃_𝐼1−𝐻,0<𝐻≤0.5,(b) 0.5(𝐻∧𝐻)≤𝜃_𝐼𝐻≤𝐻,𝐻=1−𝐻,(c) 𝜃_𝐼𝐻≤(1−(𝐻∧𝐻)²)/12.

Proof. The identity of the dual exponents for 𝐼_𝐻(𝑡) follows from [14]; the dual survival exponent exists because the dual correlation function, 𝐵_𝐼𝐻(𝑠)=(2+4𝐻)⁻¹(2+2𝐻)𝑒^𝐻𝑠+𝑒^−𝐻𝑠−𝑒^(1+𝐻)𝑠−𝑒^{−(1+𝐻)𝑠}+𝑒^𝑠/2−𝑒^{−𝑠/22𝐻+2} (2.3) is positive. The inequality (a) is a consequence of the relation 𝐵_𝐼𝐻(𝑡)≤𝐵_𝐼1−𝐻(𝑡)1,0<𝐻≤2. (2.4)
To prove (b, c), we use the correlation function of the process 𝐼̃_1/2(𝑝𝑠), that is, 𝐵_𝐼1/2=1𝑝𝑠2−𝑝|𝑠|3exp2−|𝑠|−exp3𝑝2, (2.5) and the respective exponent 𝜃=𝑝/4 (see (1.3)). The relation 𝐵_𝐼𝐻(𝑡)≤𝐵_𝐼1/21𝑝𝑡,𝐻≥2,𝑝=2(1−𝐻), (2.6) implies 𝜃_𝐼𝐻≥(1−𝐻)/2 for 𝐻≥1/2. Using (a) in addition, we come to the lower bound in (b) because 𝜃_𝐼𝐻≥𝜃_𝐼1−𝐻≥𝐻/2 for 𝐻≤1/2.
Similarly, the relation 𝐵_𝐼𝐻(𝑡)≥𝐵_𝐼1/21𝑝𝑡,𝐻≤2,𝑝=21−𝐻²3, (2.7) implies (c) for all 𝐻. A test of the purely analytical facts (2.4), (2.6), and (2.7) is given in the appendix.

Remark 2.2. Proposition 2.1(a) follows from the more informative relation: 𝑃𝐼_𝐻̃(𝑠)≤0,𝑠∈0,𝑇≤𝑃𝐼̃_1−𝐻(𝑠)≤0,𝑠∈0,𝑇. (2.8) This inequality is important for understanding the numerical result by Molchan and Khokhlov [13] represented in the form of empirical estimates of 𝜃_𝐼𝐻 in Figure 1. We can see that the empirical estimates show small but one-sided deviations from the hypothetical curve 𝜃=𝐻(1−𝐻) before and after 𝐻=1/2. The signs of these deviations are consistent with (2.8), while the amplitudes are compatible with the model 𝑃𝐼_𝐻̃(𝑠)≤0,𝑠∈0,𝑇≈𝐶𝑇^𝛼(𝐻)exp−𝐻(1−𝐻)𝑇,𝑇≫1,sgn𝛼(𝐻)=sign(𝐻−0.5), (2.9) and 𝛼(𝐻)=𝐻−0.5 (more can be found in [13]).

Figure 1 

The survival exponents 𝜃𝐼𝐻 for the integrated fractional Brownian motion in Δ𝑇=(−𝑇,𝑇): hypothetical values (parabolic line), empirical estimates (small circles, squares), and theoretical bounds (shaded zone given by Proposition 2.1(b, c)). The empirical exponents are based on the model ((2.9), 𝛼(𝐻)=0) in three time intervals of 𝑇=ln𝑇∶ ln(1/𝜀)≤𝑇(1−𝐻)𝐻≤ln(10/𝜀), where 𝜀=0.01, 0.003, and 0.001 (see more in [13]).

2.2. The Laplace Transform of White Noise

Consider the process 𝐿(𝑡)=𝑡∫^∞₀𝑒^−𝑡𝑢𝑑𝑤(𝑢), where 𝑤(𝑢) is Brownian motion. The dual stationary process 𝐿(𝑠) has the correlation function 𝐵_𝐿(𝑠)=1/cosh(𝑠/2). Using (2.5) as a majorant of 𝐵_𝐿(𝑠), we improve the lower bound of 𝜃_𝐿 as follows.

Proposition 2.3. 3^−1/2≤4𝜃_𝐿≤1.

Proof. That the exponents for the dual processes 𝐿 and 𝐿 are equal follows from Lemma 1.1 with 𝜑(𝑡)=𝑡(1+𝜀_𝑇)/(𝑡+𝜀_𝑇), where 𝜀_𝑇=1/√ln𝑇. For indeed, 𝜑(𝑡)=𝐸𝐿(𝑡)𝜂, where 𝜂=(1+𝜀_𝑇⁻¹)𝐿(𝜀_𝑇). By definition of the Hilbert space 𝐻_𝐿(Δ_𝑇), we have the desired estimate: 𝜑²_𝑇=𝐸𝜂²=𝜀_𝑇⁻¹𝜀_𝑇+1²2=𝑂√ln𝑇. (2.10) By (1.3) and Slepian’s lemma, the relation 𝐵_𝐼1/2(𝑡)≤𝐵_𝐿𝑝𝑡,𝑝≤1 (2.11) has as a consequence the estimate 4𝑝𝜃_𝐿≤1. The opposite inequality 𝐵_𝐼1/2(𝑡)≥𝐵_𝐿𝑝𝑡,𝑝²≥3, (2.12) implies 4𝑝𝜃_𝐿≥1. The test of ((2.11), 𝑝=1) and ((2.12), 𝑝=2) is very simple and yields the Li and Shao [12, 15] estimates: 0.5<4𝜃_𝐿<1. The appendix contains a proof of (2.11), (2.12) for all interesting values of 𝑝: 1,2, and √3.

Remark 2.4. The dual survival exponent of 𝐿(𝑡) is of interest as a parameter of the following asymptotic relation: 𝑃^2𝑛₀𝜉_𝑖𝑥^𝑖≠0,𝑥∈𝑅¹=(2𝑛)⁻⁴𝜃_𝐿+𝑜(1),𝑛→∞, (2.13) for random polynomials with the standard Gaussian independent coefficients [16]. A continuous analogue of the polynomial on any of four intervals 0<±𝑥^±1≤1 is the Laplace transform of white noise, which partially explains the appearance of 𝜃_𝐿 in the asymptotic relation (2.13). Simulations suggest 4𝜃_𝐿=0.76±0.03, [16] and 4𝜃_𝐿≈0.75, [17].

2.3. Fractional Slepian’s Process

We reserve this term for a Gaussian stationary process 𝑆_𝐻(𝑡) with correlation function 𝐵_𝑆𝐻(𝑡)=1−|𝑡|^2𝐻₊1,0<𝐻≤2, (2.14) because 𝑆_1/2(𝑡) is known as the Slepian process and 𝑆_𝐻(𝑡)−𝑆_𝐻(0),0<𝑡≤1, is equal in distribution to the fractional Brownian motion on the interval (0,1). By the Polya criterion, the fractional Slepian process exists because 𝐵_𝑆𝐻(𝑡) is a nonincreasing and a convex function on the semiaxis. The fact of the correlation function being nonnegative guarantees the existence of 𝜃_𝑆𝐻 in (1.4). 𝑆_𝐻(𝑡) can be useful as a reference process in estimation of the survival exponents. Therefore it is important to have accurate estimates of the exponent for 𝑆_𝐻(𝑡). The case of small 𝐻 is the most interesting because it describes a transition of 𝑆_𝐻(𝑡) to white noise. Our estimates of 𝜃_𝑆𝐻 are based on two lemmas, where we use the following notation: 𝜃𝑓,Δ=−|Δ|⁻¹log𝑃𝑥(𝑡)≤𝑓(𝑡),𝑡∈Δ. (2.15)

Lemma 2.5 (see [12]). Let 𝑥(𝑡) be a centered Gaussian stationary process with a finite nonnegative correlation function, that is, 𝐵_𝑥(𝑡)≥0 and 𝐵_𝑥(𝑡)=0 for |𝑡|≥𝑇₀. Then the limit 𝜃(𝑎)=lim_𝑇→∞𝜃(𝑎,(0,𝑇)), (2.16) exists for every 𝑎∈𝑅¹. Moreover, 11+𝑘⁻¹𝜃𝑎,𝑘Δ₀≤𝜃(𝑎)≤𝜃𝑎,𝑘Δ₀,Δ₀=0,𝑇₀. (2.17)

Remark 2.6. Lemma 1.1 was derived by Li and Shao [12] for the Slepian process, 𝑆_1/2(𝑡), but the proof remains valid for the general case. There is an explicit but very complicated formula for 𝜃_𝑆𝐻(0,Δ) with 𝐻=1/2 [18]. In case of Δ=(0,2), this result reduces to 𝑃𝑆_1/2(𝑡)()=1≤0,𝑡∈0,26−2+√38𝜋 (2.18) and gives 1.336<𝜃_𝑆1/2<2.004.

Lemma 2.7 (see [8]). Let 𝑥(𝑡) be a centered Gaussian process in an interval Δ with a correlation function 𝐵(𝑡,𝑠) and (𝐻_𝑥(Δ),‖⋅‖_Δ) be the Hilbert space with the reproducing kernel 𝐵(𝑡,𝑠) on Δ×Δ. If 0<𝜃(𝑎,Δ)<∞, then 𝜃𝑎+𝑓,Δ−𝜃(≤𝑓𝑎,Δ)^Δ2|Δ|. (2.19)

Remark 2.8. Lemma 2.7 is a version of Proposition  1.6 from the paper by Aurzada and Dereich [8]; relation (2.19) successfully supplements the original Lemma 1.1.

Proposition 2.9. The persistence exponent of process 𝑆_𝐻(𝑡) has the following estimates: −(1−𝐻)𝐻⁻¹ln(2𝐻)≤𝜃_𝑆𝐻≤49𝐻⁻², (2.20) where the left inequality holds for 0<𝐻≤𝑒⁻²/2.

Corollary 2.10. If 𝑤⁻_𝐻(𝑡)=(𝑤_𝐻(𝑡)−𝑤_𝐻(−𝑡))/2 is the odd component of the fractional Brownian motion, then 𝜃_𝑤⁻_𝐻≤(7/𝐻)²2,0<𝐻<0.5. (2.21)

Proof. The dual stationary process 𝑤⁻_𝐻 has the correlation function 𝐵_𝑤⁻_𝐻(𝑡)=𝑡cosh2^2𝐻−𝑡sinh2^2𝐻, (2.22) which is positive. Hence the exponent 𝜃_𝑤⁻_𝐻 exists. The inequality 𝐵_𝑤⁻_𝐻(2𝑡)=(cosh𝑡)^2𝐻1−(tanh𝑡)^2𝐻≥(cosh𝑡)^2𝐻1−|𝑡|^2𝐻₊≥𝐵_𝑆𝐻(𝑡), (2.23) and Proposition 2.9 immediately imply the corollary.

Remark 2.11. The following estimates of 𝜃_𝑤⁻_𝐻 are due to Krug et al. [19]: 𝜃_𝑤⁻_𝐻()≥min1−𝐻²𝐻,(1−𝐻)2^{1/(2𝐻)−1}𝜃,0<𝐻<0.5,_𝑤−𝐻≤(1−𝐻)²𝐻,0.1549<𝐻<0.5. (2.24) For small 𝐻 these estimates are one-sided only.

Remark 2.12. A considerable difference in the behavior of 𝜃_𝑤⁻_𝐻 and 𝜃_𝑤𝐻=1−𝐻 for small 𝐻 is expected. Heuristically this can be explained as follows. As 𝐻→0, the discrete processes 𝑤⁻_𝐻(𝑘Δ) and 𝑤_𝐻(𝑘Δ) have different weak limits: {𝜉_𝑘} and {𝜉_𝑘−𝜂/√2}, respectively, where {𝜉_𝑘} and 𝜂 are independent standard Gaussian variables. The probability (1.4) for the limiting processes is quite different:𝑃𝜉_𝑘<0,𝑘=1÷𝑁=2^−𝑁,𝑃𝜉_𝑘−𝜂≤0,𝑘=1÷𝑁=(𝑁+1)⁻¹. (2.25) Unfortunately, this argument fails to predict the behavior of 𝜃_𝑆𝐻 for small 𝐻, because the step Δ cannot be arbitrary and is a function of 𝐻.

2.4. Khanin’s Problem

The survival exponent for fractional Brownian motion in the intervals Δ_𝑇=(−𝑇,𝑇) is independent of the parameter 𝐻: 𝜃_𝑤𝐻=1. This interesting fact follows from both self-similarity of 𝑤_𝐻 and the stationarity of its increments [1].

In the case 𝐻<0.5, the variables 𝑤_𝐻(𝑡) and 𝑤_𝐻(−𝑡) are positive correlated. Therefore, a possible power-law asymptotics 𝑃𝑤_𝐻(𝑡)<1,−𝑤_𝐻(−𝑡)<1,𝑡∈(0,𝑇)=𝑇^{−𝜃+𝑜(1)}, (2.26) where we change sign before 𝑤_𝐻(𝑡) for negative 𝑡 only, may have a radically different exponent compared with 𝜃_𝑤𝐻=1. The question of finding bounds on the exponent 𝜃_𝜒𝐻 for the process 𝜒_𝐻(𝑡)=sign(𝑡)𝑤_𝐻(𝑡),Δ_𝑇=(−𝑇,𝑇), (2.27) was asked by K. Khanin. The next proposition contains a partial answer to this question.

Proposition 2.13. (1) In the case 0.5≤𝐻<1, the exponent 𝜃_𝜒𝐻 for Δ_𝑇=(−𝑇,𝑇) exists and admits of the following estimates: 1<𝜃_𝜒𝐻(1−𝐻)⁻¹≤2,0.5≤𝐻<1, (2.28) in addition, 𝜃_𝜒1/2=1.(2) Let 𝜃_𝜒𝐻 be the lower exponent in (2.26), then 𝜃_𝜒𝐻(1−𝐻)⁻¹≥𝐻⁻¹−1∧2^1/2𝐻−1𝜃,0<𝐻<0.25,_𝜒𝐻(1−𝐻)⁻¹≥20.25<𝐻≤0.5. (2.29)

Remark 2.14. To clarify why 𝜃_𝜒𝐻/𝜃_𝑤𝐻 is unbounded for small 𝐻 in the case Δ_𝑇=(−𝑇,𝑇), we consider again the limiting sequence for 𝑤_𝐻(𝑘Δ) as 𝐻→0. This is {(𝜉_𝑘−𝜉₀)/√2}, where the {𝜉_𝑘} are independent standard Gaussian variables. The probability (1.1) for the limit sequence is 𝑃𝜉_𝑘<𝜉₀+√2,|𝑘|≤𝑁=(2𝑁+1)⁻¹𝑙(𝑁), (2.30) where 𝑙(𝑁) is a slowly varying function, whereas for the limit sequence of 𝜒_𝐻(𝑘Δ) we have 𝑃𝜉_−𝑘−√2<𝜉₀<𝜉_𝑘+√2,0<𝑘≤𝑁≈√𝜋𝑒𝑁^−1/2Φ√2^2𝑁, (2.31) where Φ(𝑥) is the Gaussian distribution function. As in Remark 2.12, we have nontrivial exponential asymptotics where the threshold for {𝜉_𝑘} is constant or bounded. Indeed, if the event in (2.31) is true, then 𝜉₀<√∑2+max^𝑁₁𝜉_−𝑘,∑^𝑁₁𝜉_𝑘𝑁=√𝑂2+^𝑝(1)√𝑁. (2.32)

2.5. An Explicit Value of 𝜃_𝑥

We have two explicit but isolated results for the fractional Brownian motion: 𝜃_𝑤𝐻=(1−𝐻) for Δ_𝑇=(0,𝑇) and 𝜃_𝑤𝐻=1 for Δ_𝑇=(−𝑇,𝑇). These results can be combined as follows.

Proposition 2.15. If Δ_𝑇=(−𝑇^𝛼,𝑇), 0≤𝛼≤1, then 𝜃_𝑤𝐻=𝛼𝐻+(1−𝐻).

Remark 2.16. The result is based on the following properties of the position 𝑡^∗_Δ of the maximum of 𝑤_𝐻(𝑡) in Δ=[0,1]: 𝑡^∗_Δ has a continuous probability density 𝑓^∗_Δ(𝑡) in (0, 1) and 𝑓^∗_Δ(𝑡)≈𝑂(𝑡^−𝐻) as 𝑡→0. In the case of multidimensional time, the behavior of 𝑓^∗_Δ(𝑡),Δ=(0,1)^𝑑 near 𝑡=0 is a key to the survival exponent of 𝑤_𝐻(𝑡) for Δ_𝑇=(−𝑇^𝛼,𝑇)^𝑑,0<𝛼<1 and 𝐻<1. By (1.2), 𝜃_𝑤𝐻=𝑑 in the case 𝛼=1, and 𝜃_𝑤𝐻=𝛼𝑑 in the degenerate case: 𝐻=1.

3. Proofs

Proof of Proposition 2.9
Lower Bound. Let 𝑤_𝐻(𝑡) be a dual fractional Brownian motion with the parameter 𝐻, that is, a Gaussian stationary process with correlation function 𝐵_𝑤𝐻(𝑡)=cosh(𝐻𝑡)−0.5(2sinh(𝑡/2))^2𝐻. We prove in the appendix that for 0<𝐻≤𝑒⁻²/2, 𝐵_𝑤𝐻𝑝𝑡≥𝐵_𝑆𝐻(𝑡),𝑝=−𝐻⁻¹ln(2𝐻). (3.1) Applying Slepian’s lemma, one has 𝜃_𝑆𝐻≥𝑝(1−𝐻) because 𝜃_𝑤𝐻=(1−𝐻).
Upper Bound. The random variable 𝜂=∫¹₀𝑆_𝐻(𝑡)𝑑𝑡 corresponds to an element 𝑓_𝜂(𝑡) of the Hilbert space, 𝐻_𝑆(Δ),Δ=(0,1), with the reproducing kernel 𝐵(𝑡,𝑠)=1−|𝑡−𝑠|^2𝐻. By definition of 𝐻_𝑆(Δ), we have 𝑓_𝜂(𝑡)=𝐸𝑆_𝐻(𝑡)𝑡𝜂=1−^1+2𝐻+(1−𝑡)^1+2𝐻,𝑓1+2𝐻_𝜂2Δ=𝐸𝜂²=𝐻(3+2𝐻)(1+𝐻)⁻¹(1+2𝐻)⁻¹. (3.2)
It is easy to see that 𝑓_𝜂(0)≤𝑓_𝜂(𝑡)≤𝑓_𝜂(1/2). Therefore, 𝐻<𝑓_𝜂(𝑡)<2𝐻ln(2𝑒),4𝐻3<𝑓_𝜂²_Δ<3𝐻. (3.3) Let 𝑚_𝐻 be the median of the random variable 𝑀=max{𝑆_𝐻(𝑡),𝑡∈Δ}, where Δ=(0,1). Then 0.5=𝑃𝑆_𝐻(𝑡)<𝑚_𝐻,𝑡∈Δ<𝑃𝑆_𝐻(𝑡)<𝑚_𝐻𝐻⁻¹𝑓_𝜂(𝑡),𝑡∈Δ, (3.4) because 𝐻⁻¹𝑓_𝜂(𝑡)>1. Setting 𝑥(𝑡)=𝑆_𝐻(𝑡) in Lemma 2.5 and using notation (2.15), one has 𝜃𝑚_𝐻𝐻⁻¹𝑓_𝜂(𝑡),Δ<ln2,𝜃(0,Δ)<√𝑚ln2+^𝐻𝐻⁻¹𝑓_𝜂Δ√2. (3.5)
Using Lemma 1.1 and the inequality ‖𝑓_𝜂‖_Δ<√3𝐻, we have 𝜃_𝑆𝐻<𝜃(0,Δ)<√ln2+𝑚_𝐻1.5𝐻². (3.6) It is well known (see, e.g., [20]) that 𝑚_𝐻<4√2𝐷(Δ,𝜎/2), where 𝜎²=max_Δ𝐸𝑆_𝐻(𝑡) and 𝐷 is the Dudley entropy integral related to the semimetrics on Δ: 𝜌²(𝑡,𝑠)=𝐸(𝑆_𝐻(𝑡)−𝑆_𝐻(𝑠))².
In our case 𝜌(𝑡,𝑠)=√2|𝑡−𝑠|^𝐻,𝜎=1 and therefore 𝑚_𝐻<𝑐_𝐻√𝐻, (3.7) where 𝑐_𝐻=4(1−𝐻)ln2+2^3−𝐻√𝜋Φ−√11−𝐻ln4<5.36,𝐻<2, (3.8) and Φ(𝑥) is the standard Gaussian distribution. Hence, 𝜃_𝑆𝐻<√√ln2+5.361.5𝐻²<7𝐻². (3.9)

Proof of Proposition 2.13
Part (1). In the case of 𝐻≥0.5, the process 𝜒_𝐻(𝑡)=sign(𝑡)𝑤_𝐻(𝑡) has nonnegative correlations on 𝑅¹. In the standard manner, this implies the existence of 𝜃_𝜒𝐻 for Δ_𝑇=(−𝑇,𝑇). More precisely, starting from a self-similar 2D process 𝑥(𝑡)=(𝑤_𝐻(𝑡),−𝑤_𝐻(−𝑡)) on 𝑅¹₊, we consider the dual 2D stationary process 𝑥(𝑡)=𝑥(𝑒^𝑡)exp(−𝐻𝑡) whose correlation matrix has positive elements. By [12], we conclude that the exponent 𝜃_𝜒𝐻 for 𝑥(𝑡) exists.
The equality 𝜃_𝜒𝐻=𝜃_𝜒𝐻 for Δ_𝑇=(−𝑇,𝑇). We will use Lemma 1.1. By the relation 𝜒_𝐻(𝑡)=sign(𝑡)𝑤_𝐻(𝑡), the map 𝜑(𝑡)↦sign(𝑡)𝜑(𝑡) is an isometry between the Hilbert spaces 𝐻_𝜒𝐻(Δ_𝑇) and 𝐻_𝑤𝐻(Δ_𝑇) associated with 𝜒_𝐻(𝑡) and 𝑤_𝐻(𝑡) on Δ_𝑇=(−𝑇,𝑇), respectively. To prove the equality of the dual exponents, it is enough to find 𝜑(𝑡)∈(𝐻_𝑤𝐻(𝑅¹),‖⋅‖_𝑅) such that sgn(𝑡)𝜑(𝑡)≥1 for |𝑡|≥1. We can use 𝜑(𝑡)=sgn(𝑡)min(|𝑡|,1)=𝑒^𝑖𝑡𝜆−1sin𝜆𝑖𝜋𝜆²𝑑𝜆, (3.10) because 𝜑²_𝑅=𝑘_𝐻(sin𝜆)²𝜋𝜆²²|𝜆|^1+2𝐻𝑑𝜆<∞, (3.11) (see [14]).
Estimation of 𝜃_𝜒𝐻,𝐻>1/2. Since 𝐸𝜒_𝐻(𝑡)𝜒_𝐻(𝑠)≥0 for any 𝑡, 𝑠, we have, by Slepian’s lemma, 𝑝_𝑇∶=𝑃𝑤_𝐻(𝑡)<1,−𝑤_𝐻(−𝑡)<1,𝑡∈(0,𝑇)≥𝑃𝑤_𝐻(𝑡)<1,𝑡∈(0,𝑇)². (3.12) Using (1.2), one has 𝜃_𝜒𝐻≤2(1−𝐻).
Obviously, 𝑝_𝑇≤𝑃(𝑤_𝐻(𝑡)<1,𝑡∈(0,𝑇)). Therefore, 𝜃_𝜒𝐻≥(1−𝐻) for any 𝐻.
Part (2). Let 0<𝐻≤1/2, then 𝐸𝑤_𝐻(𝑡)(−𝑤_𝐻(−𝑠))≤0 for 𝑡,𝑠>0. Hence, 𝑝_𝑇≤𝑃𝑤_𝐻(𝑡)<1,𝑡∈(0,𝑇)²,𝜃_𝜒𝐻≥2(1−𝐻). (3.13) Finally, 𝑝_𝑇≤𝑃𝑤_𝐻(𝑡)−𝑤_𝐻(−𝑡)<2,𝑡∈(0,𝑇)=𝑃𝑤⁻_𝐻(𝑡)<1,𝑡∈(0,𝑇). (3.14) But then, 𝜃_𝜒𝐻≥𝜃_𝑤⁻_𝐻 for all 𝐻. If 𝜃_𝑤⁻_𝐻=𝜃_𝑤⁻_𝐻, then we get a lower bound of 𝜃_𝜒𝐻 for 0<𝐻≤1/4.
The equality 𝜃_𝑤⁻_𝐻=𝜃_𝑤⁻_𝐻.Let 𝐻_𝑤⁻_𝐻(Δ) and 𝐻_𝑤𝐻(Δ) be the reproducing Kernel Hilbert spaces associated with 𝑤^__𝐻(𝑡) and 𝑤_𝐻(𝑡), respectively. By the definition of 𝑤^__𝐻(𝑡), the map (𝜑(𝑡),𝑡>0)↦(sign(𝑡)𝜑(|𝑡|),|𝑡|<∞) is an isometric embedding of 𝐻_𝑤⁻_𝐻(𝑅¹₊) in 𝐻_𝑤𝐻(𝑅¹). To prove that the exponents are equal, it is enough to find 𝜑(𝑡),𝑡≥0 such that sign(𝑡)𝜑(|𝑡|)∈(𝐻_𝑤𝐻(𝑅¹),‖⋅‖_𝑅), 𝜑(𝑡)≥1 for 𝑡≥1, and ‖𝜑‖_𝑅<∞. As we showed above, this can be 𝜑(𝑡)=min(𝑡,1),𝑡>0.

Proof of Proposition 2.15
Consider the fractional Brownian motion in Δ_𝑇=(−𝑇^𝛼,𝑇),0≤𝛼≤1. By Lemma 1.1, we can focus on the exponent related to the position of the maximum of 𝑤_𝐻(𝑡) in Δ_𝑇,𝑡^∗_Δ𝑇.
Distribution of 𝑡^∗_Δ. We remind the main properties of the distribution function, 𝐹^∗(𝑥), of 𝑡^∗_Δ related to the normalized interval Δ=(0,1) (see [1, 14]):(i)𝐹^∗(𝑥) has a continuous density 𝑓_Δ^∗(𝑥)>0,0<𝑥<1 such that (1−𝑥)𝑓_Δ^∗(𝑥) decreases and 𝑥𝑓_Δ^∗(𝑥) increases on Δ;(ii)𝐹^∗(𝑥) have the following estimates: 𝑥^1−𝐻𝑙⁻¹(𝑥)≤𝐹^∗(𝑥)≤𝑥^1−𝐻𝑙(𝑥), (3.15) where 𝑙(𝑥)=exp(𝑐√−ln𝑥),𝑐>0.
Due to monotonicity of (1−𝑥)𝑓^∗_Δ(𝑥) and 𝑥𝑓^∗_Δ(𝑥), one has (1−𝑥)𝑓^∗_Δ(𝑥)≤𝑥^−1𝑥₀(1−𝑢)𝑓^∗_Δ(𝑢)𝑑𝑢≤𝑥⁻¹𝐹^∗(𝑥),𝑥𝑓^∗_Δ(𝑥)≥𝑥^−1𝑥_𝑥𝑞𝑢𝑓^∗_Δ(𝑢)𝑑𝑢≥𝑞𝐹^∗(𝑥)−𝐹^∗𝑥𝑞,0<𝑞<1. (3.16)(3.17) By (3.15), (3.16), 𝑓^∗_Δ(𝑥)≤𝑥^−𝐻𝑙(𝑥)(1−𝑥)⁻¹. (3.18) Using (3.15), (3.17), one has 𝑓^∗_Δ(𝑥)≥𝑞𝑥^−𝐻𝑙⁻¹(𝑥)1−𝑙(𝑥)𝑙𝑥𝑞𝑞^1−𝐻. (3.19) If we set 𝑞^1−𝐻=𝑙⁻²(𝑥)/2, then 𝑓^∗_Δ(𝑥)≥𝑞𝑥^−𝐻𝑙⁻¹(𝑥)2=𝑐_𝐻𝑥^−𝐻𝑙^−𝜈𝐻(𝑥), (3.20) where 𝜈_𝐻=(3−𝐻)/(1−𝐻), 𝑐_𝐻=2^{−(2−𝐻)/(1−𝐻)}.
Distribution of 𝑡^∗_Δ𝑇. Let 𝑇₁=𝑇₋+𝑇, where 𝑇₋=𝑇^𝛼, then the processes 𝑤_𝐻(𝑇₁𝜏−𝑇₋)−𝑤_𝐻(−𝑇₋)and 𝑤_𝐻(𝜏)𝑇^𝐻₁ on Δ=(0,1) are equal in distribution. Hence, 𝑡^∗_Δ𝑇 and 𝑇₁𝑡^∗_Δ−𝑇₋ have the same distribution as well. Therefore, 𝑝_𝑇∶=𝑃𝑡^∗_Δ𝑇≤0.5=𝑃𝑡^∗_Δ−𝑇₋𝑇₁≤0.5𝑇₁=𝑇₁⁻¹𝑓^∗_Δ𝑇₋+𝜀𝑇₁, (3.21) where |𝜀|≤0.5. We have used here the existence and continuity of 𝑓^∗_Δ(𝑥).
Exponent 𝜃̆_𝑤𝐻. Set 𝛼=1. Then (3.21) implies lim_𝑇→∞𝑇𝑝_𝑇=0.5𝑓^∗_Δ(0.5).
Let 𝛼<1, then (𝑇₋+𝜀)/𝑇₁=𝑜(1) as 𝑇→∞, and (3.20), (3.21) give a lower bound on 𝑝_𝑇: 𝑇₁𝑝_𝑇≥c_𝐻𝑎⁺_𝑇^−𝐻𝑙^−𝜈𝐻𝑎⁺_𝑇. (3.22) Here and below 𝑎^±_𝑇=(𝑇₋±0.5)/𝑇₁.
Using (3.18), (3.21), we get an upper bound on 𝑝_𝑇: 𝑇₁𝑝_𝑇=𝑓^∗_Δ𝑇^𝛼+𝜀𝑇₁≤𝑎⁻_𝑇^−𝐻𝑙𝑎⁻_𝑇𝑇₁𝑎𝑇+1≤2⁻_𝑇^−𝐻𝑙𝑎⁻_𝑇. (3.23) By substituting 𝑇₋=𝑇^𝛼, we have ln𝑎^±_𝑇=−(1−𝛼)ln𝑇+𝑂𝑇^−𝛽,𝛽=𝛼∧(1−𝛼),ln𝑙𝑎^±_𝑇=𝑂√ln𝑇. (3.24) Hence, −ln𝑝_𝑇=(1−(1−𝛼)𝐻)ln𝑇+𝑂√ln𝑇, (3.25) that is, 𝜃̆_𝑤𝐻=𝛼𝐻+(1−𝐻).
The equality 𝜃̆_𝑤𝐻=𝜃_𝑤𝐻. Consider the Hilbert space (𝐻_𝑤𝐻(𝑅¹),‖⋅‖_𝑅) related to FBM and a function 𝜑(𝑡)=min(|𝑡|,1)=𝑒^𝑖𝑡𝜆−1sin𝜆/2√2𝜋𝜆/2²𝑑𝜆. (3.26) The standard spectral representation of the kernel 𝐸𝑤_𝐻(𝑡)𝑤_𝐻(𝑠) and the representation (3.26) yield 𝜑²_𝑅=𝑘_𝐻sin𝜆/2√2𝜋𝜆/2⁴|𝜆|^1+2𝐻𝑑𝜆<∞, (3.27) where 𝑘_𝐻=∫|𝑒^𝑖𝜆−1|²|𝜆|^−1−2𝐻. Setting 𝜑_𝑇∶={𝜑(𝑡),𝑡∈Δ_𝑇}, the desired statement follows from Lemma 1.1 because 𝜑_𝑇∈(𝐻_𝑤𝐻(Δ_𝑇),‖⋅‖_𝑇) and ‖𝜑_𝑇‖_𝑇≤‖𝜑‖_𝑅.

Appendix

Relation (2.4). (𝐵_𝐼𝐻(𝑡)≤𝐵_𝐼1−𝐻(𝑡)).
By (2.3), one has for small and large 𝑡𝐵_𝐼𝐻(𝑡)=1−1−𝐻²𝑡²2+(2+4𝐻)⁻¹𝑡^2+2𝐻((1𝐵1+𝑜)),𝑡→0,_𝐼𝐻(𝑡)=(1+𝐻)(1+2𝐻)⁻¹𝑒^−𝐻𝑡1−𝑒^−𝑡+0.5(1+𝐻)𝑒^−𝐻𝑡1+𝑂𝑒^−𝑡,𝑡→∞, (A.1) where 𝐻=1−𝐻. Therefore, we have the following asymptotics for Δ(𝑡)=𝐵_𝐼𝐻(𝑡)−𝐵_𝐼𝐻(𝑡): Δ(𝑡)()𝑡=−1−2𝐻²2+𝑂𝑡^2+2𝐻Δ(𝑡)()𝐻(),𝑡→0,=−1−2𝐻2+4𝐻⁻¹𝑒^−𝐻𝑡−1−2𝐻𝐻2+4𝐻⁻¹𝑒^−𝐻𝑡+𝑂𝑒^−𝑡,𝑡→∞. (A.2) These relations support (2.4) both for small and large enough 𝑡. To verify (2.4) in the general case, we consider the following test function: (2+4𝐻)(2+4𝐻)Δ(𝑡)exp(−1.5𝑡). Using new variables: 𝑥=exp(−𝑡), 𝛼=1−2𝐻, the test function is transformed to a function 𝜓 on the square 𝑆=(0,1)×(0,1). Namely, 𝜓=𝑈(𝑥,𝛼)−𝑈(𝑥,−𝛼), where 𝑈(𝑥,𝛼)=4−𝛼²𝑥^𝛼/2(3−𝛼)^𝑥₀(𝑥−𝑢)(1−𝑢)^1−𝛼−𝑢^1−𝛼+𝑢^1−𝛼𝑑𝑢. (A.3) We have to show that 𝜓≤0. It is easy to see that 𝜓=0 at the boundary of 𝑆. By (A.1), 𝜓≤0 in vicinities of two sides of 𝑆: 𝑥=0 and 𝑥=1. The same is true for the other sides: 𝛼=0 and 𝛼=1 because 𝜕𝜓()𝜕𝛼𝑥,0=−41−𝑥²¹_1−𝑥1ln𝑢𝑑𝑢<0,𝜕𝜓()=()𝑥𝜕𝛼𝑥,11−𝑥^−1/2𝑓(𝑥)>0. (A.4) Here 𝑓(𝑥)=−𝑥(1−𝑥)+𝑥³1ln𝑥+1−𝑥³1ln().1−𝑥 (A.5) To verify 𝑓(𝑥)>0,0<𝑥<1, note that 𝑓^′(𝑥)=3𝑥²(1+𝑣+ln𝑣), where 𝑣=(1−𝑥)/𝑥. Obviously, 𝑓^′ has a single zero in (0,1), that is, 𝑓 has a single extreme point. But 𝑓(0)=0=𝑓(1) and𝑓(𝑥)>0 for small 𝑥. Therefore, 𝑓(𝑥)≥0,0<𝑥<1.
Numerical testing supports the desired inequality 𝜓<0 for interior points of 𝑆.

Comment 1. Our preliminary numerical test was concerned with points on a grid with a step of 0.005. The first derivatives of 𝜓 are uniformly bounded from above on 𝑆. This fact helps us to find the final grid step to prove 𝜓<0 for all interior points of 𝑆. The relevant analysis is cumbersome and so has been omitted.

Relation (2.6). (𝐵_𝐼𝐻(𝑡)≤𝐵_𝐼1/2(𝑝𝑡),𝐻≥1/2,𝑝=2(1−𝐻)).
To verify the inequalityΔ(𝑡)=𝐵_𝐼𝐻(𝑡)−𝐵_𝐼1/2(2(1−𝐻)𝑡)≤0, we consider the following test function: (2+4𝐻)Δ(𝑡)exp(−(1+𝐻)𝑡). Using (2.3), (2.5), and new variables (𝑥=exp(−𝑡),𝛼=2𝐻−1)∈𝑆=(0,1)×(0,1), we will have the following representation for the test function: 𝜓(𝑥,𝛼)=(3+𝛼)𝑥+𝑥^𝛼+2−1−𝑥^𝛼+3+(1−𝑥)^𝛼+3−3(𝛼+2)𝑥²+(𝛼+2)𝑥^3−𝛼. (A.6)
One has 𝜓(𝑥,𝛼)≤0 in vicinities of two sides of 𝑆: 𝑥=0 and 𝑥=1, because 𝜓()()𝑥𝑥,𝛼=−𝛼+2)(3−𝛼²2+𝑂𝑥^{(𝛼+2)∧(3−𝛼)}𝜓()()<0,𝑥→0,𝑥,𝛼=−2𝛼1−𝛼)(3−𝛼)(1−𝑥²2+𝑂(1−𝑥)³≤0,𝑥→1. (A.7)
The same is true for the other sides: 𝛼=0 and 𝛼=1.

Side 𝛼=0
One has 𝜓(𝑥,0)=0 and 𝜕𝜓()=(𝑥()𝜕𝛼𝑥,01−𝑥)1−𝑥+3𝑥²ln𝑥+(1−𝑥)²ln(1−𝑥)∶=(1−𝑥)𝜑₃(𝑥)≤0 (A.8) because 𝜑_𝑎(𝑥)=𝑥(1−𝑥)+𝑎𝑥²ln𝑥+(1−𝑥)²ln(1−𝑥)≤0,𝑎>1. (A.9) To prove (A.9), note that 𝜑_𝑎(0)=𝜑_𝑎(1)=0 and 𝜑_𝑎(𝑥)=𝑎𝑥²ln𝑥+𝑂(𝑥²)≤0 as 𝑥→0. Hence, (A.9) holds if 𝜑_𝑎(𝑥) has a single extremum in (0,1). By 𝜑_𝑎⁽⁴⁾(𝑥)=−2𝑎𝑥⁻²−2(1−𝑥)⁻²≤0, (A.10) we conclude that 𝜑_𝑎^′′(𝑥)=(3𝑎+1)+2𝑎ln𝑥+2ln(1−𝑥), (A.11) is a concave function with two zeroes in (0,1), because 𝜑_𝑎^′′(1/2)>0 and 𝜑_𝑎^′′(𝑥)→−∞ as 𝑥→0 or 1.
This means that 𝜑^′_𝑎(𝑥)=(𝑎−1)𝑥+2𝑎𝑥ln𝑥−2(1−𝑥)ln(1−𝑥), (A.12) has two extremums in (0,1) only. But 𝜑^′_𝑎(0)=0,𝜑^′_𝑎(1)=𝑎−1>0, and 𝜑^′_𝑎(𝑥)≤0 for small 𝑥 because 𝜑_𝑎^′′(𝑥)→−∞ as 𝑥→0. Hence 𝜑^′_𝑎(𝑥) has a single zero in (0,1) and 𝜑_𝑎(𝑥) has a single extremum.
We have proved that 𝜓(𝑥,𝛼)≤0 for small 𝛼.

Side 𝛼=1
Here 𝜓(𝑥,1)=0 and 𝜕𝜓()=()𝑥𝜕𝛼𝑥,11−𝑥)(3−𝑥²1ln𝑥+(1−𝑥)²𝑥+(1−𝑥)²ln(1−𝑥)≥0, (A.13) because [𝑥+(1−𝑥)²ln(1−𝑥)]≥𝑥+(1−𝑥)ln(1−𝑥)=−∫^𝑥₀ln(1−𝑢)𝑑𝑢≥0.
Hence, 𝜓(𝑥,𝛼)=𝜓^′_𝛼(𝑥,1)(𝛼−1)(1+𝑜(1−𝛼))≤0,𝛼→1.
As a result 𝜓(𝑥,𝛼)≤0 near the boundary of 𝑆=(0,1)×(0,1). Numerical testing supports the desired inequality 𝜓<0 for the interior of 𝑆 (see more in the Comment 1 from the appendix section “Relation (2.4)”).

Relation (2.7). (𝐵_𝐼𝐻(𝑡)≥𝐵_𝐼1/2(𝑝𝑡),𝐻≤1/2,𝑝=2(1−𝐻²)/3).
Let 𝜓=(2+4𝐻)(𝐵_𝐼𝐻(𝑡)−𝐵_𝐼1/2(𝑝𝑡))𝑒^{−(1+𝐻)𝑡}. By change of variables: 𝑥=exp(−𝑡) and 𝛼=2𝐻, we get a test function 𝜓(𝑥,𝛼)=(2+𝛼)𝑥+𝑥^𝛼+1−1−𝑥^𝛼+2+(1−𝑥)^𝛼+2−3(𝛼+1)𝑥^{1+(𝛼+𝑝)/2}+(𝛼+1)𝑥^{1+(𝛼+3𝑝)/2}, (A.14) on 𝑆=(0,1)×(0,1) and the relation between 𝑝 and 𝛼 is 3𝑝2²+𝛼2²=1. (A.15) One has 𝜓(𝑥,𝛼)=(2+𝛼)𝑥^1+𝛼−3(1+𝛼)𝑥^{1+(𝑝+𝛼)/2}+𝑂𝑥²𝜓()=()≥0,𝑥→0,𝑥,𝛼1−𝑥^2+𝛼+𝑂(1−𝑥)³≥0,𝑥→1. (A.16) In addition,𝜓(𝑥,0)=𝑥2−3𝑥^3−1/2+𝑥^31/2≥0. (A.17) Finally, 𝜓(𝑥,1)=0 and 𝜕𝜓()=𝜕𝛼𝑥,1𝑥𝑥𝑥+2𝑥²ln𝑥+𝑥²ln𝑥=𝑥𝜑₂(𝑥), (A.18) where 𝑥=1−𝑥. By (A.9), 𝜑₂(𝑥)≤0.
Therefore 𝜓(𝑥,𝛼)≤0 near the boundary of 𝑆=(0,1)×(0,1). The numerical testing supports this conclusion for the interior of 𝑆 (see more in the Comment 1 from the appendix section “Relation (2.4)”).

Relations (2.11), (2.12)
Consider Δ(𝑡)=𝐵_𝐼1/2(𝑡)−𝐵_𝐿(𝑝𝑡), where 𝐵_𝐿(𝑡)=1/cosh(𝑡/2) and 𝐵_𝐼1/2(𝑡) is given in (2.5). By the change of variables 𝑥=𝑒^−𝑡/2, we transform the test function 2(1+𝑒^−𝑝𝑡)Δ(𝑡) to a function 𝜓 on (0,1) such that 𝜓(𝑥)=3𝑥−𝑥³1+𝑥^2𝑝−4𝑥^𝑝. (A.19) Taking into account the asymptotics of 𝜓 near 0, we come to a necessary condition for 𝜓 to be negative, namely, 𝑝≤1. Let 𝑝=1, then 𝜓=−(1−𝑥²)²𝑥≤0, that is, 4𝜃_𝐿≤1.
The Case 𝑝>1. Here, 𝜓≥0 as 𝑥→0. An additional condition on 𝑝>1 we can get from the relation 𝜓≥0 as 𝑥→1. One has 𝜓=𝑥𝑄(𝑥), where 𝑄(𝑥)=3−𝑥²1+𝑥^2𝑝−4𝑥^𝑝−1. (A.20) By 𝑄(0)=3,𝑄(1)=𝑄^′(1)=0, we have 𝑄(𝑥)=(1−𝑥)²𝑃(𝑥) and 𝑃(1)=0.5𝑄^′′(1)=2(𝑝²−3). Thus 𝑄(1)≥0 if 𝑝²≥3.
The Case 𝑝=2. Here, 𝑃(𝑥) is a polynomial, 𝑃(𝑥)=3+2𝑥−2𝑥³−𝑥⁴, and 𝑃^′′(𝑥)=−12𝑥(1+𝑥)≤0, that is, 𝑃(𝑥) is a concave function with 𝑃(0)=3,𝑃(1)=2. Therefore, 𝑃(𝑥)≥0 and as a result, 4𝜃_𝐿≥1/𝑝=0.5.
Consider 𝑝=√3. One has 𝑄(𝑥)=8(1−𝑥)³(1+𝑜(1)),𝑥→1 and 𝑄(0)=3>0. Therefore, 𝑄(𝑥)≥0, if 𝑄(𝑥) is convex, that is, 𝑄^′′(𝑥)≥0. To verify this property, note that 0.5𝑥²𝑄^′′(𝑥)=23𝑝−5𝑥^𝑝−1+36−𝑝𝑥^2𝑝−𝑥²−7+3𝑝𝑥^2+2𝑝=7+3𝑝𝑥^2𝑝1−𝑥²+𝜌𝑥^𝑝−1+1−𝜌𝑥^2𝑝−𝑥²∶=𝜑(𝑥), (A.21) where 𝜌=6𝑝−10.
Obviously, 𝜑(𝑥)≥0 if 𝑝𝑥^𝛼−1−𝑥²≥0. This holds for 0<𝑥<𝑥₀=0.478.
For 𝑥>𝑥₀, 𝜌𝑥^𝑝−1+1−𝜌𝑥^2𝑝−𝑥²≥𝜌+1−𝜌𝑥₀^𝑝+1𝑥^𝑝−1−𝑥². (A.22) The right part here is positive for 𝑥<0.55, that is, 𝜑(𝑥)≥0 for 𝑥≤0.5.
Let 𝑥>0.5. Then 𝜑(𝑥)≥7+3𝑝2^−2𝑝1−𝑥²+𝜌𝑥^𝛼−1+1−𝜌𝑥^2𝑝−𝑥²=𝐶−(𝐶+1)𝑥²+𝜌𝑥^𝑝−1+1−𝜌𝑥^2𝑝∶=𝑢(𝑥), (A.23) where 𝐶=(7+3𝑝)2^−2𝑝. We have 𝑢(0)=𝐶,𝑢(1)=0 and 𝑢^′(𝑥)=−2(𝐶+1)𝑥+𝜌𝑝−1𝑥^𝑝−2+21−𝜌𝑝𝑥^2𝑝−1=−𝐶+1−21−𝜌𝑝𝑥^2𝑝−2𝑥−(𝐶+1)𝑥^3−𝑝−𝜌𝑝−1𝑥^𝑝−2. (A.24) It is easy to see that both terms in parentheses are positive on (0.5, 1).
Thus, 𝑢(𝑥) decreases to 𝑢(1)=0. This means that 𝑄^′′(𝑥)≥0.

Relation (3.1). (𝐵_𝑤𝐻(𝑝𝑡)≥𝐵_𝑆𝐻(𝑡),𝑝𝐻=−ln(2𝐻),0<𝐻<𝑒⁻²/2).
The difference between the correlation functions is Δ(𝑡)=cosh𝐻𝑝𝑡−0.52sinh𝑝𝑡2^2𝐻−1−|𝑡|^2𝐻₊. (A.25)
Let 𝑡>1, then Δ(𝑡)=𝐵_𝑤𝐻(𝑝𝑡)≥0.
Let 2𝐻<𝑡<1. It is enough to show that the first term, 𝜑, in the following representation: Δ(𝑡)=0.5𝑒^{−𝐻𝑝𝑡}−1+𝑡^2𝐻+0.5𝑒^𝐻𝑝𝑡1−1−𝑒^{−𝑝𝑡2𝐻}∶=𝜑+𝑅, (A.26) is nonnegative. Setting 𝐻𝑝=−ln(2𝐻),𝛼=2𝐻 one has 𝜑(𝑡)=0.5𝛼^𝑡+𝑡^𝛼−1. (A.27) Let us show that 𝜑 is decreasing. In this case 𝜑 is positive because 𝜑(1)=𝛼/2.
We have 𝜑^′(𝑡)=𝛼^𝑡1−0.5ln𝛼+𝜓(𝑡), (A.28) where 𝜓(𝑡)=𝛼^1−𝑡/𝑡^1−𝛼. The function 𝜓(𝑡) has a single extreme point in the interval: 𝑡^∗=(1−𝛼)/ln(1/𝛼). But 𝜓(𝑡^∗)=min, because 𝜓(𝑡) decreases near 𝑡=𝛼: 𝜓(𝛼)=1,𝜓^′(𝛼)=(𝛼ln(𝑒/𝛼)−1)𝛼≤0for0<𝛼<1. (A.29)
Hence, 𝜓(𝑡)≤max(𝜓(𝛼),𝜓(1))=1. As a result, 𝜑^′(𝑡)≤𝛼^𝑡1−0.5ln𝛼+1≤0. (A.30) The last inequality holds for 0<𝛼<𝑒⁻², so we have Δ(𝑡)≥0,2𝐻<𝑡<1for0<𝛼<𝑒⁻². (A.31) Let 0<𝑡<2𝐻. Use Δ(𝑡)=cosh𝐻𝑝𝑡−1+𝑡^2𝐻1−0.52𝑡⁻¹sinh𝑝𝑡2^2𝐻, (A.32) then Δ(𝑡)≥0 if 2^1/(2𝐻)≥max_(0,2𝐻)2𝑡⁻¹sinh𝑝𝑡2=𝐻⁻¹sinh𝑝𝐻=(2𝐻)⁻²−1. (A.33) This inequality holds for 0<2𝐻<1/4.
Combining the above inequalities, we get (3.1) for 2𝐻≤𝑒⁻²∧1/4.