Research Article Survival Exponents for Some Gaussian Processes

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1 International Journal of Stochastic Analysis Volume 01, Article ID 13771, 0 pages doi: /01/13771 Research Article Survival Exponents for Some Gaussian Processes G. Molchan Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Profsoyuznaya 84/3, Moscow, Russia Correspondence should be addressed to G. Molchan, molchan@mitp.ru Received May 01; Accepted 7 August 01 Academic Editor: Yaozhong u Copyright q 01 G. Molchan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 1and the integrated FBM in 0,T, T The Problem Let x t,x 0 0 be a real-valued stochastic process with the following asymptotics: P x t < 1, t Δ T T θ x 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 t, of order 0 <<1 both with one- and multidimensional time. According to Molchan 1 θ w 1 for Δ T 0,T, θ w d for Δ T T, T d. 1. Another important example is the integrated Brownian motion I t t w s ds with the 0 exponent θ I 1 4, Δ T 0,T, 1.3

2 International Journal of Stochastic Analysis Sinai. 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 t t 0 w s ds, remains an important; but as yet unsolved problem. Below we consider the survival exponents for the following Gaussian processes: I t, t 0,T ; χ t sign t w t, t T, T ; FBMinΔ 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 S t 1 t,0< 1/. 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 hs x e s, s<ln T : 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 ) ) P x s 0, 0 <s< T exp θ x T 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 ) P x t 0, t Δ T \ 1, 1 exp θ x T 1 o To formulate the simplest condition for the exponents to be equal, we define one more exponent θ x by means of the asymptotics P t T 1 ) T θ x o 1, 1.6 where t T is the position of the maximum of x t in Δ T,thatis,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 x Δ T, T be the reproducing kernel ilbert space associated with x t. Suppose that there exists such an element ϕ of x Δ T that ϕ t 1, t > 1 and ϕ T o ln T. Then θ x, θ x, and θ 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, 1. In turn, the inequality of two correlation functions, B 1 s B s, B i 0 1, implies, by Slepian s lemma, the inverted inequality for the corresponding exponents: θ 1 θ. 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.

3 International Journal of Stochastic Analysis Figure 1: The survival exponents θ I for the integrated fractional Brownian motion in Δ T T, T : hypothetical values parabolic line, empirical estimates small circles, squares, and theoretical bounds shaded zone given by Proposition.1 b, c. The empirical exponents are based on the model.9, α 0 in three time intervals of T ln T :ln 1/ε T 1 ln 10/ε, whereε 0.01, 0.003, and seemorein 13.. Examples.1. Integrated Fractional Brownian Motion Consider the process I t t 0 w s ds,.1 where w t is the fractional Brownian motion, that is, a Gaussian random process with the stationary increments: E w t w s t s, w 0 0. Molchan and Khokhlov 13, 14 analyzed the exponent θ I theoretically and numerically and formulated the following ypothesis: θ I 1 for Δ T 0,T and θ I 1 for Δ T T, T. The unexpected symmetry θ I θ I1 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 t : ρ 1 θ I θ / I 1,. 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 <1/ andδ 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 in. for <1/isany number from the interval θ, θ. The relation. can be improved as follows. Proposition.1. For Δ T 0,T, one has a θ I θ I1, 0 < 0.5, b 0.5 θ I, 1,

4 4 International Journal of Stochastic Analysis c θ I 1 /1. Proof. The identity of the dual exponents for I t follows from 14 ; the dual survival exponent exists because the dual correlation function, B I s 4 [ 1 e s e s) e 1 s e 1 s e s/ e s/) ].3 is positive. The inequality a is a consequence of the relation B I t B I1 t, 0 < 1..4 To prove b, c, we use the correlation function of the process Ĩ 1/ ps,thatis, ) 1 B I1/ ps 3 exp p s ) exp 3p s )),.5 and the respective exponent θ p/4 see 1.3.Therelation ) 1 B I t B I1/ pt,, p 1,.6 implies θ I 1 / for 1/. Using a in addition, we come to the lower bound in b because θ I θ I1 /for 1/. Similarly, the relation ) 1 B I t B I1/ pt,,p 1,.7 3 implies c for all. A test of the purely analytical facts.4,.6, and.7 is given in the appendix. Remark.. Proposition.1 a follows from the more informative relation: PĨ s 0, s )) )) 0, T PĨ1 s 0, s 0, T..8 This inequality is important for understanding the numerical result by Molchan and Khokhlov 13 represented in the form of empirical estimates of θ I in Figure 1. We can see that the empirical estimates show small but one-sided deviations from the hypothetical curve θ 1 before and after 1/. The signs of these deviations are consistent with.8, while the amplitudes are compatible with the model PĨ s 0, s )) ) 0, T C T α exp 1 T, T 1, sgn α sign 0.5,.9

5 International Journal of Stochastic Analysis 5 and α 0.5 more can be found in The Laplace Transform of White Noise Consider the process L t t 0 e tu dw u, where w u is Brownian motion. The dual stationary process L s has the correlation function B L s 1/ cosh s/. Using.5 as a majorant of B L s, we improve the lower bound of θ L as follows. Proposition / 4 θ 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/ ln T. For indeed, ϕ t EL t η, where η 1 ε 1 T L ε T. By definition of the ilbert space L Δ T, we have the desired estimate: ϕ T Eη ε 1 T ε T 1 ) O ln T..10 By 1.3 and Slepian s lemma, the relation B I1/ t B L pt ), p 1.11 has as a consequence the estimate 4p θ L 1. The opposite inequality B I1/ t B L pt ), p 3,.1 implies 4p θ L 1. The test of.11, p 1 and.1, p is very simple and yields the Li and Shao 1, 15 estimates: 0.5 < 4 θ L < 1. The appendix contains a proof of.11,.1 for all interesting values of p:1,, and 3. Remark.4. The dual survival exponent of L t is of interest as a parameter of the following asymptotic relation: P n 0 ξ i x i / 0,x R 1 ) n 4 θ L o 1, n,.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.13. Simulations suggest 4 θ L 0.76 ± 0.03, 16 and 4 θ L 0.75, 17.

6 6 International Journal of Stochastic Analysis.3. Fractional Slepian s Process We reserve this term for a Gaussian stationary process S t with correlation function B S t 1 t ), 0 < 1,.14 because S 1/ t is known as the Slepian process and S t S 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 S t is a nonincreasing and a convex function on the semiaxis. The fact of the correlation function being nonnegative guarantees the existence of θ S in 1.4. S 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 t. The case of small is the most interesting because it describes a transition of S t to white noise. Our estimates of θ S are based on two lemmas, where we use the following notation: θ f, Δ ) Δ 1 log P x t f t, t Δ )..15 Lemma.5 see 1. 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 T 0. Then the limit θ a lim T θ a, 0,T,.16 exists for every a R 1. Moreover, 1 1 k ) 1 θ a, kδ0 θ a θ a, kδ 0, Δ 0 0,T Remark.6. Lemma 1.1 was derived by Li and Shao 1 for the Slepian process, S 1/ t, but the proof remains valid for the general case. There is an explicit but very complicated formula for θ S 0, Δ with 1/ 18. In case of Δ 0,, this result reduces to P S 1/ t 0, t 0, π.18 and gives < θ S1/ <.004. Lemma.7 see 8. Let x t be a centered Gaussian process in an interval Δ with a correlation function B t, s and x Δ, Δ be the ilbert space with the reproducing kernel B t, s on Δ Δ. If 0 < θ a, Δ <,then ) f Δ θ a f, Δ θ a, Δ. Δ.19

7 International Journal of Stochastic Analysis 7 Remark.8. Lemma.7 is a version of Proposition 1.6 from the paper by Aurzada and Dereich 8 ;relation.19 successfully supplements the original Lemma 1.1. Proposition.9. The persistence exponent of process S t has the following estimates: 1 1 ln θ S 49,.0 where the left inequality holds for 0 < e /. Corollary.10. If w t w t w t / is the odd component of the fractional Brownian motion, then θ w 7/, 0 << Proof. The dual stationary process w has the correlation function B w cosh t t ) sinh t,. ) which is positive. ence the exponent θ w exists. The inequality B w t cosh t 1 tanh t ) cosh t 1 t ) B S t,.3 and Proposition.9 immediately imply the corollary. Remark.11. The following estimates of θ w are due to Krug et al. 19 : ) 1 θ w min, 1 1/ 1, 0 <<0.5, θ w 1, << For small these estimates are one-sided only. Remark.1. A considerable difference in the behavior of θ w and θ w 1 for small is expected. euristically this can be explained as follows. As 0, the discrete processes w kδ and w kδ have different weak limits: {ξ k } and {ξ k η/ }, respectively, where {ξ k } and η are independent standard Gaussian variables. The probability 1.4 for the limiting processes is quite different: P ξ k < 0, k 1 N N, P ξ k η 0,k 1 N ) N Unfortunately, this argument fails to predict the behavior of θ S for small, because the step Δ cannot be arbitrary and is a function of.

8 8 International Journal of Stochastic Analysis.4. Khanin s Problem The survival exponent for fractional Brownian motion in the intervals Δ T T, T is independent of the parameter : θ w 1. This interesting fact follows from both selfsimilarity of w and the stationarity of its increments 1. In the case <0.5, the variables w t and w t are positive correlated. Therefore, a possible power-law asymptotics P w t < 1, w t < 1, t 0,T T θ o 1,.6 where we change sign before w t for negative t only, may have a radically different exponent compared with θ w 1. The question of finding bounds on the exponent θ χ for the process χ t sign t w t, Δ T T, T,.7 was asked by K. Khanin. The next proposition contains a partial answer to this question. Proposition In the case 0.5 <1, the exponent θ χ for Δ T T, T exists and admits of the following estimates: 1 <θ χ 1 1, 0.5 <1,.8 in addition, θ χ1/ 1. Let θ χ be the lower exponent in.6, then θ χ 1 1 ) 1 1 1/ 1, θ χ < <<0.5,.9 Remark.14. To clarify why θ χ /θ w is unbounded for small in the case Δ T T, T, we consider again the limiting sequence for w kδ as 0. This is { ξ k ξ 0 / }, where the {ξ k } are independent standard Gaussian variables. The probability 1.1 for the limit sequence is { P ξ k <ξ 0 }, k N N 1 1 l N,.30 where l N is a slowly varying function, whereas for the limit sequence of χ kδ we have { P ξ k <ξ 0 <ξ k }, 0 <k N ) N, πen Φ 1/.31

9 International Journal of Stochastic Analysis 9 where Φ x is the Gaussian distribution function. As in Remark.1, we have nontrivial exponential asymptotics where the threshold for {ξ k } is constant or bounded. Indeed, if the event in.31 is true, then ξ 0 < max N 1 ξ k, N N 1 ξ k ) O p 1 N An Explicit Value of θ x We have two explicit but isolated results for the fractional Brownian motion: θ w 1 for Δ T 0,T and θ w 1forΔ T T, T. These results can be combined as follows. Proposition.15. If Δ T T α,t, 0 α 1, thenθ w α 1. Remark.16. The result is based on the following properties of the position t Δ of the maximum of w t in Δ 0, 1 : t Δ has a continuous probability density f Δ t in 0, 1 and f Δ t O t 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 t for Δ T T α,t d, 0 <α<1and<1. By 1., θ w d in the case α 1, and θ w αd in the degenerate case: Proofs Proof of Proposition.9 Lower Bound. Let w t be a dual fractional Brownian motion with the parameter, that is, a Gaussian stationary process with correlation function B w t cosh t 0.5 sinh t/. We prove in the appendix that for 0 < e /, B w pt ) BS t, p 1 ln. 3.1 Applying Slepian s lemma, one has θ S p 1 because θ w 1. Upper Bound. The random variable η 1 0 S t dt corresponds to an element f η t of the ilbert space, S Δ, Δ 0, 1, with the reproducing kernel B t, s 1 t s.by definition of S Δ, we have f η t ES t η 1 t1 1 t 1, 1 f η Δ Eη It is easy to see that f η 0 f η t f η 1/. Therefore, <f η t < ln e, 4 3 < fη Δ <

10 10 International Journal of Stochastic Analysis Let m be the median of the random variable M max{s t, t Δ}, where Δ 0, 1. Then ) 0.5 P S t <m,t Δ <P S t <m 1 f η t, t Δ, 3.4 because 1 f η t > 1. Setting x t S t in Lemma.5 and using notation.15, one has ) θ m 1 f η t, Δ < ln, θ 0, Δ < ln m 1 f η Δ. 3.5 Using Lemma 1.1 and the inequality f η Δ < 3, we have θ S < θ 0, Δ < 1.5 ln m. 3.6 It is well known see, e.g., 0 that m < 4 D Δ,σ/, where σ max Δ ES t and D is the Dudley entropy integral related to the semimetrics on Δ: ρ t, s E S t S s. In our case ρ t, s t s,σ 1 and therefore m < c, 3.7 where c 4 1 ln 3 πφ ) 1 ln 4 < 5.36, < 1, 3.8 and Φ x is the standard Gaussian distribution. ence, ln 5.36 ) 1.5 θ S < < ) Proof of Proposition.13 Part 1. In the case of 0.5, the process χ t sign t w t has nonnegative correlations on R 1. In the standard manner, this implies the existence of θ χ for Δ T T, T. More precisely, starting from a self-similar D process x t w t, w t on R 1, we consider the dual D stationary process x t x e t exp t whose correlation matrix has positive elements. By 1, we conclude that the exponent θ χ for x t exists. The equality θ χ θ χ for Δ T T, T. We will use Lemma 1.1. Bytherelation χ t sign t w t, the map ϕ t sign t ϕ t is an isometry between the ilbert spaces

11 International Journal of Stochastic Analysis 11 χ Δ T and w Δ T associated with χ t and w t on Δ T T, T, respectively. To prove the equality of the dual exponents, it is enough to find ϕ t w R 1, R such that sgn t ϕ t 1for t 1. We can use ) sin λ ϕ t sgn t min t, 1 e itλ 1 iπλ dλ, 3.10 because ϕ sin λ R k πλ λ 1 dλ <, 3.11 see 14. Estimation of θ χ,>1/. Since Eχ t χ s 0 for any t, s, we have, by Slepian s lemma, p T : P w t < 1, w t < 1, t 0,T P w t < 1, t 0,T. 3.1 Using 1., one has θ χ 1. Obviously, p T P w t < 1, t 0,T. Therefore, θ χ 1 for any. Part. Let 0 < 1/, then Ew t w s 0fort, s > 0. ence, p T P w t < 1, t 0,T, θ χ Finally, p T P w t w t <, t 0,T P w t < 1, t 0,T ) But then, θ χ θ w for all.ifθ w θ w, then we get a lower bound of θ χ for 0 < 1/4. The equality θ w θ w.let w Δ and w Δ be the reproducing Kernel ilbert spaces associated with w t and w t, respectively. By the definition of w t, the map ϕ t,t > 0 sign t ϕ t, t < is an isometric embedding of w R1 in w R 1. To prove that the exponents are equal, it is enough to find ϕ t, t 0 such that sign t ϕ t w R 1, R, ϕ t 1fort 1, and ϕ R <. As we showed above, this can be ϕ t min t, 1, t>0. Proof of Proposition.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 t in Δ T,t Δ T. 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;

12 1 International Journal of Stochastic Analysis ii F x have the following estimates: x 1 l 1 x F x x 1 l x, 3.15 where l x exp c ln x, c>0. Due to monotonicity of 1 x f Δ x and xf Δ x, one has x 1 x f Δ x x 1 1 u f Δ u du x 1 F x, xf Δ x x 1 x xq uf Δ u du q F x F xq )), 0 <q< By 3.15, 3.16, f Δ x x l x 1 x Using 3.15, 3.17, one has f Δ x qx l 1 x 1 l x l xq ) q 1 ) If we set q 1 l x /, then f Δ x qx l 1 x c x l ν x, 3.0 where ν 3 / 1, c / 1. Distribution of t Δ T.LetT 1 T T, where T T α, then the processes w T 1 τ T w T and w τ T 1 on Δ 0, 1 are equal in distribution. ence, t Δ T and T 1 t Δ T have the same distribution as well. Therefore, ) t p T : P Δ T 0.5 P t Δ T 0.5 ) T 1 T 1 T 1 1 f Δ T ε T 1 ), 3.1 where ε 0.5. We have used here the existence and continuity of f Δ x. Exponent θ w. Set α 1. Then 3.1 implies lim T Tp T 0.5f Δ 0.5. Let α<1, then T ε /T 1 o 1 as T,and 3.0, 3.1 give a lower bound on p T : ere and below a ± T T ± 0.5 /T 1. ) T 1 p T c a l ν T a T). 3.

13 International Journal of Stochastic Analysis 13 Using 3.18, 3.1, we get an upper bound on p T : T T 1 p T f α ) ε Δ T 1 ) a l T a T) T1 a T T 1 ) l ) a T. 3.3 By substituting T T α, we have ln a ± T T 1 α ln T O β), β α 1 α, ln l a ± ) ) T O ln T. 3.4 ence, ) ln p T 1 1 α ln T O ln T, 3.5 that is, θ w α 1. The equality θ w θ w. Consider the ilbert space w R 1, R related to FBM and a function ) ) ϕ t min t, 1 e itλ sin λ/ 1 dλ. πλ/ 3.6 The standard spectral representation of the kernel Ew t w s and the representation 3.6 yield ϕ ) sin λ/ 4 R k λ 1 dλ <, πλ/ 3.7 where k e iλ 1 λ 1. Setting ϕ T : {ϕ t,t Δ T }, the desired statement follows from Lemma 1.1 because ϕ T w Δ T, T and ϕ T T ϕ R. Appendix Relation.4. B I t B I1 t. By.3, one has for small and large t ) 1 t B I t t 1 o 1, t 0, B I t e t 1 e t) A e t 1 O e t)), t,

14 14 International Journal of Stochastic Analysis where 1. Therefore, we have the following asymptotics for Δ t B I t B I t : 1 t Δ t Δ t e t O t ), t 0, ) 1e 1 4) t O e t), t. A. These relations support.4 both for small and large enough t. To verify.4 in the general case, we consider the following test function: 4 4 Δ t exp 1.5t. Using new variables: x exp t, α 1, 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 α ) x α/ 3 α x 0 [ x u 1 u 1 α u 1 α) u 1 α] du. 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 0andx 1. The same is true for the other sides: α 0and α 1 because ψ α x, x ) 1 1 x ) 1 ln du < 0, u ψ α x, 1 1 x x 1/ f x > 0. A.4 ere f x x 1 x x 3 ln 1 x 1 x 3) ln 1 1 x. A.5 To verify f x > 0, 0 <x<1, note that f x 3x 1 v ln v, where v 1 x /x. Obviously, f has a single zero in 0,1, thatis,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 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.

15 International Journal of Stochastic Analysis 15 Relation.6. B I t B I1/ pt, 1/, p 1. To verify the inequalityδ t B I t B I1/ 1 t 0, we consider the following test function: 4 Δ t exp 1 t.using.3,.5, and new variables x exp t,α 1 S 0, 1 0, 1, we will have the following representation for the test function: ψ x, α 3 α x x α ) 1 x α 3 1 x α 3 3 α x α x 3 α. A.6 One has ψ x, α 0 in vicinities of two sides of S: x 0andx 1, because α 3 α x ψ x, α O x α 3 α ) < 0, x 0, α 1 α 3 α 1 x ψ x, α O 1 x 3) 0, x 1. A.7 The same is true for the other sides: α 0andα 1. Side α 0 One has ψ x, 0 0and ψ [ ] α x, 0 1 x x 1 x 3x ln x 1 x ln 1 x : 1 x ϕ 3 x 0 A.8 because ϕ a x x 1 x ax ln x 1 x ln 1 x 0, a > 1. A.9 To prove A.9,notethatϕ a 0 ϕ a 1 0andϕ a x ax ln x O x 0asx 0. ence, A.9 holds if ϕ a x has a single extremum in 0,1. By ϕ 4 a x ax 1 x 0, A.10 we conclude that ϕ a x 3a 1 a ln x ln 1 x, A.11 is a concave function with two zeroes in 0,1, because ϕ a 1/ > 0andϕ a x as x 0or1. This means that ϕ a x a 1 x ax ln x 1 x ln 1 x, A.1

16 16 International Journal of Stochastic Analysis 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. ence 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 ere ψ x, 1 0and ψ α x, 1 1 x 3 x x ln 1 x ) 1 x [ ] x 1 x ln 1 x 0, A.13 because x 1 x ln 1 x x 1 x ln 1 x x ln 1 u du 0. 0 ence, 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<0for the interior of S see more in the Comment 1 from the appendix section Relation.4. Relation.7. B I t B I1/ pt, 1/, p 1 /3. Let ψ 4 B I t B I1/ pt e 1 t. By change of variables: x exp t and α, we get a test function ψ x, α α x x α 1) 1 x α 1 x α A.14 3 α 1 x 1 α p / α 1 x 1 α 3p /, on S 0, 1 0, 1 and the relation between p and α is p ) α ) 3 1. A.15 One has ψ x, α α x 1 α 3 1 α x 1 p α / O x ) 0, x 0, ψ x, α 1 x α O 1 x 3) 0, x 1. A.16 In addition, ψ x, 0 x 3x 3 1/ x 31/) 0. A.17 Finally, ψ x, 1 0and ψ ) α x, 1 x xx x ln x x ln x xϕ x, A.18

17 International Journal of Stochastic Analysis 17 where x 1 x.by A.9, ϕ 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.4. Relations.11,.1 Consider Δ t B I1/ t B L pt, where B L t 1/ cosh t/ and B I1/ t is given in.5. By the change of variables x e t/, we transform the test function 1 e pt Δ t to a function ψ on 0, 1 such that ψ x 3x x 3) 1 x p) 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 x x 0, that is, 4θ L 1. The Case p>1. ere, ψ 0asx 0. An additional condition on p>1 we can get from the relation ψ 0asx 1. One has ψ xq x, where Q x 3 x ) 1 x p) 4x p 1. A.0 By Q 0 3, Q 1 Q 1 0, we have Q x 1 x P x and P 1 0.5Q 1 p 3. Thus Q 1 0ifp 3. The Case p. ere, P x is a polynomial, P x 3 x x 3 x 4,andP x 1x 1 x 0, that is, P x is a concave function with P 0 3,P 1. 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 1andQ 0 3 > 0. Therefore, Q x 0, if Q x is convex, that is, Q x 0. To verify this property, note that 0.5x Q x 3p 5 ) x p p ) x p x 7 3p ) x p 7 3p ) x p 1 x ) ρx p 1 1 ρ ) x p x : ϕ x, A.1 where ρ 6p 10. Obviously, ϕ x 0ifpx α 1 x 0. This holds for 0< x<x For x>x 0, ρx p 1 1 ρ ) x p x ρ 1 ρ ) ) x p 1 0 x p 1 x. A. The right part here is positive for x<0.55, that is, ϕ x 0forx 0.5. Let x>0.5. Then ϕ x 7 3p ) p 1 x ) ρx α 1 1 ρ ) x p x C C 1 x ρx p 1 1 ρ ) x p : u x, A.3

18 18 International Journal of Stochastic Analysis where C 7 3p p. We have u 0 C, u 1 0and u x C 1 x ρ p 1 ) x p 1 ρ ) px p 1 C 1 1 ρ ) px p ) x C 1 x 3 p ρ p 1 )) x p. A.4 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 w pt B S t, p ln, 0 <<e /. The difference between the correlation functions is Δ t cosh pt ) )) pt ) 0.5 sinh 1 t ). A.5 Let t>1, then Δ t B w pt 0. Let < t < 1. It is enough to show that the first term, ϕ, in the following representation: Δ t [ 0.5e pt 1 t ] 0.5e pt 1 1 e pt) ) : ϕ R, A.6 is nonnegative. Setting p ln, α one has ϕ t 0.5α t t α 1. A.7 Let us show that ϕ is decreasing. In this case ϕ is positive because ϕ 1 α/. We have ) ) 1 ϕ t α 0.5ln t ψ t, A.8 α 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. Butψ t min, because ψ t decreases near t α: ψ α 1, ψ α α ln e/α 1 α 0 for 0 <α<1. A.9 ence, ψ t max ψ α,ψ 1 1. As a result, ) ) 1 ϕ t α 0.5ln t 1 0. α A.30

19 International Journal of Stochastic Analysis 19 The last inequality holds for 0 <α<e, so we have Δ t 0, <t<1 for 0 <α<e. A.31 Let 0 <t<. Use Δ t cosh pt ) )) pt ] 1 t [1 0.5 t 1 sinh, A.3 then Δ t 0if )) pt 1/ max 0, t 1 sinh 1 sinh p ) 1. A.33 This inequality holds for 0 < <1/4. Combining the above inequalities, we get 3.1 for e 1/4. References 1 G. M. Molchan, Maximum of a fractional Brownian motion: probabilities of small values, Communications in Mathematical Physics, vol. 05, no. 1, pp , Ya. G. Sinaĭ, Distribution of some functionals of the integral of a random walk, Rossiĭskaya Akademiya Nauk. Teoreticheskaya i Matematicheskaya Fizika, vol. 90, no. 3, pp , Y. Isozaki and S. Watanabe, An asymptotic formula for the Kolmogorov diffusion and a refinement of Sinai s estimates for the integral of Brownian motion, Proceedings of the Japan Academy, Series A, vol. 70, no. 9, pp , Y. Isozaki and S. Kotani, Asymptotic estimates for the first hitting time of fluctuating additive functionals of Brownian motion, in Séminaire de Probabilités, 34, vol. 179 of Lecture Notes in Mathematics, pp , Springer, Berlin, Germany, T. Simon, The lower tail problem for homogeneous functionals of stable processes with no negative jumps, ALEA. Latin American Journal of Probability and Mathematical Statistics, vol. 3, pp , V. Vysotsky, On the probability that integrated random walks stay positive, Stochastic Processes and Their Applications, vol. 10, no. 7, pp , V. Vysotsky, Positivity of integrated random walks, 8 F. Aurzada and S. Dereich, Universality of the asymptotics of the one-sided exit problem for integrated processes, Annales de l Institut enri PoincaréB), 9 A. Dembo, J. Ding, and F. Gao, Persistence of iterated partial sums, 10 D. Denisov and V. Wachtel, Exit times for integrated random walks, v1. 11 G. Molchan, Unilateral small deviations of processes related to the fractional Brownian motion, Stochastic Processes and Their Applications, vol. 118, no. 11, pp , W. V. Li and Q.-M. Shao, Lower tail probabilities for Gaussian processes, The Annals of Probability, vol. 3, no. 1, pp. 16 4, G. Molchan and A. Khokhlov, Unilateral small deviations for the integral of fractional Brownian motion, 14 G. Molchan and A. Khokhlov, Small values of the maximum for the integral of fractional Brownian motion, Journal of Statistical Physics, vol. 114, no. 3-4, pp , W. V. Li and Q.-M. Shao, A normal comparison inequality and its applications, Probability Theory and Related Fields, vol. 1, no. 4, pp , A. Dembo, B. Poonen, Q.-M. Shao, and O. Zeitouni, Random polynomials having few or no real zeros, Journal of the American Mathematical Society, vol. 15, no. 4, pp , 00.

20 0 International Journal of Stochastic Analysis 17 T. Newman and W. Loinaz, Critical dimensions of the diffusion equation, Physical Review Letters, vol. 86, no. 13, pp , L. A. Shepp, First passage time for a particular Gaussian process, Annals of Mathematical Statistics, vol. 4, pp , J. Krug,. Kallabis, S. N. Majumdar, S. J. Cornell, A. J. Bray, and C. Sire, Persistence exponents for fluctuating interfaces, Physical Review E, vol. 56, no. 3, pp , M. A. Lifshits, Gaussian Random Functions, vol. 3 of Mathematics and Its Applications, KluwerAcademic Publishers, Dordrecht, The Netherlands, 1995.

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