Poisson statistics at the edge of Gaussian \beta-ensemble at high temperature

# Poisson statistics at the edge of Gaussian β-ensemble at high temperature

###### Abstract

We study the asymptotic edge statistics of the Gaussian -ensemble, a collection of particles, as the inverse temperature tends to zero as tends to infinity. In a certain decay regime of , the associated extreme point process is proved to converge in distribution to a Poisson point process as . We also extend a well known result on Poisson limit for Gaussian extremes by showing the existence of an edge regime that we did not find in the literature.

\@footnotetext

Key words. Random matrices, Gaussian -ensembles, Poisson statistics, Extreme value theory.\@footnotetext2010 Mathematics Subject Classification. 60B20 - 60F05 - 60G70

## 1 Introduction

The study of spectral statistics in Random Matrix Theory has gathered a consequent volume of the research attention during the last decades. For several reasons, theses statistics are considered in the asymptotic regime: as the size of the matrix (and hence the number of eigenvalues) goes to infinity. One can inquiry about the behaviour of the whole spectrum (such as linear statistics), this is called global statistics (or regime). The main object to study in this context is the empirical spectral measure and the goal is to obtain a limiting distribution and give fluctuations around this limit. On the other hand, one can seek for more subtle, precise informations, like the spacing between two consecutive eigenvalues, or the nature of the largest eigenvalues; more generally, the joint distribution of eigenvalues in an interval of length . Such statistics are called local. In this particular regime, we differentiate between the bulk and the edge statistics. The bulk regime focuses on intervals inside the support of the limiting spectral measure while the edge regime concerns about the boundary. In this article, we are mainly interested in the asymptotic local edge regime, which corresponds to the largest eigenvalues.

Among random matrix models, two matrix ensembles are distinguished: Wigner matrices and invariant ensembles. The first one indicates matrices with independent components while the second gathers matrices whose law is invariant by symmetry group action. Their intersection is known as the GOE, GUE and GSE. Their origin trace back to the pioneer Wigner. He wanted to model complex highly correlated systems with (or lacking) different kind of symmetries (see [13, 11]) and considered Hamiltonians as large random matrices. The name stems from the invariance under certain group actions. The joint density of the eigenvalues can be derived (see for example Theorem on page in ) and is proportional to:

 P(λ1,...,λn)∝exp(−14n∑i=1λ2i)|Δn(λ)|βn∏i=1dλi.

The Vandermonde determinant is noted , and . Let us mention that when , the correlation functions, which will be our prime object, describe a determinantal process (Gaudin-Mehta formula, see for example Theorem on page in ). The idea that taking different values gives rise to different models is known as the Dyson’s Threefold-Way .

We can extend the model in two directions, allowing other values of and other potentials, by writing for :

We refer this as the general -ensemble. If the potential is quadratic , it reduces to the Gaussian -ensemble which is the object of our work.

In this context, Dumitriu and Edelmann  made a major breakthrough by constructing a matrix model for such -ensemble with any , hence extending the Dyson’s Threefold-Way . It states that the Gaussian -ensemble (viewed as a density probability function) is exactly the joint law of the spectrum of a certain simple matrix. The latter is obtained from successive Househölder transformations and has a symmetric tridiagonal form. This representation of the Gaussian -ensemble by a matrix model  led the way for many progresses [10, 16, 14, 17] on the understanding of the asymptotic local eigenvalue statistics for general . In particular, the authors of , leaning on the tridiagonal structure of the Gaussian -ensemble matrix model, gave multiple indications on how renormalized random matrices can be viewed as finite difference approximations to stochastic differential operators. Notably, the renormalization focuses on the top part of the matrix where the chi’s random variables are large. This conjecture was investigated in  where the properly renormalized largest eigenvalues are shown to converge jointly in distribution to the low-lying eigenvalues of a one-dimensional Schrödinger operator, namely the stochastic Airy operator . Since Brownian motion is not differentiable, this definition remains formal (one actually needs a variational formulation). Their result writes as for fixed, denoting the largest eigenvalues of the Gaussian -ensemble matrix and the smallest eigenvalues of the stochastic Airy operator:

 (nβ)23(2−λβi√nβ)1≤i≤klaw−−→n∞(Λβi)0≤i≤k−1.

Since the minimal eigenvalue of SAO has distribution minus TW, this work thereby enlarges Tracy-Widom law to all , that is:

 (nβ)23(λβi√nβ−2)law−−→n∞TWβ.

The Tracy-Widom law (with parameter ) is qualified as universal, in the sense that such local statistics hold for various matrix models (but also for objects outside of the random matrix field) and arises from highly correlated systems (such as modeled by some random matrices).

For finite dimension , one can choose in the joint law of the Gaussian -ensemble, which displays a lack of repulsion force as the Vandermonde factor vanishes, hence the correlation decreases, which means that randomness increases. In a Gibbs interpretation (which besides makes us refer to and its counterparts as partition functions), it comes down to consider an infinite temperature in such log-gas (terminology due to Dyson ). Readily, the joint density for is the density of i.i.d. Gaussian random variables whose maximum is known  to converge weakly, as , when properly renormalized, to the Gumbel distribution, one of the three universal distributions classes of the classical Extreme Value Theory. One deduces (see [5, Th 7.1]) Poisson limit for the Gaussian (ie: when ) extreme point process as the number of particles grows to infinity. This qualitative statistic is special to us since it is essentially our purpose in this article. It also carries more information and implies the limiting Gumbel distribution.

As the Gumbel law governs the typical fluctuations of the maximum of independent Gaussian variables, which corresponds to the case , and the Tracy-Widom law stems from complicated (highly dependent) systems such as the largest particles in the case fixed and , it is thus natural to ask for an interpolation between these two phases. The authors of  answer this question by proving that the properly renormalized Tracy-Widom converges in distribution to the Gumbel law as . They use the characterization of the distribution of the bottom eigenvalues of the stochastic Airy operator in terms of the explosion times process of its associated Riccati diffusion (see ). Regarding to our motivation, they could unfortunately not prove Poissonian statistics for the minimal eigenvalues , distributed according to the Tracy-Widom law, in the limit . This procedure would exactly reverse the order of the limits considered previously. Nonetheless, the authors investigated the weak convergence of the top eigenvalues in the double limit by heuristic and numeric arguments. They alluded to the idea that one can achieve Poissonian statistics for -ensemble using the same techniques as [14, 10], at high temperature within the regime . Concerning the bulk statistics, such work has been accomplished in the regime , that is Poisson convergence of the point process with an energy level in the Wigner sea (see [7, 8]). This was achieved in  by means of correlation functions, which is also our method.

The goal of this paper is to understand the behaviour of the largest particles of the Gaussian -ensemble as the inverse temperature converges to as goes to infinity. To this purpose, we study the limiting process of the extremes of the Gaussian -ensemble. Among all possible decay rates for , we restrict ourselves to the regime . More precisely, our main result gives the convergence as of the extreme process toward a Poisson point process on . Two regimes for the extremes appear according to the asymptotic behaviour of a certain auxiliary scaling sequence . In the situation where the latter converges, the scaling focuses on the very largest particles and the limiting process is inhomogeneous. Otherwise when , it comes down to consider the top particles which are slightly more inside the bulk. Also, it gives rise to a homogeneous limiting process. Roughly speaking, the rescaled extreme eigenvalues approximate a Poisson point process which means that adjacent top particles are statistically independent. Our work also applies when is set to and de facto includes asymptotics () of extremes of Gaussian variables (). While the outcomes are identical for both cases, we want to stress out that the models are intrinsically distinct. We investigate this question in the subsequent Remark 1.2. Doing such simultaneous double scaling limit, we fulfill the corresponding task addressed by Allez and Dumaz in  within another regime mentioned in their work and by other means, namely, the correlation functions method used by Benaych-Georges and Péché in .

For and two sequences, we adopt the notation and state our main result:

###### Theorem 1.1.

Let be such that . Let a family of random variables with joint law :

with normalization constant and Vandermonde determinant .
Let a positive sequence and the modified Gaussian scaling:

 bn:=√2log(n)−12loglog(n)+2log(δn)+log(4π)√2log(n),an:=δn√2log(n).
• Assume . Then the random point process converges in distribution to an inhomogeneous Poisson point process with intensity .

• Assume such that . Then the random point process converges in distribution to a homogeneous Poisson point process with intensity .

• When , the condition on is weakened to .

Let us first discuss the assumptions and conclusions of the theorem. We prove convergence of extreme point processes

 Pn:=n∑i=1δan(λi−bn)

toward a Poisson point process on with intensity as for suitably chosen scaling sequences and intensity . This convergence occurs regardless to or although this gives rise to two different models. The scaling sequences are exactly the same in both cases and are derived from the classical Gaussian scaling (see ), except that we increase the scale by a multiplicative term and lower down the center by an additive term involving . We then observe two regimes: first, when

 δn−−→n∞δ>0,

the limiting process is an inhomogeneous Poisson process with intensity (which is a classical result in the purely Gaussian setting, when ). When , even in the purely Gaussian setting (), we obtain a result that we did not find in the literature [5, 12, 15]: in this case, even though the interval considered (centered at and with width of order ) goes to , the limiting process is a homogeneous Poisson process. An illustration of these phenomena is given in Figure 1 below.

###### Remark 1.2.

As previously mentionned, the Poissonian description of the extreme process, along with the normalizing constants which display no dependence on , is valid for both cases and . The question of how close both models are is then raised. Therefore, we need to measure the impact of the decay rate of upon the model. In this direction, one can compare the normalization constants between different regimes. This idea emerges from equilibrium statistical mechanics where the is seen as the partition function in the Gibbs interpretation. The computations show a transition: when , both models are equivalent. As soon as , the repulsion is significant. We state this result in the forthcoming Lemma 1.3 whose proof is postponed to Section 2.3. It indicates that our main theorem gains value when compelling

 n−2≪β≪(nlog(n))−1,

which corresponds to the regime where both models are truly distinct. The critical role of in this description is consistent with the fact that one can write

 log|Δn(λ)|β=exp(βn∑i

with the sum having terms. Figure 1: The centering at bn for both cases δn→δ>0 and δn→∞ are represented on the main line. We zoom in around each bn by a factor δn√2log(n) and let n go to ∞. For δn≫1, the limiting object is a Poisson point process with intensity 1. For δn−−→n∞δ, it leads to a Poisson point process with intensity e−xδ.
###### Lemma 1.3.

Let and .

• Assume , then .

• Assume and , then .

The convergence toward a Poisson process for the extreme process is a much stronger information than the limiting distribution of the maximum. Indeed, one can deduce the limiting distribution for the largest eigenvalue for fixed .

###### Corollary 1.4.

Let be such that . Let with joint law . Let from Theorem 1.1 for . Then,

 Pn,β(an(λmax−bn)≤x) −−→n∞exp(−exp(−x)).
###### Remark 1.5.

This result shows that we recover the Gumbel law as limiting distribution of the largest particle from the Poisson limit, so that we retrieve the result of  corresponding to our setup. Besides, in view of (3) in the next section, we know that the largest eigenvalue is unbounded when goes to infinity since the Gaussian distribution has unbounded support. In addition to this observation, our main result provides the explicit order and Gumbel fluctuations for the maximum eigenvalue.

The paper is organized as follows: first, we introduce and comment our model. To derive Poisson statistics, our method is the study of the correlation functions associated to the extreme point process. We refer to this as our main tool and explain how it is exploited. Since the computations involve various estimates and quantities, we exhibit them as independent claims outside the main proof. The next section is devoted to the precise proof of our result. We give a tractable expression of the correlation functions. Then, we prove the conditions needed to provide inhomogeneous Poisson limit. Our work transposes to the homogeneous limit with ease so we merge both cases in our statements. Finally, we give a peculiar proof of the statement when . This is done by other means and displays a wider asymptotic regime for the perturbation , so we present it as an independent result.

###### Remark 1.6.

We consider two cases: and . For the second case, the assumption required is such that . It means that with . Note that the perturbation by corresponds to an increase of the zoom around the Gaussian center from first case minus a negligible factor. Nonetheless, most of our results remain valid under both regimes and with a weaker growth restriction. For this reason, in this text, the reader will encounter a less restrictive hypothesis on , namely . It ensures that is equivalent to for any such as goes to infinity.

###### Remark 1.7.

One may inquire about the extra factor in our growth condition in comparaison with the original regime . Indeed, we also expect the result to hold when . The reasons will become apparent along the paper. We will specially mention each time such restriction occurs. We also add that it seems rather difficult to overstep this technical limitation with our method.

Acknowledgements: I would like to express my gratitude to my supervisor Florent Benaych-Georges for his guidance throughout this work, his careful reading of the paper and the numerous advices he brought to me.

## 2 General model of the Gaussian β-ensemble for β≪1 and α>0

### 2.1 Background and preliminaries

For any , , and , we define:

 Zn,α,β:=∫Rnexp(−α2n∑i=1λ2i)|Δn(λ)|βn∏i=1dλi (1)

with the Vandermonde determinant factor:

 |Δn(λ)|β:=n∏i

and consider an exchangeable family of random variables with joint law

 Pn,α,β(dλ1,...,dλn) :=1Zn,α,βexp(−α2n∑i=1λ2i)|Δn(λ)|βn∏i=1dλi. (2)

When , we adopt the following notation:

 Zn,β:=∫Rnexp(−12n∑i=1λ2i)|Δn(λ)|βn∏i=1dλi

In the sequel, the parameter is always assumed to be except in some specific cases which will be mentionned. The reason of this choice shall be clear after incoming explanations.

###### Remark 2.1.

For , we retrieve the density of i.i.d. Gaussian random variables, which form a system of uncorrelated particles. The partition function in this case is just . Allowing , the Vandermonde factor vanishes when and acts as a (long range) repulsion force between the particles, which thereby constitutes a correlated system. The smaller is, the weaker repulsion operates.

From the crucial matrix model of Dumitriu and Edelman , we endow the Gaussian -ensemble with a matrix structure. Recall that where a -distributed random variable has density on . We state the corresponding result for our setup:

###### Theorem 2.2 (Dumitriu, Edelman, ).

Let the tridiagonal symmetric random matrix defined as:

 1√α⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝g11√2Xn−11√2Xn−1g21√2Xn−21√2Xn−2g31√2Xn−3⋱⋱⋱⋱⋱1√2X11√2X1gn⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠,

with i.i.d. sequence, an independent sequence such that and independent overall entries up to symmetry.

For any , , the joint law of the eigenvalues of is .

It makes the connection between the particles of law and the spectrum of . By trace invariance, we can easily access to further information: when , the empirical spectral distribution of has asymptotic first moment and second moment . In our setting , it reduces to consider .

In , with the choice , the authors proved under the assumption of simultaneous limit as , a continuous asymptotic interpolation for the empirical spectral measure between the Wigner semicircle law and the Gaussian distribution . The latter case is of our interest and particularly to the setting , they proved that:

 1nn∑i=1f(λi)P−−→n∞∫R1√2πf(x)e−x22dx,∀f∈Cb(R). (3)

This convergence also justifies the choice in our model for .

Such transition from Gaussian to Wigner distribution is furthermore investigated in [1, 7]. The limiting empirical eigenvalue density in the double limit is derived as a family of densities with parameter . Each of the papers [1, 4, 7] although provides different computations, hence giving rise to new identities which seem difficult to prove directly.

### 2.2 Correlation functions and Poisson convergence

The theorem we intend to prove will stem from the following result, which thereby makes it the cornerstone of our demonstration. It ensures that under pointwise convergence of the correlation functions and some uniform bound on it, the initial point process converges to a Poisson process.

###### Proposition 2.3 (Benaych-Georges, Péché, ).

Let be a locally compact Polish space and a Radon measure on . Let be an exchangeable random vector taking values in with density with respect to . For , we define the -th correlation function on :

 Rnk(x1,...,xk):=n!(n−k)!∫(xk+1,...,xn)∈Xn−kρn(x1,...,xn)dμ⊗(n−k)(xk+1,...,xn). (4)

Suppose there exists independent of such that:

• For fixed integer, on , we have the pointwise convergence:

 Rnk(x1,...,xn)−−→n∞θk.
• For each compact , there exists such that for any integer large enough, any integer , on , we have:

 1{k≤n}Rnk(x1,...,xk) ≤θkK.

Then, the point process converges in distribution to a Poisson point process with intensity as .

###### Remark 2.4.

The proof can be found in [4, Prop.5.6] where the scheme is successfuly applied to the bulk regime when . In this paper, we inspect the edge regime by using Proposition 2.3 for the rescaled -ensemble in two ways as we chose the real measure to be or just the Lebesgue measure. The two induced densities differ only by a -dependent term. By integral linearity, the same goes for the correlation functions. We derive the mandatory conditions in both cases, leading to two types of Poisson limit, but proofs are similar.

### 2.3 Partition functions

In this section, we list some identities, bounds and asymptotics involving partition fonctions. They will be used from time to time in the sequel of the text.

First, we give the main formula for the partition functions. From this, we will be able to compute several asymptotics of partition functions ratio.

###### Lemma 2.5.

For any and , the following identity holds:

 Zn,α,β=(2π)n2(n!)α−βn(n−1)4−n2n−1∏i=0Γ((i+1)β2)Γ(β2). (5)

If , one has also:

 Zn,α,β =(2π)n2α−βn(n−1)4−n2n∏i=1Γ(1+iβ2)Γ(1+β2). (6)
###### Proof.

Let . By the Selberg integral theorem in , we have:

 ∫Rn|Δn(x)|βe−12∑ni=1x2idx1...dxn=(n!)(2π)n2n−1∏i=0Γ((i+1)β2)Γ(β2).

By the change of variable , we get the fundamental identity on partition functions

 Zn,α,β =(2π)n2(n!)α−βn(n−1)4−n2n−1∏i=0Γ((i+1)β2)Γ(β2) =(2π)n2α−βn(n−1)4−n2n∏i=1Γ(1+iβ2)Γ(1+β2).

The case is easily treated. ∎

We are now ready to prove several results needed later.

###### Lemma 2.6.

Fix an integer and a real number . Let such that . Let any positive real sequence such that . Then,

 Zn−k,α,βZn,α,β =(1+o(1))(2π)−k2αk2 (7)
 Zn−k,α−kβ4b2n,βZn−k,α,β =1+o(1). (8)
###### Proof.

For , recall the equivalence of the Gamma function near the origin:

 Γ(u)=1u(1+o(1))≫1.

Using equation (5) of Lemma 2.5, we compute the ratio (7) for :

 Zn−k,α,βZn,α,β =(2π)−k2(n−k)!n!αk2+β4(2nk−k(k+1))n−1∏i=n−kΓ(β2)Γ((i+1)β2) =(1+o(1))(2π)−k2αk2.

If , the identity claimed is readily computed from equation (6).

Let us show the asymptotic (8). For , using (5), we have:

 Zn−k,α−kβ4b2n,β =(2π)n−k2(n−k!)α−β(n−k)(n−k−1)4−n−k2(1−kβ4αb2n)−β(n−k)(n−k−1)4−n−k2n−k−1∏i=0Γ((i+1)β2)Γ(β2).

Thus by a Taylor expansion of around :

 Zn−k,α−kβ4b2n,βZn−k,α,β

The last term converges to under our hypothesis. The case is easily treated. ∎

###### Lemma 2.7.

Assume . Let any positive real sequence such that . Fix a positive real number . There exists a sequence converging to , such that for large enough,

 Zn−1,αb2n−β4,βZn,α,β ≤cn√α2π(bn)−β(n−1)(n−2)2−n+1. (9)
###### Proof.

Note that our assumptions imply that the partition function is well defined since . From the identity (6), we have:

 Zn−1,αb2n−β4,β =(2π)n−12(αb2n−β4)−β(n−1)(n−2)4−n−12n−1∏i=1Γ(1+iβ2)Γ(1+β2)

and we can compute the ratio:

 Zn−1,αb2n−β4,βZn,α,β =(bn)−β(n−1)(n−2)2−n+1α(n−1)β2+12(1−β4αb2n)−β(n−1)(n−2)4−n−12Γ(1+β2)√2πΓ(1+nβ2).

We apply the following inequality:

 11−x≤4x,x∈[0,12], (10)

with . This inequality is true when is large enough. Thus,

 Zn−1,b2n−β4,βZn,β ≤cnΓ(1+β2)√2πΓ(1+nβ2)(bn)−β(n−1)(n−2)2−n+1√α,

where we have set:

 cn:=exp((β2(n−1)(n−2)16αb2n+β(n−1)8αb2n)log4+(n−1)β2logα).

It is clear that the latter sequence converges to from our hypothesis on .

Besides, the Gamma function has local minimum at with value , it follows that for ,

 Γ(1+β2)√2πΓ(1+nβ2)≤Γ(1+β2)√2π≤Γ(2)√2π=1√2π.

The next result states uniform bounds over for ratios of partition functions in connection with second condition of Proposition 2.3.

###### Lemma 2.8.

Let and positive real sequence such that . Assume . Let an integer such that . Then for large enough,

 Zn−k,1−kβ4b2n,βZn−k,β ≤4k (11)
 Zn−k,βZn,β ≤(√2π)k. (12)
###### Proof.

Since the case can be easily treated, we only consider . From our hypothesis, is less than when is large enough and:

 kβ4b2n≤12⟺kβ≤2b2n% which is true.
 β4b2n(β(n−k)(n−k−1)4+n−k2)≤(nβ)216b2n+nβ8b2n≤1.

Using (6) to compute the ratio and applying inequality (10), we have:

 Zn−k,1−kβ4b2n,βZn−k,β =(1−kβ4b2n)−β(n−k)(n−k−1)4−n−k2 ≤exp(kβ4b2n(β(n−k)(n−k−1)4+n−k2)log(4))≤4k.

We prove the second statement. From the identity (6) with , we get:

 Zn−k,βZn,α,β =(2π)−k2n∏i=n−k+1Γ(1+β2)Γ(1+iβ2).

The Gamma function has local minimum at with value , it follows that for any , since ,

 12≤Γ(1+iβ2)≤Γ(1+β2)≤1.

Hence,

 n∏i=n−k+1Γ(1+β2)Γ(1+iβ2)≤2k.

At last, we prove the previously stated lemma which compares the partition functions between different regime of :

###### Proof of Lemma 1.3.

Denoting the Euler constant, recall that for :

 logΓ(1+x)=−γx+π212x2+o(x3).

Remark that for any , one has:

 nkβk−1≫nk+1βk.

We compute the ratios:

 Zn,βZn,0 =n∏i=1Γ(1+iβ2)Γ(1+β2)=exp(−nlogΓ(1+β2)+n∑i=1logΓ(1+iβ2))

and,

 Zn,βZn,β′ =n∏i=1Γ(1+iβ2)Γ(1+iβ′2)=exp(−n∑i=1logΓ(1+iβ′2)+n∑i=1logΓ(1+iβ2)).

Since the quantity converges to uniformly in , we deduce that for some , thus verifying for any ,

 n∑i=1logΓ(1+iβ2) =−γβ2n∑i=1i+π248β2n∑i=1i2+n∑i=1yi,n =−γ8n2β(1+o(1)).

We deduce that:

 Zn,βZn,0=exp(−γ8n2β(1+o(1))),Zn,βZn,β′=exp(γ8n2(β′−β)(1+o(1))).

### 2.4 Estimates: bulk and largest eigenvalues

In the section, we establish some estimates on the eigenvalues of , which are -distributed. Since the particles are exchangeable, every estimate will concern .

We give exponential type bound on the probability of a scaled eigenvalue to be larger than any arbitrary value. Same-wise, an exponential estimate for the probability of to be as close as we want to any value is given.

These estimates will be crucial for the analysis of the integral term , which presents itself as the expectation of some functional of . The link is made through to the identity

 E|X|=∫+∞0P(|X|≥t)dt.

We begin with a technical but fundamental lemma.

###### Lemma 2.9.

For any and , one has:

 |a+b|β≤2βeβa2+b28. (13)
###### Proof.

First recall two inequalities:

 |x|≤2ex216,(x+y)2≤2x2+2y2.

Applying the first inequality with , then using the second one give:

 |a+b|β ≤(2e(a+b)216)β≤(2ea2+b28)β.

This inequality is of interest because it roughly allows to gain quadratic sum bound from a quantity of type . It provides an useful algebraic mean to upper-bound the integral term with a ratio of partition functions.

Next, we show an estimate on the scaled top eigenvalue. This result is also established in  but in another form, more appropriate to the bulk regime. For the sake of completeness, we give its proof since our version is slightly different.

###### Lemma 2.10.

Let such that . There exists a constant such that for any and , any fixed and any large enough,

 Pn,α,β(∣∣∣u−λ1bn∣∣∣≥t) ≤CMbβ(n−1)−2nexp⎛⎝−α2(b2n−nβ4α