The random matrix hard edge: rare events and a transition

We study probabilities of various rare events for the limiting point process that appears at the random matrix hard edge. We also show a transition from hard edge to bulk behavior. Asymptotic events studied include a central limit theorem and large deviation result for the number of points in a growing interval $[0,\lambda]$ as $\lambda \to \infty$. We study these results for the square root of the hard edge process. In this setting many of these behaviors mimic those of the Sine$_\beta$ process.


Introduction
In the study of classical Hermitian random matrix ensembles three distinct types of local behavior have been observed. The Gaussian Unitary Ensemble (GUE) exhibits one type of behavior in the interior, or bulk, of its spectrum and another at its edge. Scaling the n × n model and passing to the n → ∞ limit one obtains the Sine 2 and Airy 2 processes respectively. The Laguerre (also called Wishart) and Jacobi (MANOVA) ensembles exhibit the same behavior in the bulk, but depending on the choice of parameters may exhibit two different types of behavior at the edge. In one case the limit process at the edge is again the Airy 2 process. This is referred to as a soft edge. In the other case the limit process at the edge is a family of determinantal point processes where the determinant is defined in terms of Bessel functions J α , This type of limiting behavior will occur when the eigenvalues of the random matrix are pushed against some hard constraint, and so will be referred to as hard-edge behavior.
The Laguerre and Jacobi ensembles may be generalized to one parameter families of point processes called β-ensembles defined through their joint distribution. In particular the β-Laguerre ensemble has joint density p n,m,β (λ 1 , λ 2 , . . . , λ n ) = 1 Z β,n,m Here β may be any value greater than 0, m ≥ n, and Z β,n,m is an explicitly computable normalizing constant. With a slight abuse of terminology we refer to the points of the β-Laguerre ensemble as its eigenvalues. The local limits of these β-ensembles may again be studied, but can no longer be described by a determinantal point process.
As in the classical case, the β-Laguerre ensemble can exhibit two different types of limiting behavior at the lower edge of the spectrum. For m/n → γ = 1 the lower edge of the spectrum exhibits soft-edge behavior. In this case the appropriately rescaled lower edge of the β-Laguerre ensemble converges to the Airy β process [16]. In the case where instead m = n + a n and a n → a the lower edge of the spectrum exhibits hard-edge behavior [14].
In the intermediate regime where a n → ∞, a n /n → 0 it is expected that the behavior is soft edge and so the limiting process will be Airy β . For this regime there is a partial result in the case β = 2 for a n ∼ c √ n by Deift, Menon, and Trogdon (see [4]), but otherwise the problem remains open. Similar soft and hard edge scaling results were shown for the β-Jacobi ensemble [7]. Later universality results extended the soft edge limit to a wide class of β-ensembles [3,11], and recent work by Rider and Waters did the same for the hard edge [18].
Let λ 0 < λ 1 < λ 2 < ... be the ordered eigenvalues of the β-Laguerre ensemble. For the hard edge regime when a n → a the set {nλ 0 , nλ 1 , ..., nλ k } converges to the first k eigenvalues of the Stochastic Bessel operator introduced by Ramírez and Rider in [14]. The operator acts on functions R + → R and is given by: with Dirichlet boundary conditions at 0 and Neumann conditions at infinity, where b(x) is a Brownian motion, a > −1 and β > 0. Moreover, it can be shown that the spectrum defines a simple point process which will be referred to as the 'hard edge process'. For further discussion of the Stochastic Bessel operator see Ramírez and Rider [14].
The square root of the hard edge process gives a point process description for the singular values of G β,a . This scale is natural one for studying the transition from the edge to the bulk, and moreover, in this setting the asymptotic likelihood of rare events mimic those of the Sine β process. We will denote the singular value process by Bess a,β in honor of the Bessel functions present in the determinantal description. The results for the Bess a,β process will be stated in terms of its counting function M a,β (λ) which we define to be the number of points of the Bess a,β process in the interval [0, λ].
For a bit of amplification on the choice of the singular value process consider the following: we may perform the change of variables y = √ x in the Marchenko-Pastur distribution, the resulting distribution shows that the mean spacing after the change is the same order for both the edge and the bulk. This is confirmed by the work Edelman and LaCroix [5]. They show that the singular values of a GUE are distributed as the union of the singular values of two independent Laguerre ensembles with hard edge type distribution. A similar decomposition may also be done for the GOE [2].
In the bulk of the spectrum, with the appropriate centering and rescaling, Jacquot and Valkó showed that the eigenvalues of the β-Laguerre ensemble converge to the Sine β process [10]. This process was first introduced introduced as the limit of the β-Hermite ensemble by Valkó and Virág [19]. In this paper we make use of tools developed for the Sine β process to study the Bess a,β process and show a transition from Bess a,β to Sine β . The Sine β process may be described via its counting function in the following way: let α λ be a one parameter family of diffusions indexed by λ that satisfy where Z t = X t + iY t with X and Y standard Brownian motions and α λ (0) = 0. The α λ are coupled through the noise term. Define N β (λ) = 1 2π lim t→∞ α λ (t), then N β (λ) is the counting function for Sine β .
We might expect that as we move away from the edge of the Bessel process the effects of the edge will lessen and it will begin to behave like the bulk process. In this paper we show that this is indeed the case; there is a transition from Bess a,β to Sine β as we move from the edge (near 0) out towards ∞ in the Bess a,β process for a > 0. We also show two results on the asymptotic probability of various rare events for Bess a,β . The first is a central limit theorem for the number of points in the interval [0, λ] as λ → ∞. The second is a large deviation result on the asymptotic density of points in a large interval [0, λ]. We expect to see roughly 2λ/π many point in a large interval. We consider the asymptotic probability of seeing roughly ρλ many points for ρ = 2/π.

Results
We begin with the transition between the hard edge process and Sine β . Theorem 1. Let a > 0 and β > 0 fixed, then This can be understood by thinking of this as the distribution of the points in any neighborhood of λ scaled down by 4 converges as λ → ∞ to the distribution of Sine β in a neighborhood of 0. The centering of the neighborhood in the Sine β process is irrelevant since the process is translation invariant.
We now give the two results on the asymptotic behavior of M a,β (λ) as λ → ∞. The first of these gives a central limit theorem for the number of points in the interval.
A similar result except with limiting variance 2 βπ 2 was shown using a different method for the counting function of the Sine β process by Kritchevski, Valkó, and Virág [12].
The next result describes the large deviation behavior of the counting function. Before stating the result we introduce certain special functions that are used in the statement. We will use for the complete elliptic integrals of the first and second kind, respectively. Note that there are several conventions denoting these functions, we use the modulus notation from [1]. We also introduce the following function for m < 1: Theorem 3. Fix β > 0, a > −1. The sequence of random variables 1 λ M a,β (λ) satisfies a large deviation principle with scale λ 2 and good rate function βI Bess a,β (ρ) with where γ (−1) denotes the inverse of the continuous, strictly decreasing function given by Roughly speaking, this means that the probability of seeing close to ρλ points in [0, λ] for a large λ is asymptotically e −λ 2 βI Bess a,β (ρ) .
This result is closely related to the analogous result for the counting function N β (λ) of the Sine β process. There we consider the sequence 1 λ N β (λ) and we find an LDP with rate λ 2 and rate function βI Sine where I Bess a,β (ρ) = 32I Sine (ρ/4) [8]. Moreover we can check that the central limit theorem and the large deviation result are at least formally consistent. Lastly, observe that the large deviation result is also consistent with the tail behavior of the lowest For a bit of clarification on why we see consistency between results on the bulk and hard-edge processes we introduce the following characterization of Bess a,β . ϕ a,λ be the diffusion that satisfies the stochastic differential equation with initial condition ϕ a,λ (0) = 2π. Then Moreover, for a > 0 we have that lim t→∞ ⌊(ϕ a,λ (t) − 2π)/4π⌋ = lim t→∞ ⌊ϕ a,λ (t)/4π⌋ almost surely.
We also make the observation that for a fixed λ the α λ diffusion used in the characterization of the bulk process satisfies the SDE where B t is a standard Brownian motion. The Brownian motion B t that appears depends on the λ parameter.
Notice now that for λ large the ϕ a,λ diffusion will be rapidly increasing until time on the order of log λ. On this region the finite variation terms involving sin( ϕ a,λ 2 ) and sin ϕ a,λ will be rapidly oscillating and so have a minimal contribution. Essentially these terms are not felt in the λ → ∞ limit and so they vanish in asymptotic results. The results on oscillatory integrals involving ϕ a,λ will turn out to be the key component in the proof of all 3 main results and will be given in section 2.
It is worth noting that from the characterization in Theorem 4 it seems likely that one could show other results for Bess a,β related to existing results on the Sine β process. In particular we anticipate it would not be difficult to determine the asymptotic probability of overcrowding (P (M a,β (λ) ≥ n) ∼? as n → ∞, see [9]).
The remainder of the paper will be organized as follows: Section 2 will give the proof of Theorem 4 as well as several results on the ϕ a,λ diffusion. Section 3 will give the proof of the transition from Bess a,β to Sine β . Section 4 will give the proof of the central limit theorem. Section 5 will give the proof of the large deviation result.
Acknowledgements: The author would like to thank Benedek Valkó for helpful comments and corrections.

The counting function of Bess a,β
Before giving the proof of Theorem 4 we recall an existing description of Bess a,β that characterizes the process via diffusions rather then an operator. We consider the 'Riccati diffusion' for G β,a , given by the stochastic differential equation with initial condition p(0) = +∞, which it leaves instantaneously. Note that there is a positive probability of explosion to −∞.
In other words the counting function of the process is the number of times that p λ (t) hits 0, and may be denoted by M a,β ( √ λ), where M a,β is the counting function of Bess a,β .
We do a similar change of variables for p λ < 0. Take X 2 (t) = log(−p λ (βt/4)) + βt/8 − log λ/2. This gives us The boundary condition p λ (βt/4) = 0 gives X 2 (t) = −∞, and for p λ (βt/4) = −∞ we get To find the ϕ a,λ diffusion given in Theorem 4 we work back from X 1 and X 2 to ϕ a, √ λ . Notice that the zeros of p λ describe the eigenvalue process of G β,a and so the resulting The conditions X 1 = −∞ and X 1 = +∞ become ϕ = −2π and ϕ = 0 respectively. For X 2 take ϕ = 4 arctan e X 2 . This gives The conditions X 2 = −∞ and X 2 = +∞ become ϕ = 0 and 2π respectively. Now notice that this diffusion is invariant under 4π spacial shifts, so for a fixed λ with initial condition ϕ a, Lastly we use that ⌊ϕ a, √ λ ⌋ 4π is monotone nondecreasing in t to rewrite the supremum as a limit.
For the final statement when a > 0 we appeal to the erratum for the original convergence result on the hard edge, [15]. Observe that in this case counting 0 of the p λ diffusion is equivalent to counting explosions to −∞. Therefore any time ϕ a,λ passes a multiple of 4π is must pass the next 2π multiple as well.
It will be useful to consider what is the relationship between two diffusions that satisfy stochastic differential equations of the form (8) with two different λs which are coupled through their noise terms. Let ψ a,λ,x = ϕ a,λ+x − ϕ a,λ , then the SDE for ψ a,λ,x is with initial condition ψ a,λ,x (0) = 0. This follows from standard Itô techniques together with the application of angle addition formulas. This rather ugly formula can be made more palatable by the observation that the oscillatory terms may be well controlled.
Proposition 6. Let ϕ a,λ and ψ a,λ,x be defines as above, then for T ≤ 8 β log λ there exists a constants M and γ (uniform in λ and T ) such that In particular in the case where T is fixed this gives sup 0≤s≤T s 0 sin (cϕ a,λ ) dt → 0 in L 1 (and hence in probability) as λ → ∞ (and similarly for cos(cϕ a,λ ) and integrals related to (16)).

Moreover
A similar bound on the tail of the integral appearing in (16) also holds.
Proof of Proposition 6. We write ϕ a,λ (t) in its integrated form (dropping the subscripts) We break this into two pieces, the first term we will write λH(t) = λ8 1 − e − β 8 t and the remaining terms will be grouped together as the process We use the following version of Itô's formula to extract the main term. Let f, g be continuously differentiable functions and let G denote the antiderivative of g. Then for X and Itó Now observe that Λ(t) may be bounded in the following way: In absolute value the final integral is bounded by 1 λcβ (e β 8 t − 1) which gives us that From this we get that the dt terms in the dE t integral may be bounded in absolute value Lastly, for the martingale term we break it into its real and complex parts and use Doob's martingale inequality on the associated exponential submartingales. We show the argument for the imaginary part. The complex part may be done the same way. Let then N t is a true martingale because it has L 1 bounded quadratic variation. Therefore exp(ξN t ) is a positive submartingale and so P (sup 0≤t≤T exp(ξN t ) ≥ x) ≤ E(exp(ξN T ))/x.
From this we get that To compute E exp(ξN t ) we make use of the martingale . This gives us that Optimizing our choice of ξ we get P sup This gives the necessary bound for (17). Integrating in the C variable gives which completes the proof of equation (15).
To extend this to the case where we have the additional e icψ λ,f,g − 1 multiplier in the integral we use the same decomposition of ξ f,a,λ,0 and work with the integral.
An application of a slightly modified Itó's lemma leads us to the same type of analysis as before and give the necessary bounds for (16). In particular for u, v, w continuously differentiable functions with V, W the antiderivatives of v and w we get All of the finite variation terms the integrand may be bounded in absolute value byM Λ(t) for some constantM . The martingale terms may be handled the same way as before.

From the hard edge to the bulk
In order to show the transition we need the following two results on limits of martingales and stochastic integrals:  For a proof see e.g. Theorem 7.1.4 in Ethier and Kurtz [6].
The following proposition gives conditions under which a sequence of diffusions X n satisfying the stochastic integral equations converge to a limiting process. Here we take M n : R + → (C[0, ∞)) d to be a d−dimensional martingale and V n : R + → R d×d is a finite variation process. The following is a specialization of Theorem 5.4 from from [13]. Brownian motion and V (t) = tI. Suppose that the diffusion X satisfies and that (26) has a unique strong solution. Then X n ⇒ X.
Lastly we will need a property of the diffusion α λ (t).
Proof of Theorem 1. We study the difference M a,β (λ + x) − M a,β (λ) as λ → ∞ by way of the SDE characterization. For convenience we use the diffusion ψ a,λ,x = ϕ a,λ+x − ϕ a,λ defined in section 2 (see (14)). The proof breaks into two pieces. The first is to show that for any finite set {x 1 , x 2 , ..., x k } the family of diffusions {ψ a,λ,x i } i=1,...,k converge weakly to a family of ..,k on compact sets [0, T ]. We then show that for T sufficiently large this convergence is enough, that is that the tail of the diffusion is well behaved. These together will be sufficient for process convergence because they will show that the finite marginals of M a,β (λ + x) − M a,β (λ) will converge to those of N β (x/4) where N β is distributed as the counting function of Sine β .
Recall that ψ a,λ,x satisfies the SDE in (14). In order to study the limit as λ → ∞ we start by showing that the martingales converge in distribution to two independent Brownian motions W 1 (t) and W 2 (t) as λ → ∞.
We then use these limits to show the convergence of the vector (ψ a,λ,x 1 (t), ..., ψ a,λ,x k (t)) to (ψ x 1 (t), . . . ,ψ x k (t)), which is a vector of time and space changed versions of α x , for x = Proposition 8 gives us that if this SDE has a strong solution, thenψ x d = lim λ→∞ ψ a,λ,x exists and satisfies (28). To show that (28) has a strong solution we will show this is equivalent to the SDE for α λ having a strong solution. Apply time change t/2 → s and space changê whereW 1 andW 2 here are the time changed W 1 and W 2 . Notice that the time change is independent of x and soW 1 andW 2 are the same driving Brownian motions for x 1 , ..., x k .
Therefore the diffusionsψ x i are coupled through their Brownian terms. Lastly observe that ψ x i d = α 4x i and α λ has a strong solution [19].
For the next step we need to show that convergence of the diffusions implies convergence of the processes Bess a,β to Sine β . To do this we show convergence of the finite dimensional marginals of the counting function. That is we show that for any finite collection ..,k jointly in law as λ → ∞.
We begin with the following lemma, which gives us that it is enough to study the diffusion ψ a,λ,x .
Lemma 10. For ψ a,λ,x and M a,β (λ) defines as above we have For convenience we introduce the notation ⌊y⌋ 4π = 4π⌊y/(4π)⌋. We now show that for any ε > 0 we can choose λ and T sufficiently large so that where evaluation at ∞ should be understood as a limit. Since we have that this will be sufficient to complete the proof.
Together these bounds give us that where ε and δ may be chose arbitrarily small.
To start we note that this problem is equivalent to studying the diffusions restarted at Before moving forward we make the following observation: For large enough time S (depending on δ, assuming x/λ ≤ 1) This follows from the fact that ⌊φ(t)⌋ 4π is a monotone increasing function with an almost surely finite limit. Then by using continuous dependence on parameters and initial conditions we also have that for small enough δ and large enough λ P |φ a, 1 Proof of Lemma 10. We need the final statement in Theorem 4. This gives us that for a fixed λ and large enough T we will have that for t ≥ T , ϕ a,λ (t) ∈ (M a,β (λ)4π+2π, M a,β (λ)4π+4π).

The central limit theorem
The proof of the central limit theorem may be done in a manner similar to the proof for the Sine β process which was done by Kritchevski, Valkó, and Virág in [12], but here we will get the result as an easy consequence of Proposition 6.
Proof of Theorem 2. First notice that the processφ a,λ (t) = ϕ a,λ (t + T ) with T = 8 β log(λ) satisfies the same SDE (8) with λ = 1 with a random initial condition. Since the equation is 4π periodic in the ϕ variable and we wish to consider the difference with its limit we may shift the process down so that the initial condition is in the interval [0, 4π]. That iŝ ϕ a,λ (∞) −φ a,λ (t) =φ a,λ (∞) −φ a,λ (t) whereφ a,λ (t) =φ a,λ (t) − ⌊φ a,λ (0)⌋ 4π . Here we use ⌊·⌋ 4π to denote rounding down to the next multiple of 4π as in section 3. From this we get in distribution and hence in probability. Because of this it is sufficient to consider the weak Written in its integrated form (and dropping the a, λ subscripts), the SDE for ϕ a,λ gives us We will show that when scaled down by √ log λ the first two terms vanish in the limit, then show that the martingale term has the appropriate variance. An application of Proposition 6 gives that the expected value of the first two integrals is finite for all λ, and so when scaled down by √ log λ we get convergence to 0 in probability.
We now turn our attention to the last remaining term in (41). We rewrite this as for some standard Brownian motionB t which depends on λ. By Proposition 6 this final integral term goes to 0 in probability. Therefore may be made arbitrarily small. This is enough to give the desired convergence in distribution, and so completes the proof.

Large Deviations
In [8] a large deviation result was proved for the Sine β process on growing intervals [0, λ]. The proof of the large deviation for the Bess a,β process is similar with a few notable differences.
The details of the proof will be largely omitted, but we will give an outline of the proof and fill in the details in the steps where the proof differs significantly from the one for Sine β .
Outline of the proof of Theorem 3 The proof of the large deviation principle is done by first proving a large deviation principle for the path of the ϕ a,λ diffusion. We use this together with the contraction principle to prove the large deviation result for the end point of the diffusion. The proof of the LDP for the path begins with the observation that ⌊ϕ a,λ ⌋ 2π := 2π⌊ϕ a,λ /2π⌋ is a monotone nondecreasing function for λ > 0. Because of this it is enough to understand the time it take ϕ a,λ to traverse an interval of the form [2πk, 2π(k + 1)]. Bounds on these travel times will be done using a Girsanov change of measure and for various reasons are easier to compute when the β 4 λe − β 4 t dt term is replaced with a constant (or piecewise constant) drift. What follows is a more detailed outline of how to prove Theorem 3.
Step 1: An LDP for a modified diffusion To start we define a diffusion with no time dependence in the drift. Letφ a,λ satisfy dφ a,λ = β 2 (a + 1 2 ) sin ϕ a,λ 2 dt + λdt + sin ϕ a,λ 2 dt + 2 sin withφ a,λ (0) = 2π. We prove a path large deviation principle for this diffusion on finite time intervals. To prove this we take the following steps: 1. We can see that whenφ a,λ is a multiple of 2π all of the terms vanish except for the λdt term. From this we get that ⌊φ a,λ ⌋ 2π is nondecreasing when λ > 0. Define We make use of the same change of variables and Girsanov arguments as the Sine β to get: for A < 1, then for τ = τ 2 , τ 3 we have Let t A = 4K(A) and fix 0 < ε < |t A − 2π|, then where lim λ→∞ C(ε, λ, A) = 1 for fixed a, ε. See Proposition 8 in [8] for the idea of the proof.
2. From the jump bounds we get the following estimates for comparing the diffusion with a linear path: There exist a constant c so that for λ > 2 we have Moreover, there are absolute constants c 0 , c 1 so that if qtλ, q and λq log q are all bigger than c 0 then 3. The bounds from the previous step may be used to derive a path deviation result for ϕ a,λ . Step 2: From a path deviation forφ a,λ to one for ϕ a,λ We use the path deviation onφ a,λ , together with some bounds on the behavior of the diffusion for large t to get the following path diffusion for ϕ a,λ .
Theorem 13. Fix β > 0 and let ϕ a,λ (t) be the process defined in (8)    in the case where g(0) = 0 and g is absolutely continuous with non-negative derivative g ′ . In all other cases J Bess a,β (g) is defined as ∞.

Approximation 1: Truncation
Fix T > 0, the value of which will go to infinity later. Define Then for T sufficiently large (not depending on λ), lim sup λ→∞ a,λ −ϕ a,λ ≥ δλ) ≤ c 1 T δ 2 . The proof of this for the bulk is done in Proposition 13 of [8] with the exception of the very last bound used, which bounds P (ϕ a,λ (∞) − ϕ a,λ (T λ) ≥ δλ/2). This bound is replaced by Proposition 14 which will be given below.
The method of proof is the same as in [8], but there bound used in line (56) does not hold for theφ a,λ diffusion and is replaced by Lemma 16 given below.

Approximation 3: Piecewise constant path
Let π M N be the projection of a path onto a piecewise linear path defined by and linear in between these values. Define This approximation may be treated in the same manor as α λ defined in [8] for both determining the probability of it being close to some particular path, as well as the probability that it is close to ϕ (2) a,λ .

Lower bound
The proof of the lower bound uses similar ideas. We will essentially reuse approximations ϕ (1) a,λ and ϕ (2) a,λ . We show that ϕ (2) a,λ stays close to a particular path by making use of the path deviation result onφ a,λ . The approximation bounds to show that this is sufficient are the same as those in the upper bound. For more on the argument we refer the reader to [8].
Step 3: From a path deviation to the endpoint The final step in the proof of Theorem 3 is to use the contraction principle to go from the path deviation to a large deviation result on the end point. We use the existing analysis in section 7 of [8] and a relationship between the Bess a,β and Sine β rate functions to draw our conclusion. Consider the J Bess a,β rate function given in Theorem 13, we apply the change of variables x = (1 − e − 8 β t ) to get the modified rate functioñ J Bess a,β (g) = 8β ). The path large deviation rate function for the Sine β process [8] is related to this one byJ Bess a,β (g) = 32J Sine (g/8).
Lastly note that we want to optimize over g with endpoint 4πρ where in the Sine β case we had an endpoint of 2πρ. This gives us that I Bess a,β (ρ) = 32I Sine (ρ/4).
The details for step 2 part 1 approximations 1 and 2 The following proposition replaces a tail bound used for the α λ diffusion that does not exist for the ϕ a,λ diffusion because of the diffusion dependent drift terms. Proposition 14. Let ϕ a,λ be the diffusion defined in (8), T > 0 and ǫ > 0 fixed, then where c β is some explicitly computable constant depending only on β.
We will start with the case a ≥ −1/2. Recall the diffusion X 1 from section 2 defined in (12). We are working with the singular value process, therefore we work with a diffusion where √ λ has been replaced with λ. In particular we study X λ which satisfies Notice that in order to have ϕ a,λ (∞) ≥ k4π we must have that it crosses an interval of the type [ℓ4π − 2π, ℓ4π] at least k times. On this interval we can do a change of variables so this is equivalent to X 1,λ ℓ exploding for ℓ ≤ k where λ ℓ is λe − β 4 t ℓ where t ℓ is the hitting time of ℓ4π − 2π. Notice that as λ decreases the likelihood of explosion decreases, so we may use the following bound In order to study the probability that Xλ explodes we define Z = Xλ + B t , then Z explodes at the same time as Xλ and satisfies the differential equation In its integrated form we get We now observe that cosh(Z −B) ≤ 2 cosh Z cosh B, therefore since the remaining drift term is less than or equal to 0 for a ≥ −1/2 and we get that Using this we get the following bounds on our second integral term: Putting the bounds from (47) and (48) together into the bound in line (46) completes the proof of the proposition for a ≥ −1/2. For a < −1/2 we may do a similar analysis using the X 2 diffusion in (13).
Following the proof of Lemma 10 in [8] this can be used to show the following bound.
These two propositions are the only major changes needed to adapts the proof of the large deviation result for Sine β given in [8] to the Bess a,β process.
Sketch of proof of Proposition 15. The proof of the jump time bound for σ k is done through a coupling argument. Let T k = σ 1 + · · · + σ k−1 , we make the change of variablesψλ ,λ (T k + t) = 8 arctan(e Y (t) ) on the interval [4πk, 4π(k + 1)). We get that (dropping the subscripts) Y satisfies the stochastic differential equation with initial condition Y (0) = −∞ and Y explodes at the hitting time σ k . These oscillatory integrals are not the same as the ones that appear in Proposition 6, but they are amenable to the same type of analysis. We can check that the quadratic variation of this process is Let Osc Y,t denote the finite variation terms involving sin(cϕ) or cos(cϕ). We can check that P (| Let W t = t 0 g(Y, ϕ)dB s and consider the diffusionŶ given byŶ t = Y t − W t . Then dŶ =λ 4 coshŶ cosh W t − sinhŶ sinh W t dt + 1 2 √ 2 dB t − 1 8 tanh Y dt + Osc Y,t dt withŶ (0) = −∞. Notice that W t is finite almost surely, and soŶ explodes at the same time as Y . This explosion time is σ k . Therefore proving bounds on the explosion time ofŶ is enough.
We can choose δ small enough, so that we get The explosion time ofŶ t will be bounded between the explosion times of Z ± (on the set where the oscillatory integrals are small). Now the Z ± diffusions may be treated using the same methods as Proposition 8 in [8].