Extremes of a class of non-stationary Gaussian processes and maximal deviation of projection density estimates

Abstract

In this paper, we consider the distribution of the supremum of non-stationary Gaussian processes, and present a new theoretical result on the asymptotic behaviour of this distribution. We focus on the case when the processes have finite number of points attaining their maximal variance, but, unlike previously known facts in this field, our main theorem yields the asymptotic representation of the corresponding distribution function with exponentially decaying remainder term. This result can be efficiently used for studying the projection density estimates, based, for instance, on Legendre polynomials. More precisely, we construct the sequence of accompanying laws, which approximates the distribution of maximal deviation of the considered estimates with polynomial rate. Moreover, we construct the confidence bands for densities, which are honest at polynomial rate to a broad class of densities.

Appendices

Appendix A: choice of the parameter δ

In this section, we provide an example on the choice of the parameter δ in Theorem 1. We will concentrate on the case of the Gaussian process

$$ \begin{array}{@{}rcl@{}} {\Upsilon}(t) = {\sum}_{j=0}^{J} \psi_{j} (t) Z_{j}, \end{array} $$

(61)

where Z₀,Z₁,... are the Legendre polynomials. As it is explained in Section 2.2, this case corresponds to the item (i) in Theorem 1: t₀ = A = − 1, $r_{10}(t_{0},t_{0})=\frac {1}{2}(\sigma ^{2}(t_{0}))^{\prime }<0$ (the behaviour at the point t₀ = B = 1 is completely the same). Denote

$$ D(\delta):=\max_{t\in \widetilde{\mathcal{M}}(\delta)}\bigl[-(\sigma^{2}(t))^{\prime}\bigr], $$

where ~(δ) = (δ) ∩ [− 1, 0]. In what follows, we assume that δ is such that ~(δ) = [A,b] for some b ∈ (− 1, 0). In the considered case, the absolute value of (σ²(t))^′ decays in some right vicinity of the point t₀ = − 1 (for instance, the plot of σ²(t) for J = 4 is given on Fig. 3) and therefore we can take δ such that

$$ D(\delta)=-(\sigma^{2}(A))^{\prime}=2|r_{10}(A,A)|. $$

(62)

Fig. 3

Plot of the variance σ²(t) = Var(Υ(t)) for J = 4. Note that S = maxσ²(t) = (J + 1)²/2 = 25/2, the set (δ) := {t : σ²(t) > S/(1 + δ)} is an interval for any δ < 4.44

Now we aim to find a lower bound for $\min \limits _{s,t\in \widetilde {{\mathscr{M}}}(\delta )}r(s,t)$. The two-dimensional mean value theorem yields

$$ \begin{array}{@{}rcl@{}} r(s,t) =r(A,A) +{{\int}_{0}^{1}}\left[(s-A)r_{10}(A+h(s-A),A+h(t-A))\right.\\ \left.+(t-A)r_{01}(A+h(s-A),A+h(t-A))\right]dh. \end{array} $$

We get

$$ r(s,t)\geq S-2(b-A)D_{1}(\delta),\qquad s,t\in \widetilde{\mathcal{M}}(\delta), $$

where D₁(δ) = maxs,t∈~(δ)(−r₀₁(s,t)). The inequality (35) reads as

$$ \chi<\chi_{1}(\delta):=\min\left\{ \delta,\frac{S-(b-A)D_{1}(\delta)}{(b-A)D_{1}(\delta)}\right\}. $$

(63)

Note that for the Legendre polynomials, A = − 1 and

$$ \begin{array}{@{}rcl@{}} D_{1}(\delta) = \max_{s,t\in \widetilde{\mathcal{M}}(\delta)} \Bigl(- {\sum}_{j=0}^{J} \psi_{j}(s) \psi^{\prime}_{j}(t) \Bigr)= - {\sum}_{j=0}^{J} \psi_{j}(-1) \psi^{\prime}_{j}(-1), \end{array} $$

(64)

because for δ small enough it holds

$$ \begin{array}{@{}rcl@{}} - \psi_{j}(s) \psi^{\prime}_{j}(t) \geq 0 \qquad \forall s,t \in \widetilde{\mathcal{M}}(\delta), \forall j=0,1,.. \end{array} $$

(65)

and

$$\operatornamewithlimits{arg max}_{s \in \widetilde{\mathcal{M}}(\delta)} |\psi^{\prime}_{j}(s)| = \operatornamewithlimits{arg max}_{t \in \widetilde{\mathcal{M}}(\delta)} |\psi_{j}(t)| = -1,$$

see, e.g., Section 5.1 from Konakov and Panov (2016b). The last expression in Eq. 64 and S can be directly computed:

$$ \begin{array}{@{}rcl@{}} - {\sum}_{j=0}^{J} \psi^{\prime}_{j}(-1) \psi_{j}(-1) = \frac{1}{4} {\sum}_{j=0}^{J} j(j+1)(2j+1)&=&\frac{1}{8}J(J+1)^{2}(J+2),\\ S = {\sum}_{j=0}^{J} {\psi_{j}^{2}}(-1)={\sum}_{j=0}^{J} \frac{2j+1}{2} &=& \frac{(J+1)^{2}}{2}. \end{array} $$

Therefore, we conclude that

$$ \begin{array}{@{}rcl@{}} \chi_{1}(\delta) = \min\left\{ \delta,\frac{4}{(b-A)J(J+2)} - 1 \right\} \end{array} $$

The second restriction on χ arrises due to Eq. 39:

$$ \chi\leq\chi_{2}(\delta):=\min_{t\in \widetilde{\mathcal{M}}(\delta)} \mathbb{P}si(t), \qquad \mathbb{P}si(t) := \frac{r_{10}^{2}(t,t)} {r_{11}(t,t)}. $$

(66)

This function cann’t be simplified in the considered particular case, and will be analysed numerically later.

The last restriction on χ appears due to the Corollary 1. Applying Theorem 8.1 from Piterbarg (1996), we get

$$ \begin{array}{@{}rcl@{}} P_{2u}(X(s)+X(t),\widetilde{\mathcal{M}}(\delta) \times (\mathcal{M}(\delta)\setminus \widetilde{\mathcal{M}}(\delta)))\leq C(b-A)^{2}ue^{-2u^{2}/R(\delta)} \end{array} $$

(67)

with some C > 0 and

$$R(\delta) =\max_{ \begin{smallmatrix} s\in \widetilde{\mathcal{M}}(\delta), \\ t \in \mathcal{M}(\delta)\setminus \widetilde{\mathcal{M}}(\delta) \end{smallmatrix}} \text{Var}({\Upsilon}_{s}+{\Upsilon}_{t}).$$

We have

$$ \text{Var} ({\Upsilon}_{s}+{\Upsilon}_{t}) =r(s,s)+r(t,t)+2r(s,t) = {\sum}_{j=0}^{J} \bigl(\psi_{j} (s) + \psi_{j}(t) \bigr)^{2}. $$

For the Legendre polynomials, it holds ψ_j(t) = (− 1)^jψ_j(−t), and therefore

$$ \begin{array}{@{}rcl@{}} R(\delta) = \max_{s,t \in \widetilde{\mathcal{M}}(\delta)} \mathcal{R}(s,t), \qquad \text{where} \quad \mathcal{R}(s,t):={\sum}_{j=0}^{J} \bigl(\psi_{j} (s) + (-1)^{j} \psi_{j}(t) \bigr)^{2}. \end{array} $$

Empirically, we get that the maximum of (s,t) is attained at the point (− 1,− 1) (see Fig. 4), and therefore

$$ \begin{array}{@{}rcl@{}} R(\delta) = \begin{cases} (J+1)(J+2), & J \text{is even},\\ J(J+1), & J \text{is odd}, \end{cases} \end{array} $$

Fig. 4

Plot of the function (s,t) for J = 4. Maximum is attained at the point (− 1,− 1)

for any δ, which guarantees that the set (δ) is an interval. Therefore, from Eq. 67 we get the last restriction on χ :

$$ \chi<\chi_{3}(\delta):=\frac{4S}{R(\delta)}-1 = \begin{cases} J/ (J+2), & J \text{is even},\\ (J+2)/J, & J \text{is odd}. \end{cases} $$

(68)

Finally, we conclude that for the optimisation of χ, we should find as follows

$$ \chi_{opt}=\max_{\delta} m(\delta), \qquad \text{where} \quad m(\delta):=\min\{\chi_{1}(\delta),\chi_{2}(\delta),\chi_{3}(\delta)\}. $$

This optimisation procedure is illustrated on Fig. 5. The left picture presents the plots of the functions χ₁(δ),χ₂(δ),χ₃(δ), while the right picture depicts the minimum between them. It turns out, that the maximum over δ is equal to 2/3, and this value is attained for any δ ∈ (0.71, 2.13).

Fig. 5

Plots of the function χ₁(t),χ₂(t),χ₃(t) (left) and of the function m(δ) = min(χ₁(t),χ₂(t),χ₃(t)) (right) for J = 4. The maximal value of m(δ) is equal to J/(J + 2) = 2/3, and this value is attained with any δ ∈ (0.71,2.13)

Appendix B: SBR-type theorem for projection density estimates

In this subsection, we briefly discuss the SBR-type theorem for the estimate (19). The next theorem shows that the distribution of _n converges to the Gumbel distribution. Nevertheless, the rate of this convergence is very slow, of logarithmic order.

Theorem 3

Assume that p ∈_q,H,β with some q,H > 0,β∈ (0, 1], and the basis functions ψ_j(x),j = 0, 1, 2,... satisfy the assumptions (A1) and (A2). Let M = M_n = ⌊n^λ⌋ with λ ∈ (0, 1).

• (i) For any x ∈ ℝ, it holds

  $$ \begin{array}{@{}rcl@{}} \mathbb{P} \Bigl\{\sqrt{\frac{n}{M_{n}}}\mathcal{R}_{n} \leq u_{M}(x) \Bigr\} =e^{- e^{-x}} \left( 1 +e^{-x} {\Lambda}_{M} (1+o(1)) \right), \end{array} $$

  (69)

  as n →∞, where

  $$ \begin{array}{@{}rcl@{}} {\Lambda}_{M} = \frac{\bigl(\log \log (M) \bigr)^{2}}{16 \log (M)} \end{array} $$

  and

  $$ \begin{array}{@{}rcl@{}} u_{M}(x) &=& a_{M} + \frac{xS}{a_{M}}, \end{array} $$

  (70)
  $$ \begin{array}{@{}rcl@{}} a_{M} &=& \bigl(2S \log(M) \bigr)^{1/2} - \frac{S^{1/2}}{2^{3/2}} \frac{\log\bigl(\bigl(8 \pi^{2} S / \mathfrak{c}_{0} \bigr) \log(M) \bigr) } {\bigl(\log(M) \bigr)^{1/2} } \end{array} $$

  (71)

  with S = maxσ²(t), and c₀ defined by Eq. 15 for X(t) = Υ(t).

• (ii) In Eq. 69, _n can be changed to _n, provided λ ∈ (1/(2β + 1), 1).

Proof

(i). Note that the process Υ(x) defined by Eq. 21 has zero mean and variance equal to Eq. 17. From Eqs. 23 and 15, we get for u →∞,

$$ \begin{array}{@{}rcl@{}} \mathbb{P} \Bigl\{\sqrt{\frac{n}{M}}\mathcal{R}_{n} \leq u \Bigr\} \!& \leq &\! \left[ \mathbb{P} \Bigl\{ \max|{\Upsilon}(x)| \leq u + \gamma_{n,M} \Bigr\} \right]^{M} + \mathcal{C}_{1} n^{-\kappa}\\ \!&=&\! \exp\Bigl\{M \log \Bigl(1 - \frac{ 1 }{2 \pi } e^{-\breve{u}_{n,M}^{2}/(2 \S)} \breve{u}_{n,M}^{-1} \bigl(\mathfrak{c}_{0} + \mathfrak{c}_{1} \breve{u}_{n,M}^{-2} + o(\breve{u}_{n,M}^{-2}) \bigr) \Bigr) \Bigr\} \\ && + \mathcal{C}_{1} n^{-\kappa} \\ \!&=&\! \exp\Bigl\{ - \frac{ M }{2 \pi } e^{-\breve{u}_{n,M}^{2}/(2 \S)} \breve{u}_{n,M}^{-1} \Bigl(\mathfrak{c}_{0} + \mathfrak{c}_{1} \breve{u}_{n,M}^{-2} + o(\breve{u}_{n,M}^{-2}) \Bigr)\Bigr\} \\ && + \mathcal{C}_{1} n^{-\kappa} \end{array} $$

(72)

with ŭ_n,M = u + γ_n,M. Next, let us substitute u = a_M + xS/a_M − γ_n,M with some a_M →∞ as M →∞. We have

$$ \begin{array}{@{}rcl@{}} \mathbb{P} \Bigl\{\sqrt{\frac{n}{M}}\mathcal{R}_{n} \leq a_{M} + \frac{xS}{a_{M}} -\gamma_{n,M} \Bigr\} \\ \leq \exp\Bigl\{ -e^{-x} \cdot \mathcal{G}_{M} \cdot \Bigl(1+a_{M}^{-2} \bigl(-x^{2}(S/2) - xS +\mathfrak{c}_{1}/\mathfrak{c}_{0} \bigr) +o(1) \Bigr) \Bigr\} + \mathcal{C}_{1} n^{-\kappa}, \end{array} $$

as M →∞, where

$$ \begin{array}{@{}rcl@{}} \mathcal{G}_{M} = \frac{M \mathfrak{c}_{0}}{2 \pi e^{{a_{M}^{2}}/(2 \S)} a_{M}}. \end{array} $$

(73)

Now we specify a_M such that _M ≍ 1. More concretely, let us find a_M in the form a_M = c_M − d_M/c_M, where c_M,d_M →∞,d_M/c_M → 0 as M →∞. This form leads to the equalities

$$ \begin{array}{@{}rcl@{}} M = e^{{c_{M}^{2}}/(2S)}, \qquad e^{d_{M}/S} \mathfrak{c}_{0} = 2\pi c_{M}, \end{array} $$

which suggest

$$ \begin{array}{@{}rcl@{}} c_{M} = \sqrt{2S \log(M)}, \qquad d_{M} = S \log(2 \pi c_{M}/\mathfrak{c}_{0}). \end{array} $$

Under this choice, we get

$$ \begin{array}{@{}rcl@{}} \mathcal{G}_{M} = 1-\frac{{d_{M}^{2}}}{2S {c_{M}^{2}}} (1+o(1)). \end{array} $$

(74)

Therefore,

$$ \begin{array}{@{}rcl@{}} &&\mathbb{P} \left\{\sqrt{\frac{n}{M_{n}}}\mathcal{R}_{n} \leq a_{M} + \frac{xS}{a_{M}} -\gamma_{n,M}\right\}\\ &\leq& \exp\left\{ -e^{-x} \cdot \left( 1 - \frac{{d_{M}^{2}}}{2S {c_{M}^{2}}} (1 + o(1) \right) \cdot \left( 1 + c_{M}^{-2} \left( -x^{2}(S/2) - xS +\mathfrak{c}_{1}/\mathfrak{c}_{0} \right) + o(1) \right) \right\} \\&&+ \mathcal{C}_{1} n^{-\kappa}\\ &=& \exp\left\{ -e^{-x} \left( 1 - {\Lambda}_{M} (1+o(1)) \right.\right\} + \mathcal{C}_{1} n^{-\kappa} = e^{-e^{-x} } \left( 1 + e^{-x} {\Lambda}_{M} (1+o(1)) \right). \end{array} $$

After the change x by x + γ_n,Ma_M/S, we get

$$ \begin{array}{@{}rcl@{}} \mathbb{P} \Bigl\{\sqrt{\frac{n}{M_{n}}}\mathcal{R}_{n} \leq u_{M}(x)\Bigr\} &\leq& e^{-e^{-x} } \bigl(1 + e^{-x} {\Lambda}_{M} (1+o(1)) \bigr),\quad n\to \infty,\end{array} $$

because γ_n,Ma_M converges to zero at polynomial rate (here we use the assumption M = n^λ, λ ∈ (0, 1) at the first time, see Remark 5). The proof of the inverse inequality follows from the second statement of the Proposition 1.

(ii). It holds

$$ \begin{array}{@{}rcl@{}} \bigl| \hat{p}_{n}(x) - p(x) \bigr| \leq \bigl| \hat{p}_{n}(x) - {\mathbb E} \hat{p}_{n}(x) \bigr| + \bigl| {\mathbb E} \hat{p}_{n}(x) - p(x) \bigr|. \end{array} $$

(75)

Let us show that the second summand can be upper bounded by an expression of order M^− 1. In fact, for any x ∈ [A,B],

$$ \begin{array}{@{}rcl@{}} 0 &=& {\sum}_{m=1}^{M}{\int}_{I_{m}} \psi_j^{(m)}(y) dy \cdot \psi_j^{(m)}(x), \qquad j=1,2,...,\\ 1 &=& {\sum}_{m=1}^{M}{\int}_{I_{m}} \psi^{(m)}_{0}(y) dy \cdot \psi^{(m)}_{0}(x). \end{array} $$

Both equalities are obtained by direct calculations; e.g., the first equality follows from

$$ \begin{array}{@{}rcl@{}} {\int}_{I_{m}} \psi_j^{(m)}(y) dy &=& M^{-1/2} \sqrt{\frac{2j+1}{2}} {\int}_{-1}^{1} \psi_{j}(x) dx \\&=& M^{-1/2} \sqrt{\frac{2j+1}{2}} \frac{1}{2^{n} n!} \frac{d^{n-1}}{dx^{n-1}}(x^{2}-1)^{n} |_{-1}^{1} =0. \end{array} $$

Therefore,

$$ \begin{array}{@{}rcl@{}} {\mathbb E}\hat{p}_{n}(x) - p(x) = {\sum}_{m=1}^{M} {\sum}_{j=0}^{J} \left[ {\int}_{I_{m}} \psi_j^{(m)}(y) \left( p(y) - p(x) \right) dy \cdot \psi_j^{(m)}(x) \right]. \end{array} $$

Applying the Cauchy-Schwarz inequality for the second sum, we get

$$ \begin{array}{@{}rcl@{}} \left| {\mathbb E}\hat{p}_{n}(x) - p(x) \right| &\leq& {\sum}_{m=1}^{M} \left( {\sum}_{j=0}^{J} \left( {\int}_{I_{m}} \psi_j^{(m)}(y) \left( p(y) - p(x) \right) dy \right)^{2} \right)^{1/2}\\ &&\left( {\sum}_{j=0}^{J} (\psi_j^{(m)}(x))^{2} \right)^{1/2}. \end{array} $$

Next, we apply the Cauchy-Schwarz inequality for the integral in Eq. 76:

$$ \begin{array}{@{}rcl@{}} \left| {\mathbb E}\hat{p}_{n}(x) - p(x) \right| &\leq& {\sum}_{m=1}^{M} \left( {\int}_{I_{m}} \left( p(y)- p(x) \right)^{2} dy \cdot {\sum}_{j=0}^{J} {\int}_{I_{m}} \left( \psi_j^{(m)}(y)\right)^{2} dy \right)^{1/2} \\ &&\cdot \left( {\sum}_{j=0}^{J} \left( \psi_j^{(m)}(x)\right)^{2} \right)^{1/2}. \end{array} $$

Now we will use that

$$ \begin{array}{@{}rcl@{}} {\int}_{I_{m}} \left( \psi_j^{(m)}(y)\right)^{2} dy=1, \forall j,m, \qquad \qquad {\sum}_{j=0}^{J} \left( \psi_j^{(m)}(x)\right)^{2} \leq C_{1} M, \end{array} $$

with some C₁ > 0 depending on J. We have

$$ \begin{array}{@{}rcl@{}} {\int}_{I_{m}} \left( p(y)- p(x) \right)^{2} dy \leq C_{2} M^{-(2\upbeta+1)}, \qquad \forall x \in I_{m}. \end{array} $$

We arrive at

$$ \begin{array}{@{}rcl@{}} \left| {\mathbb E} \hat{p}_{n}(x) - p(x) \right| \leq C_{3} M^{-\upbeta}, \qquad \forall x \in [A,B].\end{array} $$

with some constant C₃ > 0. Substituting this result into Eq. 75, we get

$$ \begin{array}{@{}rcl@{}} \sqrt{\frac{n}{M}} \mathcal{D}_{n} \leq \sqrt{\frac{n}{M}} \mathcal{R}_{n} + C_{4} n^{1/2} M^{-\upbeta-1/2} \end{array} $$

where C₄ = C₃q^− 1. On the other side, we have

$$ \begin{array}{@{}rcl@{}} \left| \hat{p}_{n} (x) - p(x) \right| \geq \left| \hat{p}_{n} (x) - {\mathbb E} \hat{p}_{n} (x) \right| - \left| {\mathbb E} \hat{p}_{n} (x) - p(x) \right|, \end{array} $$

and therefore

$$ \begin{array}{@{}rcl@{}} \sqrt{\frac{n}{M}} \mathcal{D}_{n} \geq \sqrt{\frac{n}{M}} \mathcal{R}_{n} - C_{4} n^{1/2} M^{-\upbeta-1/2}. \end{array} $$

We conclude that

$$ \begin{array}{@{}rcl@{}} \mathbb{P} \left\{ \left| \sqrt{\frac{n}{M}} \mathcal{D}_{n} - \sqrt{\frac{n}{M}} \mathcal{R}_{n} \right| \leq C_{4} n^{1/2} M^{-\upbeta-1/2} \right\}=1. \end{array} $$

By Lemma 1, for any x ∈ ℝ,

$$ \begin{array}{@{}rcl@{}} \mathbb{P} \left\{ \sqrt{\frac{n}{M}} \mathcal{R}_{n} \leq x - C_{4} n^{1/2} M^{-\upbeta-1/2} \right\} &\leq& \mathbb{P} \left\{ \sqrt{\frac{n}{M}} \mathcal{D}_{n} \leq x \right\} \\ &\leq& \mathbb{P} \left\{ \sqrt{\frac{n}{M}} \mathcal{R}_{n} \leq x + C_{4} n^{1/2} M^{-\upbeta-1/2} \right\}. \end{array} $$

Substituting u_M(x) defined by Eqs. 70–71 instead of x, we get that the left-hand side in Eq. 76 can be transformed as follows

$$ \begin{array}{@{}rcl@{}} &&\mathbb{P} \left\{ \sqrt{\frac{n}{M}} \mathcal{R}_{n} \leq u_{M}(x) - C_{2} n^{1/2} M^{-\upbeta-1/2} \right\} \\ &=& \mathbb{P} \left\{ \sqrt{\frac{n}{M}} \mathcal{R}_{n} \leq u_{M}\bigl(x-C_{2} n^{1/2} M^{-\upbeta-1/2}a_{M}/S \bigr) \right\}\\ &=& e^{- e^{-x}} \left( 1 +e^{-x} {\Lambda}_{M} (1+o(1)) \right), \end{array} $$

provided n^1/2M^{−β− 1/2}a_M converges to 0 at polynomial rate. The last condition is fulfilled for any α ∈ (1/(2β + 1), 1). The same argument holds for the right-hand side of Eq. 76, and the desired result follows. □

Appendix C: one technical lemma

Lemma T 1

Denote

$$ {\Theta}_{n,M}(x) := A_{M}(x+w_{n,M}) - A_{M}(x),$$

where w_n,M > 0 converges to zero as n,M →∞. Then

$$ \begin{array}{@{}rcl@{}} \sup_{x \in \mathbb{R}} {\Theta}_{n,M}(x) \leq c_{1} M^{\theta_{1}} w_{n,M} + c_{2} M^{-\theta_{2}} \end{array} $$

for some c₁,c₂ > 0 and any 𝜃₁,𝜃₂ > 0.

Proof

For x < c_M − w_n,M, we have Θ_n,M(x) = 0.

If x ≥ c_M,

$$ \begin{array}{@{}rcl@{}} {\Theta}_{n,M}(x) &\leq& \sup_{x\geq c_{M}} \bigl[ A_{M}^{\prime}(x) \bigr] w_{n,M}= \sup_{x\geq c_{M}} \Bigl[ - A_{M}(x) {\sum}_{i=1}^{k} \mathscr{P}^{\prime}_{i}\bigl(x \bigr) \Bigr] M w_{n,M}. \end{array} $$

Note that for x ≥ c_M, we have

$$ \begin{array}{@{}rcl@{}} 0 \leq - {\sum}_{i=1}^{k} \mathscr{P}^{\prime}_{i}\bigl(x \bigr) = \frac{2k}{\sqrt{S}} \phi\Bigl(\frac{x}{\sqrt{S}}\Bigr) < \frac{2k}{\sqrt{2\pi S}} e^{-{c_{M}^{2}}/(2S)}\lesssim M^{-1} e^{\sqrt{2S\log(M)}}. \end{array} $$

Since $e^{\sqrt {2S\log (M)}}\lesssim M^{\theta _{1}}$ for any $(\theta _{1}>0,)$ we conclude that

$$ \begin{array}{@{}rcl@{}} \sup_{x\geq c_{M}} {\Theta}_{n,M}(x) \lesssim c_{1} M^{\theta_{1}} w_{n,M}, \qquad n,M \to \infty \end{array} $$

(76)

for any 𝜃₁ > 0. Finally, if x ∈ (c_M − w_n,M,c_M), we have

$$ \begin{array}{@{}rcl@{}} {\Theta}_{n,M}(x) = A_{M}(x+ w_{n,M}) < {\Theta}_{n,M}(c_{M}) + A_{M}(c_{M}), \end{array} $$

where the first term is bounded by the expression in the right-hand side of Eq. 76. Now let us consider the second term:

$$ \begin{array}{@{}rcl@{}} A_{M}(c_{M}) &=& \exp \Bigl\{ -\frac{M \mathfrak{c}_{0}}{ 2\pi c_{M}} e^{-{c_{M}^{2}} / (2S)} (1 +o(1)) \Bigr\}\\ &=& \exp \Bigl\{ -\frac{ \mathfrak{c}_{0} e^{-S/2}}{ 2\pi} \frac{ e^{(2S \log M)^{1/2}}}{(2S \log(M))^{1/2}} (1 +o(1)) \Bigr\}. \end{array} $$

Applying (60), we conclude that A_M(c_M) converges to zero at polynomial rate with respect to M. This observation completes the proof of Lemma T.□