3.1. Cramér-type moderate deviation for linear random fields

Let X = {X_j, j ∈ ℤ^d} be a linear random field defined on a probability space (Ω, $\mathcal F$, ℙ) by

\begin{align*} X_j=\sum_{i\in \mathbb{Z}^d}a_i \varepsilon_{j-i}, \quad\quad j\in\mathbb{Z}^d, \end{align*}

where the innovations ε_i, i ∈ ℤ^d, are i.i.d. random variables with mean zero and finite variances σ ², and where {a_i, i ∈ ℤ^d} is a sequence of real numbers that satisfy $\sum_{i \in \mathbb{Z}^d} a_i^2 \lt \infty$.

Linear random fields have been extensively studied in probability and statistics. We refer to [Reference Sang and Xiao41] for a brief review on studies in limit theorems, large and moderate deviations for linear random fields, and to [Reference Koul, Mimoto and Surgailis24, Reference Lahiri and Robinson25], and the reference therein for recent developments in statistics.

By applying Theorem 2.1, we establish the following moderate deviation result for linear random fields with short or long memory, under Cramér’s condition on the innovations ε_i, i ∈ ℤ^d. Compared with the moderate deviation results in [Reference Sang and Xiao41], our Theorem 3.1 below gives a more precise convergence rate which holds on a much wider range for x.

Suppose that there is a disc centered at z = 0 within which the cumulant generating function $L(z) =L_{\varepsilon_i}(z)=\log {\rm {\mathbb E}} \text {e}^{z\varepsilon_i}$ of ε_i is analytic and can be expanded in a convergent power series

\begin{align*} L(z)=\sum_{k=1}^\infty \frac{\gamma_k}{k!}z^{\,k}, \end{align*}

where γ_k is the cumulant of order k of the random variables ε_i, i ∈ ℤ^d. We have $\gamma_1= {\rm {\mathbb E}}\varepsilon_i=0$ and $\gamma_2 = {\rm {\mathbb E}}\varepsilon_i^2=\sigma^2, i\in\mathbb{Z}^d$.

We write

(3.1)\begin{align}\label{partialsum} S_n=\sum_{j\in\Gamma^d_n }X_j=\sum_{j\in\mathbb{Z}^d} b_{nj}\varepsilon_j, \end{align}

where $b_{nj}=\sum_{i\in\Gamma^d_n}a_{i-j}$. In the setting of Section 2, we have X_nj = b_njε_j, j ∈ ℤ^d. Then it can be verified that, for all n ≥ 1 and j ∈ ℤ^d, X_nj satisfy condition (2.1) for suitably chosen H_n. In the notation of Section 2, we have

\begin{align*} B_n=\sigma^2\sum_{j\in\mathbb{Z}^d} b_{nj}^2, \quad\quad F_n(x)={\rm {\mathbb P}}(S_n\lt x \sqrt{B_n}). \end{align*}

Hence, we can apply Theorem 2.1 to prove the following theorem.

Theorem 3.1. Assume that the linear random field X = {X_j, j ∈ ℤ^d} has short memory, i.e.

\begin{align*} A:= \sum_{i\in \mathbb{Z}^d}|a_i|\lt\infty, \quad\quad a:= \sum_{i\in \mathbb{Z}^d}a_i\ne 0, \end{align*}

or long memory with coefficients

(3.2)\begin{align} \label{Eq:a} a_i=l(|i|)b\Big(\frac{i}{|i|}\Big)|i|^{-\alpha}, \quad\quad i\in \mathbb{Z}^d,\,|i|\ne 0, \end{align}

where α ∈ (d/2, d) is a constant, l(·): [1, ∞) → ℝ is a slowly varying function at ∞, and b(·) is a continuous function defined on the unit sphere ${\mathbb S}_{d-1}$ Suppose that there exist positive constants H and C such that

(3.3)\begin{align}\label{cgf cond lm} |L(z)|\lt C \end{align}

in the disc |z| < H. Then, for all x ≥ 0 with x = o(n^{d /2}), we have

(3.4)\begin{align}\label{Eq:lMD} \frac{1-F_n(x)}{1-\Phi(x)}=\exp\Big\{\frac{x^3}{n^{d/2}}\lambda_n\Big(\frac{x}{n^{d/2}}\Big)\Big\} \Big(1+O\Big(\frac{x+1}{n^{d/2}}\Big)\Big), \end{align}

where

\begin{align*} \lambda_n(t)=\sum_{k=0}^\infty \beta_{kn}t^k \end{align*}

is a power series that stays bounded uniformly in n for sufficiently small values of |t|, and the coefficients β_kn only depend on the cumulants of ε_i and on the coefficients a_i of the linear random field.

To the best of our knowledge, Theorem 3.1 is the first result that gives the exact tail probability for partial sums of random fields with dependence structure under the Cramér condition.

Due to its preciseness, Theorem 3.1 can be applied to evaluate the performance of approximation of the distribution of linear random fields by truncation. We often use the random variable $X_j^m=\sum_{i\in \Gamma_m^d} a_i \varepsilon_{j-i}$ with finite terms to approximate the linear random field $X_j=\sum_{i\in \mathbb{Z}^d}a_i \varepsilon_{j-i}$ in practice. For example, the moving average with finite terms MA(m) is applied to approximate the linear process (moving average with infinite terms). In this case, Theorem 3.1 also applies to the partial sum $S_n^m = \sum_{j\in\Gamma^d_n }X_j^m=\sum_{j\in\mathbb{Z}^d} b_{nj}^m\varepsilon_j$. Here only finite terms $b_{nj}^m$ are nonzero. Denote

\begin{align*} B_n^m=\sigma^2\sum_{j\in\mathbb{Z}^d} (b_{nj}^m)^2, \quad\quad F_n^m(x)={\rm {\mathbb P}}(S_n^m\lt x\sqrt{B_n^m}). \end{align*}

Then, for all x ≥ 0 with x = o(n^{d/ 2}), we have

\begin{align}\frac{1-F_n^m(x)}{1-\Phi(x)}=\exp\Big\{\frac{x^3}{n^{d/2}}\lambda_n^m \Big(\frac{x}{n^{d/2}}\Big)\Big\} \Big(1+O\Big(\frac{x+1}{n^{d/2}}\Big)\Big), \end{align}

where

\begin{align*}\lambda_n^m(t)=\sum_{k=0}^\infty \beta_{kn}^mt^k,\end{align*}

and where the coefficients $\beta_{kn}^m$have a similar definition as β_kn. To see the difference between the two tail probabilities of the partial sums, we have

\begin{align*} \frac{1-F_n(x)}{1-F_n^m(x)}&=\exp\Big\{\frac{x^3}{n^{d/2}}\Big[\lambda_n \Big(\frac{x}{n^{d/2}}\Big)-\lambda_n^m\Big(\frac{x}{n^{d/2}}\Big)\Big]\Big\} \Big(1+O\Big(\frac{x+1}{n^{d/2}}\Big)\Big) \\ &=\exp\Big\{\frac{x^3}{n^{d/2}}\Big[\beta_{0n}-\beta_{0n}^m+\sum_{k=1}^\infty (\beta_{kn}-\beta_{kn}^m)\Big(\frac{x}{n^{d/2}}\Big)^k\Big]\Big\} \Big(1+O\Big(\frac{x+1}{n^{d/2}}\Big)\Big), \end{align*}

here, as in the proof of Theorem 3.1, we take M_n = max_j∈ℤ^d|b_nj|, H_n = H/2M_n, $M_n^m=\max_{j\in\mathbb{Z}^d}|b_{nj}^m|$, $H_n={H}/{2M_n^m}$,

\begin{align*} \beta_{0n}&=\frac{H_n}{6B_n}\sum_{j\in\mathbb{Z}^d}\gamma_{\,3nj} =\frac{H\gamma_3}{12M_nB_n}\sum_{j\in\mathbb{Z}^d}(b_{nj})^3, \\ \beta_{0n}^m&=\frac{H_n^m}{6B_n^m}\sum_{j\in\mathbb{Z}^d}\gamma_{\,3nj}^m =\frac{H\gamma_3}{12M_n^mB_n^m}\sum_{j\in\mathbb{Z}^d}(b_{nj}^m)^3. \end{align*}

If γ ₃ ≠ 0, ${(1-F_n(x))}/{(1-F_n^m(x))}$ is dominated by exp$\{({x^3}/{n^{d/2}})(\beta_{0n}-\beta_{0n}^m)\}$. If γ ₃ = 0, then $\beta_{0n}=\beta_{0n}^m=0$ and ${(1-F_n(x))}/{(1-F_n^m(x))}$ can be dominated by exp$\{({x^4}/{n^{d}})(\beta_{1n}-\beta_{1n}^m)\}$ which depends on whether γ ₄ = 0. In general, Theorem 3.1 can be applied to evaluate whether the truncated version $X_j^m$ is a good approximation to X_j in terms of the ratio ${(1-F_n(x))}/{(1-F_n^m(x))}$ for x in different ranges which depends on the property of the innovation ε and the sequence {a_i i ∈ ℤ^d }.

Theorem 3.1 can be applied to calculate the tail probability of the partial sum of some well-known dependent models. For example, the autoregressive fractionally integrated moving average FARIMA(p, β, q) processes in the one-dimensional case introduced by [Reference Granger and Joyeux16, Reference Hosking19], which is defined as

$$\phi(B)X_n=\theta(B)(1-B)^{-\beta}\varepsilon_{n}.$$

Here p, q are nonnegative integers, ϕ(z) = 1 − ϕ ₁ z − · · · − ϕ _pz ^p is the AR polynomial, and θ(z) = 1 + θ ₁ z + · · · + θ_qz ^q is the MA polynomial. Under the conditions that ϕ(z) and θ(z) have no common zeros, the zeros of ϕ(·) lie outside the closed unit disk and $-\tfrac12\lt\beta\lt\tfrac12$, the FARIMA(p, β, q) process has linear process form ${X_n} = \sum\nolimits_{i = 0}^\infty {{a_i}{\varepsilon _{n - 1}}} $, $\;n\in\mathbb{N}$ with a_i = θ(1)i^{β −1}/ϕ(1) Γ(β) + O(i ⁻¹). Here (·) is the gamma function.

3.2. Approximation of risk measures

Theorem 3.1 can be applied to approximate the risk measures such as quantiles and tail conditional expectations for the partial sums S_n in (3.1) of the linear random field X = {X_j, j ∈ ℤ^d }. Given the tail probability α ∈ (0, 1), let Q_α,n be the upper αth quantile of S_n. Namely, ℙ(S_n ≥ Q_α,n) = α. By Theorem 3.1, for all x ≥ 0 with x = o(n^{d /2}),

\begin{align*}{\rm {\mathbb P}}(S_n \gt x\sqrt{B_n})=\exp\Big\{\frac{x^3}{n^{d/2}}\lambda_n\Big(\frac{x}{n^{d/2}}\Big)\Big\}(1-\Phi(x))(1+o(1)).\end{align*}

We approximate Q_α,n by $x_{\alpha}\sqrt{B_n}$ where x = x_α = o(n^{d /2}) can be solved numerically from

\begin{align*} \exp\Big\{\frac{x^3}{n^{d/2}}\lambda_n\Big(\frac{x}{n^{d/2}}\Big) \Big\}(1-\Phi(x))=\alpha. \end{align*}

The tail conditional expectation is computed as

\begin{align}{\rm {\mathbb E}}(S_{n}\mid S_{n} \geq Q_{\alpha,n}) &= \frac{Q_{\alpha,n}{\rm {\mathbb P}} (S_{n}\geq Q_{\alpha,n})+{\int_{Q_{\alpha,n}}^{\infty}{\rm {\mathbb P}}(S_{n}\geq w)\text d w}}{{{\rm {\mathbb P}}(S_{n}\geq Q_{\alpha,n})}} \\ &= Q_{\alpha,n}+\frac{\sqrt{B_n}}{\alpha} {\int_{Q_{\alpha,n}/\sqrt{B_n}}^{\infty}\exp\Big\{\frac{y^3}{n^{d/2}} \lambda_n\Big(\frac{y}{n^{d/2}}\Big)\Big\}(1-\Phi(y))\text d y}, \end{align}

which can be solved numerically. The quantile and tail conditional expectations, which are also called value at risk (VaR) or expected shortfall (ES) in finance and risk theory, are important measures to model the extremal behavior of random variables in practice. The precise moderate deviation results in this paper provide a vehicle in the computation of these two measures of time series or spatial random fields. See [Reference Peligrad, Sang, Zhong and Wu31] for a brief review of VaR and ES in the literature, and a study of them when a linear process has pth moment p > 2 or has a regularly varying tail with exponent t > 2.

3.3. Nonparametric regression

Consider the regression model

\begin{align*} Y_{n,j}=g(z_{n,j})+X_{n,j}, \quad\quad j\in \Gamma_n^d, \end{align*}

where g is a bounded continuous function on ℝ^m, z_n,j are fixed design points over $\Gamma_n^d \subseteq \mathbb{Z}^d$ with values in a compact subset of ℝ^m, and X_n,j = Σ_{i∈ ℤ}^d a_iε_{n,j −i} is a linear random field over ℤ^d,where the i.i.d. innovations ε_n,i satisfy the same conditions as in Subsection 3.1. The kernel regression estimation for the function g on the basis of sample pairs (z_n,j, Y_n,j), $j\in \Gamma_n^2 \subset \mathbb{Z}^2$ has been studied by [Reference Sang and Xiao41] under the condition that the i.i.d. innovations ε_n,i satisfy ‖ε_n,i‖ _p < ∞ for some p > 2, and (or) the innovations have regularly varying right tail with index t > 2. See [Reference Sang and Xiao41] for more references in the literature for regression models with independent or weakly dependent random field errors.

We study the kernel regression estimation for the function g on the basis of sample pairs (z_n,j, Y_n,j), $j\in \Gamma_n^d$, when the i.i.d. innovations ε_n,i satisfy the conditions as in Subsection 3.1. As in [Reference Sang and Xiao41] and the other references in the literature, the estimator that we consider is given by

$$g_n(z)=\sum_{j\in\Gamma_n^d} w_{n,j}(z)Y_{n, j},$$

where the weight functions w_n,j(·) on ℝ^m have the form

\begin{align*} w_{n,j}(z)=\frac{K({(z-z_{n,j})}/{h_n})} {\sum_{i\in\Gamma_n^d} K({(z-z_{n,i})}/{h_n})}. \end{align*}

Here K : ℝ^m → ℝ⁺ is a kernel function and h_n is a sequence of bandwidths which goes to 0 as n → ∞. Note that the weight functions satisfy the condition $\sum_{j\in\Gamma_n^d} w_{n,j}(z) = 1$.

For a fixed z ∈ ℝ^m, let

$$S_n(z) = g_n(z)-\mathbb{E}g_n(z) = \sum_{j\in\Gamma_n^d} w_{n,j}(z)X_{n, j} =\sum_{j\in\mathbb{Z}^d} b_{n,j}(z)\varepsilon_{n,j},$$

where $b_{n,j}(z)=\sum_{i\in\Gamma_n^d} w_{n,i}(z) a_{i-j}.$ Let $B_n(z)=\sigma^2\sum_{j\in \mathbb{Z}^d}b_{n,j}^2(z)$ and $M_n(z)=\max_{j\in\mathbb{Z}^d}| b_{nj}(z)|$. By the same analysis as in the proof of Theorem 3.1, we take H_n ∝ M_n(z)⁻¹ and derive a moderate deviation result for $S_n(z)=g_n(z)-\mathbb{E}g_n(z)$. That is, if $B_n (z)H_n^2\to \infty$ as n → ∞, x ≥ 0, $x=o(H_n\sqrt{B_n(z)})$, then

(3.5)$$\matrix{ {{\rm {\mathbb P}}({S_n}(z) > x\sqrt {{B_n}(z)} )} \hfill \cr {\quad \;\quad = \left( {1 - \Phi (x)} \right)\exp \left\{ {{{{x^3}} \over {{H_n}\sqrt {{B_n}(z)} {\kern 1pt} }}{\lambda _n}\left( {{x \over {{H_n}\sqrt {{B_n}(z)} {\kern 1pt} }}} \right)} \right\}\left( {1 + O\left( {{{x + 1} \over {{H_n}\sqrt {{B_n}(z)} {\kern 1pt} }}} \right)} \right).} \hfill \cr } $$

A similar bound can be derived for ${\rm {\mathbb P}}(|S_n(z)|>x\sqrt{B_n(z)} )$. Note that these tail probability estimates are more precise than those obtained in [Reference Sang and Xiao41], where an upper bound for the law of the iterated logarithm of $g_n(z)-\mathbb{E}g_n(z)$ was derived. With the more precise bound on the tail probability in (3.5) and certain assumptions on g and the fixed design points {z_{n ,j}} (cf. [Reference Gu and Tran17]), we can construct a confidence interval for g(z).

More interestingly, our method provides a way for constructing confidence bands for the function g(z) when z ∈ T, where T ∈ ℝ^m is a compact interval. Observe that, for any z, z′ ∈ T, we can write

$$S_n(z) - S_n(z') = \sum_{j\in\mathbb{Z}^d} \big(b_{n,j}(z) - b_{n,j}(z')\big) \varepsilon_{n,j}.$$

Under certain regularity assumptions on g and the fixed design points {z_{n ,j}} (cf. [Reference Gu and Tran17]), we can apply the argument in Subsection 3.1 to derive an exponential upper bound for the tail probability ${\rm {\mathbb P}} (|S_n(z) - S_n(z')|>x\sqrt{B_n(z, z')})$, where $B_n(z, z') = \sigma^2\sum_{j\in \mathbb{Z}^d}(b_{n,j}(z) - b_{n,j}(z'))^2$. Such a sharp upper bound, combined with the chaining argument (cf. [Reference Talagrand46]) would allow us to derive an exponential upper bound for

\begin{align*}{\rm {\mathbb P}}\Big( \sup_{z, z' \in T} \frac{|S_n(z) - S_n(z')|}{\sqrt{B_n(z, z')}}\gt x\Big),\end{align*}

which can be applied to derive the uniform convergence rate of g_n(z) → g(z) for all z ∈ T and to construct a confidence band for the function g(z), z ∈ T. It is nontrivial to carry out this project rigorously and the verification of the details is a little lengthy. Hence, we will have to consider it elsewhere.

4.1. Proof of Theorem 2.1

Since γ _1nj = 0, the cumulant generating function L_nj(z) of X_nj can be written as

\begin{align*} L_{nj}(z)=\log {\rm {\mathbb E}} \text e^{z\:X_{nj}}=\sum_{k=2}^\infty \frac{\gamma_{knj}}{k!}z^{\,k}. \end{align*}

Cauchy’s inequality for the derivatives of analytic functions together with the condition (2.2) yields

(4.1)\begin{align}\label{cumulant} |\gamma_{knj}|\lt \frac{k!c_{nj}}{H_n^k}. \end{align}

By following the conjugate method (cf. [Reference Petrov34, Reference Petrov35]), we now introduce an auxiliary sequence of independent random variables {X̅_nj}, j ∈ ℤ^d, with the distribution functions

\begin{align*} \overline{V}_{nj}(x)=\text e^{-L_{nj}(z)}\int_{-\infty}^{x}\text e^{zy}\text d V_{nj}(y), \end{align*}

where V_nj(y) = ℙ(X_nj < y) and z ∈ (− H_n, H_n) is a real number whose value will be specified later.

Denote

\begin{align*} \overline{m}_{nj}= {\rm {\mathbb E}}\overline{X}_{nj}, \quad\quad \overline{\sigma}_{nj}^2= {\rm {\mathbb E}}(\overline{X}_{nj}-\overline{m}_{nj})^2, \quad\quad \overline{S}_{m,n}=\sum_{j\in\Gamma^d_m}\overline{X}_{nj}, \\ \overline{S}_n=\sum_{j\in\mathbb{Z}^d}\overline{X}_{nj}, \quad\quad \overline{M}_{n}=\sum_{j\in\mathbb{Z}^d}\overline{m}_{nj}, \quad\quad \overline{B}_{n}=\sum_{j\in\mathbb{Z}^d}\overline{\sigma}_{nj}^2 \end{align*}

and

\begin{align*} \overline{F}_n(x)={\rm {\mathbb P}}\Big(\overline{S}_n\lt \overline{M}_{n}+ x{2pt}{$\sqrt{\overline{B}_{n}}$}\Big). \end{align*}

Note that, in the above and below, we have suppressed z for simplicity of notation.

We shall see in the later analysis that the quantities S̅_n, M̅_n, and B̅_n are well defined for every n and z ∈ ℝ with |z| < aH_n, where a < 1 is a positive constant which is independent of n. Throughout the proof we will obtain some estimates holding for the values of z satisfying |z| < bH_n, where the positive constant b < 1 may vary but is always independent of n. We will then take a to be the smallest one among those constants b. The selection of the constants does not affect the proof since the z = z_n we need in the later analysis has property z = o(H_n).

Also, the change of the order of summation of double series presented in the proof is justified by the absolute convergence of those series in the specified regions.

Step 1: Representation of ℙ(S_n < x) in terms of the conjugate measure. First note that by Equation (2.11) of [Reference Petrov35, p. 221], for any m ∈ ℕ, we have

(4.2)\begin{align}\label{partial} {\rm {\mathbb P}}(S_{m,n}\lt x)=\exp\Big\{\sum_{j\in\Gamma^d_m}L_{nj}(z)\Big\} \int_{-\infty}^{x}\text e^{-zy}\text d{\rm {\mathbb P}}(\overline{S}_{m,n}\lt y). \end{align}

Note that the condition (2.3) implies that C_n < ∞, n ∈ ℕ. From (4.1), it follows that, for any w with $|w|\lt \tfrac{2}{3}H_n$ and for any m ∈ ℕ, we have

(4.3)\begin{align} \Big|\sum_{j\in\Gamma^d_m}L_{nj}(w)\Big| & = \Big|\sum_{j\in\Gamma^d_m}\sum_{k=2}^\infty \frac{\gamma_{knj}}{k!}w^k\Big| \\ &\leq \sum_{j\in\Gamma^d_m}\sum_{k=2}^\infty \frac{|\gamma_{knj}|}{k!}|w|^k\\ &\leq \sum_{j\in\mathbb{Z}^d}\sum_{k=2}^\infty \frac{c_{nj}}{H_n^k}|w|^k \\ &\leq \frac{4}{3}\sum_{j\in\mathbb{Z}^d}c_{nj} \\ &=\frac{4}{3}C_n \\ &\lt \infty.\label{cgf partial} \end{align}

Therefore, for any v with $|v|\lt \tfrac{1}{2}H_n$ and z with $|z|\lt \tfrac{1}{6}H_n$,

(4.4)\begin{align} {\rm {\mathbb E}} \exp\{v\overline{S}_{m,n}\} &= \prod_{j\in\Gamma^d_m} {\rm {\mathbb E}}\exp\{v \overline{X}_{nj}\} \\ &=\prod_{j\in\Gamma^d_m}\int_{-\infty}^{\infty}\text e^{vx}\text d \overline{V}_{nj}(x) \\ &= \prod_{j\in\Gamma^d_m}\int_{-\infty}^{\infty}\text e^{vx}\text e^{-L_{nj}(z)} \text e^{zx}\text d V_{nj}(x) \\ &=\prod_{j\in\Gamma^d_m}\text e^{-L_{nj}(z)}\int_{-\infty}^{\infty} \text e^{(v+z)x}\text d V_{nj}(x) \\ &= \prod_{j\in\Gamma^d_m}\text e^{-L_{nj}(z)}\text e^{L_{nj}(v+z)} \\ &\rightarrow\exp\Big(\sum_{j\in\mathbb{Z}^d} [L_{nj}(v+z)-L_{nj}(z)]\Big) \\ &\lt \infty \quad\text{as } m\rightarrow \infty.\label{mgf partial} \end{align}

Hence, S̅_n is well defined and S̅_m,n converges to S̅_n in distribution or, equivalently, in probability or almost surely as m→ ∞.

For the x in ℙ(S_n < x), let f (y) = exp{−zy}1{y < x} and M > 0. By Markov’s inequality, we have

\begin{align*} &{\rm {\mathbb E}} \{f(\overline{S}_{m,n})\textbf{1}\{ |f(\overline{S}_{m,n})|>M\}\} \\ &\quad\quad\leq {\rm {\mathbb E}} \{\exp\{\!-z\overline{S}_{m,n}\} \textbf{1}\{ \exp\{\!-z\overline{S}_{m,n}\}>M\}\} \\ &\quad\quad\leq\Big[{\rm {\mathbb E}} \Big\{\!\exp\{\!-2z\overline{S}_{m,n}\}\Big\} \Big]^{{1}/{2}}\Big[{\rm {\mathbb E}} \Big\{ \textbf{1}\{ \exp\{\!-z\overline{S}_{m,n}\}>M\}\Big\} \Big]^{{1}/{2}} \\ &\quad\quad\leq \Big[ \prod_{j\in\Gamma^d_m}\text e^{-L_{nj}(z)}\text e^{L_{nj}(\!-z)} \Big]^{{1}/{2}}\Big[\frac{1}{M}{\rm {\mathbb E}} \Big\{\!\exp\{\!-z\overline{S}_{m,n}\}\Big\}\Big]^{{1}/{2}} \\ &\quad\quad=\frac{1}{\sqrt{M}\,}\Big[ \prod_{j\in\Gamma^d_m}\text e^{-L_{nj}(z)}\text e^{L_{nj}(\!-z)}\Big]^{{1}/{2}} \Big[ \prod_{j\in\Gamma^d_m}\text e^{-L_{nj}(z)} \text e^{L_{nj}(0)}\Big]^{{1}/{2}}. \end{align*}

Hence, by (4.3) we have, for $|z|\lt \tfrac{1}{6}H_n$,

\begin{align*} \lim_{M\to\infty}\limsup_{m\to\infty}{\rm {\mathbb E}} \{f(\overline{S}_{m,n}) \textbf{1}\{ |f(\overline{S}_{m,n})|>M\}\}=0. \end{align*}

Applying Theorem 2.20 of [Reference Van Der Vaart47], we have

$$\int_{-\infty}^{x}\text e^{-zy}\text d {\rm {\mathbb P}}(\overline{S}_{m,n}\lt y)\rightarrow \int_{-\infty}^{x}\text e^{-zy}\text d {\rm {\mathbb P}}(\overline{S}_n\lt y) \quad {\text {as}}\space m\rightarrow \infty.$$

And taking into account that

\begin{align*} {\rm {\mathbb P}}(S_{m,n}\lt x)\rightarrow {\rm {\mathbb P}}(S_n\lt x) \end{align*}

and

\begin{align*} \exp\Big\{\sum_{j\in\Gamma^d_m}L_{nj}(z)\Big\}\rightarrow \exp\Big\{\sum_{j\in\mathbb{Z}^d}L_{nj}(z)\Big\} \quad\text{as $m\rightarrow \infty,$ } \end{align*}

we obtain, from (4.2),

(4.5)\begin{align} \label{cdf1} {\rm {\mathbb P}}(S_n\lt x)=\exp\Big\{\sum_{j\in\mathbb{Z}^d}L_{nj}(z)\Big\} \int_{-\infty}^{x}\text e^{-zy}\text d{\rm {\mathbb P}}(\overline{S}_n\lt y). \end{align}

Step 2: Properties of the conjugate measure. From the calculation of (4.4), it follows that the cumulant generating function L̅_nj(v) of the random variable X̅_nj exists when |v| is sufficiently small and we have

(4.6)\begin{align} \label{Lbar}\overline{L}_{nj}(v)=-L_{nj}(z)+L_{nj}(v+z),\end{align}

j ∈ ℤ^d. Denoting by $\overline{\gamma}_{knj}$ the cumulant of order k of the random variable X̅_nj, we obtain

\begin{align*} \overline{\gamma}_{knj}=\Big[\frac{\text d^k\overline{L}_{nj}(v)} {\text d v^k}\Big]_{v=0}=\frac{\text d^kL_{nj}(z)}{\text d z^{\,k}}. \end{align*}

Setting k = 1 and k = 2, we find that

(4.7)\begin{align} \label{Eq:mbar} \overline{m}_{nj}=\frac{\text d L_{nj}(z)}{\text d z}= \sum_{\ell =2}^\infty \frac{\gamma_{\ell nj}}{(\ell -1)!}z^{\ell-1}, \end{align}

and

(4.8)\begin{align} \label{Eq:sbar} \overline{\sigma}_{nj}^2=\frac{\text d^2L_{nj}(z)}{\text d z^2}=\sum_{\ell=2}^\infty \frac{\gamma_{\ell nj}}{(\ell-2)!}z^{\ell-2}. \end{align}

Hence, for $|z|\lt \tfrac{1}{2}H_n$, (4.7) implies that

(4.9)\begin{align} \label{M_nn} |\overline{M}_{n}|&=\Big|\sum_{j\in\mathbb{Z}^d}\overline{m}_{nj} \Big | \\ &= \Big|\sum_{j\in\mathbb{Z}^d}\sum_{k=2}^\infty \frac{\gamma_{knj}}{(k-1)!}z^{k-1}\Big| \\ &\leq\sum_{j\in\mathbb{Z}^d}\sum_{k=2}^\infty \frac{k!\,c_{nj}}{H_n^k}\frac{|z|^{k-1}}{(k-1)!} \\ &\leq\frac{3}{H_n}\sum_{j\in\mathbb{Z}^d}c_{nj} \\ &=\frac{3C_n}{H_n}, \end{align}

which means that M̅_n is well defined and, as a function of z ∈ ℂ, is analytic in $|z|\lt \tfrac{1}{2}H_n$.

Also, without loss of generality, we assume that

(4.10)\begin{align} \label{1} \limsup_n\frac{C_n}{B_n H_n^2}\leq 1. \end{align}

By the definition of M̅_n and (4.7), we have

(4.11)\begin{align}\label{sum for M_n} \overline{M}_{n} =z\sum_{j\in\mathbb{Z}^d} \gamma_{2nj}+\sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty \frac{\gamma_{knj}}{(k-1)!}z^{k-1} =zB_n+\sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty \frac{\gamma_{knj}}{(k-1)!}z^{\,k-1}. \end{align}

It follows from (4.1) that

\begin{align*} \Big|\sum_{k=3}^\infty \frac{\gamma_{knj}}{(k-1)!}z^{\,k-1}\Big| &\leq |z|\sum_{k=3}^\infty \frac{k!\,c_{nj}}{H_n^k}\frac{|z|^{k-2}}{(k-1)!} =\frac{|z|c_{nj}}{H_n^2}\sum_{k=3}^\infty k\Big| \frac{z}{H_n}\Big|^{k-2} \leq \frac{|z|c_{nj}}{2H_n^2} \end{align*}

for |z| < b ₁H_n and a suitable positive constant b ₁ < 1 which is independent of j and n. This together with (4.11) implies that, for |z| < b ₁H_n,

\begin{align*} |z|\Big(B_n-\frac{C_n}{2H_n^2}\Big) \leq|\overline{M}_{n}|\leq |z|\Big(B_n+\frac{C_n}{2H_n^2}\Big). \end{align*}

Taking into account condition (4.10), we obtain

(4.12)\begin{align}\label{speed of M_n} \overline{M}_{n}\propto |z|B_n. \end{align}

Moreover, (4.11) implies that, for $|z|\lt \frac{1}{2}H_n$,

(4.13)\begin{align}\label{M_n-zB_n} \big|\overline{M}_{n}-zB_n \big| \leq\sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty \frac{k!\,c_{nj}}{H_n^k}\frac{|z|^{k-1}}{(k-1)!}\leq\frac{|z|^2}{H_n^3}\sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty kc_{nj}\frac{|z|^{k-3}}{H_n^{k-3}}\leq \frac{8|z|^2C_n}{H_n^3}. \end{align}

Also, by the definition of B̅_n and (4.8), we have

(4.14)\begin{align}\label{sum for B_n}\overline{B}_{n} =\sum_{j\in{\rm \mathbb{Z}}^d}\gamma_{2nj}+\sum_{j\in{\rm \mathbb{Z}}^d}\sum_{k=3}^\infty\frac{\gamma_{knj}}{(k-2)!}z^{\,k-2}=B_n+\sum_{j\in{\rm \mathbb{Z}}^d}\sum_{k=3}^\infty\frac{\gamma_{knj}}{(k-2)!}z^{\,k-2}.\end{align}

It follows from (4.1) that

\begin{align*} \Big|\sum_{k=3}^\infty \frac{\gamma_{knj}}{(k-2)!}z^{\,k-2}\Big| \leq\sum_{k=3}^\infty \frac{k!\, c_{nj}}{H_n^k}\frac{|z|^{k-2}}{(k-2)!}\leq\frac{c_{nj}}{2H_n^2} \end{align*}

for |z| < b ₂H_n and a suitable positive constant b ₂ < 1 which is independent of j and n. This together with (4.14) implies that, for |z| < b ₂ H_n, B̅_n is well defined and

\begin{align*} B_n-\frac{C_n}{2H_n^2} \leq|\overline{B}_{n}|\leq B_n+\frac{C_n}{2H_n^2}. \end{align*}

Condition (4.10) then implies that

(4.15)\begin{align} \label{speed of B_n} \overline{B}_n\propto B_n. \end{align}

Furthermore, (4.14) and (4.1) imply that, for $|z|\lt \frac{1}{2}H_n$,

(4.16)\begin{align} \label{B_n tail} | \overline{B}_{n}-B_n| \leq\sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty \frac{k!\,c_{nj}}{H_n^k}\frac{|z|^{k-2}}{(k-2)!} \leq\frac{|z|}{H_n^3}\sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty k(k-1)c_{nj}\frac{|z|^{k-3}}{H_n^{k-3}}\leq \frac{28|z|C_n}{H_n^3}. \end{align}

Step 3: Selection of z. Let z = z_n be the real solution of

(4.17)\begin{align} \label{x} x=\frac{\overline{M}_{n}}{\sqrt{B_n}\,}, \end{align}

and let

(4.18)\begin{align} \label{t} t=t_n=\frac{x}{H_n\sqrt{B_n}\,}. \end{align}

Then

(4.19)\begin{align}\label{t1} t=\frac{\overline{M}_{n}}{H_nB_n} =\frac{1}{H_nB_n}\sum_{j\in\mathbb{Z}^d}\sum_{k=2}^\infty \frac{\gamma_{knj}}{(k-1)!}z^{\,k-1}. \end{align}

By (4.9), we know that M̅_n/H_nB_n is analytic in a disc $|z|\lt \frac{1}{2}H_n$ and

\begin{align*} \Big|\frac{\overline{M}_{n}}{H_nB_n}\Big|\leq\frac{3C_n}{H_n^2B_n} \end{align*}

in that disc. It follows from Bloch’s theorem (see, e.g. [Reference Privalov39, p. 256]) that (4.19) has a real solution which can be written as

(4.20)\begin{align} \label{inverse} z=\sum\limits_{m=1}^\infty a_{mn}t^m \end{align}

for

\begin{align*} |t|\lt \Biggl(\sqrt{\frac{1}{2}+\frac{3C_n}{H_n^2B_n}}- \sqrt{\frac{3C_n}{H_n^2B_n}} \Biggr)^2. \end{align*}

Moreover, the absolute value of that sum in (4.20) is less than $\frac{1}{2}H_n$. Condition (2.3) implies that there exists a disc with center at t = 0 and radius R that does not depend on n within which the series on the right-hand side of (4.20) converges.

It can be checked from (4.19) and (4.20) that

(4.21)\begin{align}\label{a1a2} a_{1n}=H_n \quad\text{and}\quad a_{2n}=-\frac{H_n^2}{2B_n}\sum_{j\in\mathbb{Z}^d}\gamma_{\,3nj}. \end{align}

Cauchy’s inequality implies that, for every m ∈ ℕ,

\begin{align*} |a_{mn}|\leq\frac{H_n}{2R^m}. \end{align*}

Therefore, as t → 0, a _1nt becomes the dominant term of the series in (4.20). Hence, for sufficiently large n, we have

\begin{align*} \frac{1}{2}tH_n\leq z\leq 2tH_n,\quad\quad z=o(H_n), \end{align*}

and taking into account (4.18) we obtain

(4.22)\begin{align} \label{z} \frac{x}{2\sqrt{B_n}\,} \leq z\leq \frac{2x}{\sqrt{B_n}\,}. \end{align}

It follows from (4.3) and (4.9) that for $z\lt \frac{1}{2}H_n$,

\begin{align*} \Big|z\overline{M}_{n}-\sum_{j\in\mathbb{Z}^d}L_{nj}(z)\Big|\leq \frac{3|z|}{H_n}C_n+\frac{4}{3}C_n\lt 3C_n. \end{align*}

For the solution z of (4.17), we also have

(4.23)\begin{align} z\overline{M}_{n}-\sum_{j\in\mathbb{Z}^d}L_{nj}(z)&=\sum_{j\in \mathbb{Z}^d}\sum_{k=2}^\infty \frac{\gamma_{knj}} {(k-1)!}z^{\,k}-\sum_{j\in\mathbb{Z}^d}\sum_{k=2}^\infty \frac{\gamma_{knj}}{k!}z^{\,k} \\ &=\sum_{j\in\mathbb{Z}^d} \sum_{k=2}^\infty \frac{(k-1)\gamma_{knj}}{k!}\Bigg(\sum\limits_{m=1}^\infty a_{mn}t^m \Bigg)^k \\ &:= \sum_{j\in\mathbb{Z}^d}\frac{\gamma_{2nj}}{2}a_{1n}^2t^2- \sum_{k=3}^\infty b_{kn}t^k \\ & =\frac{H_n^2B_nt^2} {2}-H_n^2B_nt^3\sum_{k=3}^\infty\frac{ b_{kn}}{H_n^2B_n}t^{k-3} \\ &=\frac{H_n^2B_nt^2}{2}-H_n^2B_nt^3\lambda_n(t),\label{lambda} \end{align}

where $\lambda_n(t) =\sum_{k=0}^\infty \beta_{kn}t^k$ with $ \beta_{kn} = b_{(k+3)n}(H_n^2B_n)^{-1}$.

Recall that the series $\sum_{m=1}^\infty a_{mn} t^m$ converges in the disc centered at t = 0 with radius R > 0 that does not depend on n, and the absolute value of this sum is less than $\frac{1}{2}H_n$. We see from (4.23) that the function λ_n(t) is obtained by the substitution of $\sum_{m=1}^\infty a_{mn}t^m$ in a series that converges on the interval $(-\frac{1}{2}H_n,\frac{1}{2}H_n)$. It follows from Cauchy’s inequality that

\begin{align*} \big|\beta_{kn} \big| \leq\frac{3C_n}{H_n^2B_nR^{k+3}}\leq \frac{3}{R^{k+3}}, \quad\quad k\geq 0, \end{align*}

which means that, for $|t|\lt \frac{1}{2}R, \lambda_n(t)$ stays bounded uniformly in n. In particular, by (4.21) and (4.23), we have β _0n = (H_n/6B_n) Σ_j∈ℤ^d γ 3nj.

From now on we will assume that z is the unique real solution of (4.17).

Step 4: The 0 ≤ x ≤ 1 case. Now we prove the theorem for the 0 ≤ x ≤ 1 case using the method presented in [Reference Petrov and Robinson37]. Throughout the proof, C denotes a positive constant which may vary from line to line, but is independent of j, n, and z. If f_n(s) is the characteristic function of $S_n/\sqrt{B_{n}}$ then, for $|s|\lt H_n \sqrt{B_{n}}/2$,

\begin{align*} f_n(s) =\int\limits_{-\infty}^\infty \text e^{isu}\text d{\rm {\mathbb P}}(S_n\leq u\sqrt{B_{n}}) =\int_{-\infty}^\infty \text e^{isy/\sqrt{B_{n}}}\text d {\rm {\mathbb P}}(S_n\leq y) =\exp\Big\{\sum_{j\in\mathbb{Z}^d}L_{nj}\Big(\frac{is}{\sqrt{B_{n}}\,}\Big)\Big\}. \end{align*}

Then

\begin{align*} \log f_n(s)&= \sum_{j\in\mathbb{Z}^d}L_{nj}\Big(\frac{is}{\sqrt{B_{n}}\,}\Big) \\ &=\sum_{j\in\mathbb{Z}^d}\sum_{k=2}^\infty \frac{\gamma_{knj}}{k!}\Big(\frac{is}{\sqrt{B_{n}}\,}\Big)^k \\ &=-\sum_{j\in\mathbb{Z}^d}\frac{\gamma_{2nj}}{2}\frac{s^2}{B_{n}}+ \sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty \frac{\gamma_{knj}}{k!}\Big(\frac{is}{\sqrt{B_{n}}\,}\Big)^k \\ &=-\frac{s^2}{2}+\sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty \frac{\gamma_{knj}}{k!}\Big(\frac{is}{\sqrt{B_{n}}\,}\Big)^k. \end{align*}

Thus, using (4.1), we obtain, for $|s|\lt \delta H_n \sqrt{B_{n}}/2$ with 0 < δ < 1,

\begin{align*}\Big|\log f_n(s)+\frac{s^2}{2}\Big|\leq\sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty c_{nj}\Big(\frac{|s|}{H_n\sqrt{B_{n}}\,}\Big)^k\leq C_n\Big(\frac{|s|}{H_n\sqrt{B_{n}}\,}\Big)^3(1-\delta)^{-1}\end{align*}

Then, for an appropriate choice of δ, we have

\begin{align*}|f_n(s)-\text e^{-s^2/2}|\lt C\frac{\text e^{-s^2/4}|s|^3C_n}{H_n^3\sqrt{B_{n}}^3} \lt C\frac{\text e^{-s^2/4}|s|^3}{H_n\sqrt{B_{n}}} \quad\text{for $|s|\lt \delta H_n \sqrt{B_{n}}/2$.}\end{align*}

Now applying Theorem 5.1 of [Reference Petrov36] with b = 1/π and $T=\delta H_n \sqrt{B_{n}}/2$, we obtain

(4.24)\begin{align} \label{P} \sup\limits_x|F_n(x)-\Phi(x)|\lt \frac{C}{H_n\sqrt{B_{n}}\,}. \end{align}

Since 0 ≤ x ≤ 1, $B_nH_n^2\rightarrow \infty$ as n → ∞, and $\lambda_n({x}/{H_n\sqrt{B_n}})$ is bounded uniformly in n, we have

\begin{align*} \exp\Big\{\frac{x^3}{H_n\sqrt{B_n}\,}\lambda_n\Big(\frac{x} {H_n\sqrt{B_n}\,}\Big)\Big\}=1+O(H_n^{-1}B_n^{-1/2}). \end{align*}

Together with condition (2.3), to have (2.4) in the 0 ≤ x ≤ 1 case, it is sufficient to show that 1

\begin{align*} \frac{1-F_n(x)}{1-\Phi(x)}=1+O\Big(\frac{C}{H_n\sqrt{B_{n}}\,}\Big), \end{align*}

which is given by (4.24), since $\tfrac12\le \Phi(x)\le \Phi(1)$ for $0\le x\le 1$.

So we will limit the proof of the theorem to the case x > 1, $x=o(H_n\sqrt{B_n})$.

Step 5: The case x > 1, $x=o(H_n\sqrt{B_n})$}. Making a change of variables $ y\rightsquigarrow \overline{M}_n+y\sqrt{\overline{B}_n}$ and applying (4.17), we can rewrite (4.5) as

(4.25)\begin{align}\label{cdf3} 1-F_n(x)&=\exp\Big\{\!-z\overline{M}_n+\sum_{j\in\mathbb{Z}^d}L_{nj}(z) \Big\}\int_{(x\sqrt{B_n}-\overline{M}_n)/\sqrt{\overline{B}_n}}^{\infty} \exp\{-zy\sqrt{\overline{B}_n}\}\text d\overline{F}_n(y)\notag \\ &=\exp\Big\{\!-z\overline{M}_n+\sum_{j\in\mathbb{Z}^d}L_{nj}(z)\Big\} \int_0^{\infty} \exp\Big\{-zy\sqrt{\overline{B}_n}\Big\} \text d\overline{F}_n(y). \end{align}

Denote r_n(x) = F̅_n(x) − Φ(x) and we show that, for sufficiently large n,

(4.26)\begin{align}\label{r_n} \sup\limits_x|r_n(x)|\leq \frac{C}{H_n\sqrt{B_{n}}\,}. \end{align}

Let f̅_n(s) be the characteristic function of $\[({\bar S_n} - {\bar M_n})/\sqrt {{{\bar B}_n}} \]$. We then have

\begin{align*} \overline{f}_n(s) &=\int_{-\infty}^\infty \text e^{isu}\text d {\rm {\mathbb P}}(\overline{S}_n\leq u\sqrt{\overline{B}_{n}}+\overline{M}_{n}) \\ &=\int_{-\infty}^\infty \text e^{is(y- \overline{M}_{n})/\sqrt{\overline{B}_{n}}} \text d{\rm {\mathbb P}}(\overline{S}_n\leq y) \\ &=\exp\Big\{\!-\frac{is\overline{M}_{n}}{\sqrt{\overline{B}_{n}}\,}- \sum_{j\in\mathbb{Z}^d}L_{nj}(z)\Big\}\int_{-\infty}^\infty \text e^{(z+is/\sqrt{\overline{B}_{n}})y}\text d{\rm {\mathbb P}}(S_n\leq y) \\ &=\exp\Big\{\!-\frac{is\overline{M}_{n}}{\sqrt{\overline{B}_{n}}\,}- \sum_{j\in\mathbb{Z}^d}L_{nj}(z)+\sum_{j\in\mathbb{Z}^d}L_{nj} \Big(z+\frac{is}{\sqrt{\overline{B}_{n}}\,}\Big)\Big\}. \end{align*}

Then, by (4.6), for $|z|\lt \frac{1}{2} H_n$ and $|s|\lt H_n \sqrt{\overline{B}_{n}}/6$, we have

\begin{align*} \log\overline{f}_n(s) &= -\frac{is\overline{M}_{n}} {\sqrt{\overline{B}_{n}}\,}+ \sum_{j\in\mathbb{Z}^d}\overline{L}_{nj}(\frac{is} {\sqrt{\overline{B}_{n}}\,}) \\ &= -\frac{1}{2}s^2+\frac{1}{6}\big(\frac{is} {\sqrt{\overline{B}_{n}}\,}\big)^3 \Big[\frac{\text d^3\sum_{j\in\mathbb{Z}^d} \overline{L}_{nj}(y)}{\text d y^3} \Big]_{y=\theta is/\sqrt{\overline{B}_{n}}}, \end{align*}

where 0 ≤ |θ| ≤ 1. For $|z|\lt \frac{1}{2} H_n$ and $|s|\lt \delta H_n \sqrt{\overline{B}_{n}}/6$ with 0 < δ < 1, we have

\begin{align*} \Big|\Big[\frac{\text d^3\sum_{j\in\mathbb{Z}^d}\overline{L}_{nj}(y)} {\text d y^3}\Big]_{y=\theta is/\sqrt{\overline{B}_{n}}}\Big|&=\Big|\Big[\frac{\text d^3}{\text d y^3} \sum_{j\in\mathbb{Z}^d}\sum_{k=1}^\infty \frac{\overline{\gamma}_{knj}}{k!}y^k\Big]_{y=\theta is/\sqrt{\overline{B}_{n}}}\Big| \\ &=\Big|\sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty \frac{\overline{\gamma}_{knj}}{(k-3)!}\Big(\frac{\theta is}{\sqrt{\overline{B}_{n}}\,}\Big)^{k-3}\Big| \\ &\leq \sum_{j\in\mathbb{Z}^d}\sum_{k=3}^\infty k(k-1)(k-2)\frac{c_{nj}}{(H_n/2)^k} \Bigg(\frac{s}{\sqrt{\overline{B}_{n}}\,}\Bigg)^{k-3} \\ &=\frac{48C_n}{H_n^3}\Big(1-\frac{s/\sqrt{\overline{B}_{n}}}{H_n/2} \Big)^{-4} \\ &\leq \frac{48C_n}{H_n^3}(1-\delta)^{-4}. \end{align*}

Thus,

\begin{align*} \Big|\log\overline{f}_n(s)+\frac{s^2}{2}\Big|\lt \frac{8|s|^3C_n}{H_n^3 \sqrt{\overline{B}_{n}}^3}(1-\delta)^{-4}. \end{align*}

Then, for an appropriate choice of δ, we have

\begin{align*} |\overline{f}_n(s)-\text e^{-s^2/2}|\lt C\frac{\text e^{-s^2/4}|s|^3C_n}{H_n^3 \sqrt{\overline{B}_{n}}^3}\lt C\frac{\text e^{-s^2/4}|s|^3}{H_n \sqrt{\overline{B}_{n}}} \quad\text{for $|s|\lt \delta H_n \sqrt{\overline{B}_{n}}/6$.} \end{align*}

Now applying (4.15) and Theorem 5.1 of [Reference Petrov36] with b = 1/π and $T=\delta H_n \sqrt{\overline{B}_{n}}/6$, we have (4.26).

By (4.26), we have

(4.27)\begin{align} \label{int} &\int_0^{\infty}\exp\{-zy\sqrt{\overline{B}_n}\} \text d\overline{F}_n(y) \\ &\quad\quad=\frac{1}{\sqrt{2\pi}\,} \int_0^{\infty}\exp \biggl\{\!-zy\sqrt{\overline{B}_n} -\frac{y^2}{2}\biggr\}\text d y-r_n(0)+z\sqrt{\overline{B}_n}\int_0^{\infty}r_n(y)\exp \{\!-zy\sqrt{\overline{B}_n}\}\text d y \\ &\quad\quad=\frac{1}{\sqrt{2\pi}\,}\int_0^{\infty}\exp \Big\{\!-zy\sqrt{\overline{B}_n}-\frac{y^2}{2}\Big\}\text d y+\alpha_n, \end{align}

where $|\alpha_n|\leq {C}/{H_n\sqrt{B_{n}}}$.

Denote

\begin{align*} I_1=\int_0^{\infty}\exp\Big\{\!-zy\sqrt{\overline{B}_n}-\frac{y^2}{2}\Big\} \text d y=\psi(z\sqrt{\overline{B}_n}) \end{align*}

and

\begin{align*} I_2=\int_0^{\infty}\exp\Big\{\!-\frac{\overline{M}_{n}}{\sqrt{B_n}\,}- \frac{y^2}{2}\Big\}\text d y=\psi(\overline{M}_{n}B^{-{1}/{2}}_n), \end{align*}

where

\begin{align*} \psi(x)=\frac{1-\Phi(x)}{\Phi '(x)}=\text e^{{x^2}/{2}}\int_x^{\infty} \text e^{-{t^2}/{2}}\text d t \end{align*}

is the Mills ratio which is known to satisfy

\begin{align*} \frac{x}{x^2+1}\lt \psi(x)\lt \frac{1}{x} \quad\text{for all $x>0$. } \end{align*}

Hence, by (4.22) and (4.15), we obtain

\begin{align*} \frac{\alpha_n}{xI_1} &=\frac{\alpha_nz\sqrt{\overline{B}_n}}{x}+ \frac{\alpha_n}{xz\sqrt{\overline{B}_n}\,} \\ &\leq C\Bigg(\frac{z\sqrt{B_n}}{xH_n\sqrt{B_n}\,}+\frac{1}{H_n\sqrt{B_n}xz \sqrt{B_n}\,}\Bigg) \\ &\leq C\Bigg(\frac{1}{H_n\sqrt{B_n}\,}+\frac{1}{H_n\sqrt{B_n}x^2}\Bigg) \\ &\leq \frac{C}{H_n\sqrt{B_n}\,}. \end{align*}

Hence,

(4.28)\begin{align} \label{alpha_n} \alpha_n=I_1O\Big(\frac{x}{H_n\sqrt{B_n}\,}\Big). \end{align}

For every y ₁ < y ₂, we have ψ(y ₂) − ψ(y ₁) = (y ₂ − y ₁)ψ′(u), where y ₁ < u < y ₂. As for u > 0, |ψ′(u)| < u ⁻², then using (2.3), (4.12), (4.13), (4.15), (4.16), and (4.22), we obtain

\begin{align*} |I_2-I_1| &=|\psi '(u)||\overline{M}_{n}B^{-{1}/{2}}_n-z \sqrt{\overline{B}_n}| \\ &\leq \frac{1}{u^2\sqrt{B_n}\,}|\overline{M}_{n}-z\sqrt{B_n} \sqrt{\overline{B}_n}| \\ &\leq\frac{1}{u^2\sqrt{B_n}\,}\Big(|\overline{M}_{n}-zB_n|+ |zB_n-z\sqrt{B_n}\sqrt{\overline{B}_n}|\Big) \\ &\leq \frac{C}{(x/{4})^2\sqrt{B_n}\,}\Big(\frac{z^2C_n}{H_n^3}+ z\sqrt{B}_n|\sqrt{B_n}-\sqrt{\overline{B}_n}|\Big) \\ &\leq \frac{C}{x^2\sqrt{B_n}\,}\Big(\frac{x^2C_n}{B_nH_n^3}+\frac{x|B_n -\overline{B}_n|}{\sqrt{B_n}+\sqrt{\overline{B}_n}\,}\Big) \\ &\leq \frac{C}{x^2\sqrt{B_n}\,}\Big(\frac{x^2C_n}{B_nH_n^3}+\frac{xzC_n} {H_n^3\sqrt{B_n}}\Big) \\ &\leq \frac{C}{x^2\sqrt{B_n}}\Big(\frac{x^2C_n}{B_nH_n^3}+\frac{x^2C_n} {H_n^3B_n}\Big) \\ &=\frac{CC_n}{B_n^{{3}/{2}}H_n^3} \\ &\leq \frac{C}{H_n\sqrt{B_n}\,}. \end{align*}

Hence,

\begin{align*} \frac{|I_2-I_1|}{xI_2}\leq \frac{C}{xH_n\sqrt{B_n}\psi(\overline{M}_{n} B^{-{1}/{2}}_n)}=\frac{C} {xH_n\sqrt{B_n}\psi(x)}\lt \frac{C}{xH_n\sqrt{B_n}\,} \frac{x^2+1}{x}\lt \frac{C}{H_n\sqrt{B_n}\,}, \end{align*}

which means that

(4.29)\begin{align} \label{I_2} I_1=I_2\Big(1+O\Big(\frac{x}{H_n\sqrt{B_n}\,}\Big)\Big). \end{align}

Finally, combining (4.17), (4.18), (4.23), (4.25), (4.27), and (4.28), we obtain

\begin{align*} 1-F_n(x) &=\exp\Big\{\!-\frac{H_n^2B_nt^2}{2}+H_n^2B_nt^3\lambda_n(t)\Big\} \int_{0}^{\infty}\exp\Big\{\!-zy\sqrt{\overline{B}_n}\Big\} \text d\overline{F}_n(y) \\ &=\exp\Big\{\!-\frac{x^2}{2}+\frac{x^3}{H_n\sqrt{B_n}\,}\lambda_n \Big(\frac{x}{H_n\sqrt{B_n}\,}\Big)\Big\}\Big(\frac{1}{\sqrt{2\pi}\,} I_1+\alpha_n\Big) \\ &=\exp\Big\{\!-\frac{x^2}{2}+\frac{x^3}{H_n\sqrt{B_n}\,}\lambda_n \Big(\frac{x}{H_n\sqrt{B_n}\,}\Big)\Big\}\frac{1}{\sqrt{2\pi}\,}I_1 \Big(1+O\Big(\frac{x}{H_n\sqrt{B_n}\,}\Big)\Big). \end{align*}

By (4.29) and the fact that I ₂ = ψ(x), we see that

\begin{align*} \frac{1-F_n(x)}{1-\Phi(x)}=\exp\Big\{\frac{x^3}{H_n\sqrt{B_n}\,} \lambda_n\Big(\frac{x}{H_n\sqrt{B_n}\,}\Big)\Big\}\Big(1+O \Big(\frac{x}{H_n\sqrt{B_n}\,}\Big)\Big). \end{align*}

This proves (2.4). The proof of (2.5) follows the same pattern and is therefore omitted.

4.2. Proof of Theorem 3.1

Since γ ₁ = 0, we see that the cumulant generating function L_nj(z) of the random variable b_njε_j, j ∈ ℤ^d, is given by

\begin{align*} L_{nj}(z)=\log {\rm {\mathbb E}} \text e^{zb_{nj}\varepsilon_j}=\sum_{k=2}^\infty \frac{\gamma_k b_{nj}^k}{k!}z^{\,k}. \end{align*}

Cauchy’s inequality for the derivatives of analytic functions together with the condition (3.3) yields

(4.30)\begin{align} \label{cumulant lm} |\gamma_k|\lt \frac{k!\,C}{H^k}. \end{align}

Denote $M_n=\max\limits_{j\in\mathbb{Z}^d}| b_{nj}|$. Then by (4.30), for any H_n with 0 < H_n ≤ H/2M_n and for any z with |z| < H_n, we have

\begin{align*} \big|L_{nj}(z)\big| \leq\sum_{k=2}^\infty \frac{|\gamma_k|| b_{nj}|^k}{k!}|z|^k \leq C\sum_{k=2}^\infty \frac{| b_{nj}H_n|^k}{H^k} = \frac{C}{H} \frac{b_{nj}^2H_n^2}{H-| b_{nj}H_n|} \leq \frac{2Cb_{nj}^2H_n^2}{H^2}. \end{align*}

Hence,

\begin{align*} C_n=\sum_{j\in\mathbb{Z}^d} \frac{2Cb_{nj}^2H_n^2}{H^2}=\frac{2CB_nH_n^2}{\sigma^2H^2}. \end{align*}

Then by Theorem 2.1, if $B_n H_n^2\to \infty$ as n → ∞, we have

(4.31)\begin{align}\label{result1} \frac{1-F_n(x)}{1-\Phi(x)}=\exp\Big\{\frac{x^3}{H_n\sqrt{B_n}\,} \lambda_n\Big(\frac{x}{H_n\sqrt{B_n}\,}\Big)\Big\} \Big(1+O\Big(\frac{x+1}{H_n\sqrt{B_n}\,}\Big)\Big) \end{align}

for x ≥ 0, $x=o(H_n\sqrt{B_n})$.

If the linear random field has long memory then we have (see [Reference Surgailis45, Theorem 2]) B_n ∝ n ^3d−2αl ²(n). As the function b(·) is bounded, then, for $j\in\Gamma^d_n$, we have

\begin{align*} |b_{nj}| \leq C_1\sum_{i\in\Gamma^d_n}l(|i-j|)|i-j|^{-\alpha}\leq C_1\sum\limits_{k=1}^{2dn}k^{d-1}l(k)k^{-\alpha} \propto n^{d-\alpha}l(n), \end{align*}

where we have used the fact (see [Reference Bingham, Goldie and Teugels6] or [Reference Seneta42]) that, for a slowly varying function l(x) defined on [1, ∞) and for any θ > −1,

\begin{align*} \int_{1}^{x}y^{\theta}l(y)dy\mathbb{\sim}\frac{x^{\theta+1}l(x)} {\theta+1} \quad\text{as } x\rightarrow\infty. \end{align*}

It follows from the definition of a_i in (3.2) that (for sufficiently large n) $M_n=\max\limits_{j\in\mathbb{Z}^d}| b_{nj}|$ is attained at some $j\in \Gamma^d_n$. Hence, M_n = O(n^{d −α}l(n)). We take H_n ∝ n ^−d+αl ⁻¹(n) which yields

\begin{align*} H_n\sqrt{B_n}\propto n^{d/2}. \end{align*}

Then the result follows from (4.31).

If the linear random field has short memory, i.e. A := Σ_i∈ℤ^d |a_i| < ∞, a := Σ_i∈ℤ^d a_i ≠ 0, we can take M_n = A and H_n = H/2A. Moreover, we also have

\begin{align*} \sum_{j\in\mathbb{Z}^d}|b_{nj}|\leq \sum_{j\in\mathbb{Z}^d} \sum_{i\in\Gamma^d_n}|a_{i-j}|=(2n+1)^d \sum_{i\in \mathbb{Z}^d}|a_i|=A(2n+1)^d \end{align*}

and

\begin{align*} \sum_{j\in\mathbb{Z}^d}|b_{nj}|\geq |\sum_{j\in\mathbb{Z}^d} \sum_{i\in\Gamma^d_n}a_{i-j}| =(2n+1)^d|\sum_{i\in \mathbb{Z}^d}a_i|=|a|(2n+1)^d, \end{align*}

which means that Σ_j∈ℤ^d |b_nj| ∝ n^d.

As for all n ∈ ℕ, we have |b_nj| ≤ A by the definition of A, then

\begin{align*} \sum_{j\in\mathbb{Z}^d}b_{nj}^2\leq A\sum_{j\in\mathbb{Z}^d}|b_{nj}| \leq A^2(2n+1)^d. \end{align*}

On the other hand, for $j\in\Gamma^d_{\left \lfloor{n/2}\right \rfloor}$, we have |b_nj| > |a|/2 for sufficiently large n. Hence,

\begin{align*} &\sum_{j\in\mathbb{Z}^d}b_{nj}^2\geq \sum_{j\in\Gamma^d_{\left \lfloor{n/2}\right \rfloor }}b_{nj}^2\geq \frac{a^2}{4} \Big( 2\left \lfloor{\frac{n}{2}}\right \rfloor+1\Big)^d. \end{align*}

Thus, $\sum_{j\in\mathbb{Z}^d}b_{nj}^2\propto n^d$ and the result follows from (4.31).