Large deviation probabilities for sums of lattice random vectors with heavy-tailed distribution

A. Yu. Zaigraev, A. V. Nagaev, A. Jakubowski^{∗}

Nicholas Copernicus University, Toruń

The large deviation problem for sums of i. i. d. random vectors with values from R^{d} is considered. It is assumed that the underlying distribution has the heavy tails. The special attention is paid to the role which is played by the density of the distribution support at infinity. The proven local and integral theorems generalize until now known results concerned with heavy-tailed distributions.

UDK 519.2

1. Introduction

Let ξ, ξ_{1}, ξ_{2}, … be i. i. d. random vectors having a lattice distribution. The latter means that ξ takes values from the set {x : x = Am + λ, m ∈ Z^{d}}, where the vector λ ∈ R^{d} and non-degenerate d × d-matrix A are fixed. Consider the partial sums S_{n} = ξ_{1} + ⋯ + ξ_{n}, n = 1, 2 …. With no loss in generality we assume that λ = 0 and A is the identity matrix.

Let D be the support of the underlying distribution, i. e.,

D = {k ∈ Z^{d} : p_{k} = P (ξ = k) > 0}.

We suppose that the distribution span equals to 1, i. e., D is not contained in any coset with respect to any proper subgroup of Z^{d}. Analitically this assumption is written as

{t ∈ R^{d} : |Ee^{2πi⟨t,ξ⟩}| = 1} = {t ∈ R^{d} : Ee^{2πi⟨t,ξ⟩} = 1} = Z^{d}.  (1)

∗Supported by Komitet Badań Naukowych under grant PB 591/P03/95/08

The statements concerned with the asymptotic properties of the probability P (S_{n} = k) as n → ∞ are called the local limit theorems. The classical local limit theorem is formulated as follows (see, e. g., Spitzer (1969, p.99)).

Proposition 1. If the condition (1) is fulfilled, Eξ = a and E|ξ|^{2} < ∞, then as n → ∞,

sup_{k∈Z^{d}} |n^{d/2}P (S_{n} = k) − ϕ_{B} (n^{−1/2}(k − na))| = o(1),

where B denotes the covariance matrix of ξ and ϕ_{B} is the density of the normal distribution in R^{d} with the zero expectations and the covariance matrix B.

From Proposition 1 it follows, in particular, that as |k − na| = O(n^{1/2})

P (S_{n} = k) ∼ n^{−d/2}ϕ_{B} (n^{−1/2}(k − na)) .  (2)

If n^{−1/2}|k − na| → ∞ then Proposition 1 states only that

P (S_{n} = k) = o(n^{−d/2}).  (3)

Values k, under which (3) is valid, are called large deviations for S_{n} in contrast to the normal deviations k for which (2) holds.

The asymptotics of the probability of large deviations for sums of lattice random variables is studied rather completely in the one-dimensional case (see Petrov (1961,1963), S. Nagaev (1962), A. Nagaev (1967,1968)). The case d > 1 is considerably less advanced. It is worth mentioning here the works written by Rihter (1958) and Anorina (1971). The closest results to the content of this paper are those obtained in A. Nagaev (1970) and Anorina (1971). In particular, the next statement is given in A. Nagaev (1970).

Proposition 2. Let d = 1. Suppose that

1) Eξ = 0, E|ξ|^{2} < ∞;

2) there exists k_{0} such that p_{k} > 0 for any k ≥ k_{0};

3) as k → ∞

p_{k} = k^{−γ}l(k),  (4)

where γ > 3 and l(·) is slowly varying at infinity.

Then as n → ∞, k ≥ n^{1/2} ln n,

P (S_{n} = k) = np_{k}(1 + o(1)).

The next statement is contained in Anorina (1971).

Proposition 3. Let d = 2. Suppose that

1) Eξ = 0, E|ξ|^{2} < ∞;

2) there exists r_{0} such that p_{k} > 0 for any |k| ≥ r_{0};

3) as |k| → ∞

p_{k} = |k|^{−γ}(1 + w(|k|)), γ > 4,

where w(t) → 0 as t → ∞.

Then as n → ∞, |k| ≥ n^{1/2} ln n,

P (S_{n} = k) = np_{k}(1 + o(1)).

The stated Propositions give rise the following questions.

1. How could look an analogue of condition (4) in the multidimensional case?

2. To what extent conditions 2) in Propositions 2 and 3 are essential? ^{1} The basic goal of this paper is to answer these questions. Incidentally we establish

a limit distribution for the conditional distribution of the maximal, in a certain sense, summand under the condition S_{n} = k, where k enjoys a large deviation. Note that from the local theorem various forms of the integral ones could be obtained, i. e., the theorems which describe the behaviour of the probability P (S_{n} ∈ A) when A lies within a large deviation region.

The paper consists of four sections. In Section 2 the main results are formulated. Section 3 is devoted to their proof. In the last section we give hints how to reproduce the omitted proofs and an example illustrating the results obtained.

2. Main Results

Let as before p_{k} = P (ξ = k) and p_{n}(k) = P (S_{n} = k). Denote by e_{x} the unit vector having the same direction as x ∈ R^{d}, i. e., e_{x} = |x|^{−1}x.

The following generalization of condition (4) can be proposed.

We say that the distribution {p_{k}} satisfies condition R if

p_{k} = r(|k|)(h(e_{k}) + w(|k|)), k ∈ D,

1Let us note that in cited papers conditions 2) is not specified but in the proofs of Propositions 2 and 3 they are implied.

where r(t) is regularly varying as t → ∞ with exponent −γ, γ > d, the non-negative function h(e) ≢ 0 is continuous on the unit sphere S^{d−1} and w(t) → 0 as t → ∞.

Condition R gives no answer to the second question that was put in Section 1. That is why, let us consider the following kinds of the density of the underlying distribution support at infinity.

We say that support D of the distribution {p_{k}} is of filled tails or simply filled if as |k| → ∞, L → ∞, L = o(|k|)

#{l ∈ Z^{d} : ∥l∥ < L, k − l ∉ D} = o(L^{d}),  (5)

where ∥ · ∥ is the max-norm, i. e., ∥(x_{1}, …, x_{d})∥ = max{|x_{1}|, …, |x_{d}|}. It is easy to see that (5) together with condition R lead to the regular variation of the tails of the underlying distribution understood as (see, e. g., Rvacheva (1962))

P (|ξ| ≥ t, e_{ξ} ∈ E) = r_{1}(t)(∫_{E} h(e)λ(de) + o(1)), t → ∞,  (6)

where r_{1}(t) is regularly varying as t → ∞ with exponent d − γ, E is a Borel subset of S^{d−1} and λ is the uniform measure on the sphere. If (6) holds the distribution of ξ is called heavy tailed.

We say that the support is of sparse tails or simply sparse if as |k| → ∞, L → ∞, L = o(|k|)

#{l ∈ Z^{d} : ∥l∥ < L, k − l ∈ D} = o(L^{d}).  (7)

If a distribution satisfies condition R and has a sparse support then instead of (6) we have

P (|ξ| ≥ t) = o(r_{1}(t)), t → ∞.

So, practically nothing is known about heaviness of the tails.

The intermediate case we obtain if instead of (5) or (7) for some fixed c ∈ (0, 1)

#{l ∈ Z^{d} : ∥l∥ < L, k − l ∈ D} ∼ c(2L)^{d}.  (8)

We call such a support regularly sparse. In contrast to (6) a regularly sparse support together with R imply

P (|ξ| ≥ t, e_{ξ} ∈ E) = cr_{1}(t)(∫_{E} h(e)λ(de) + o(1)), t → ∞,  (9)

where the function r_{1}(t) is the same as in (6). So, such distributions also possess the heavy tails.

Let S denote the support of the function h and S_{ϵ} = {e ∈ S^{d−1} : h(e) ≥ ϵ}. In what follows we suppose that the conditions of Proposition 1 are fulfilled. It is also taken for granted that n → ∞ and ϵ is an arbitrarily small fixed number.

Theorem 1. Let the underlying distribution satisfy condition R with γ > d + 2. If D is filled then for any ϵ > 0

p_{n}(k) = nr(|k − na|)h(e_{k−na})(1 + o(1))  (10)

uniformly in e_{k−na} ∈ S_{ϵ} and

p_{n}(k) = o(nr(|k − na|))  (11)

uniformly in e_{k−na} ∉ S. But if D is sparse then (11) holds. In both the relations o(·) is understood uniformly in |k − na| ≥ n^{1/2} ln n.

On the basis of Theorem 1 various forms of the integral limit theorem can be established. Consider one of them. Let A_{ρ} = {k ∈ Z^{d} : |k| < ρ(e_{k})}, where ρ : S^{d−1} → R is a positive continuous function which is separated from zero, i. e.,

inf_{e∈S^{d−1}} ρ(e) > 0.

Corollary 1. If the underlying distribution satisfies condition R with γ > d + 2 and its support is filled then

P (S_{n} − na ∉ tA_{ρ}) = nP (ξ ∉ tA_{ρ}) (1 + o(1)), t ≥ n^{1/2} ln n.

The next result highlights the crucial role played by the maximal summand when a large deviation occurs (cf Theorem 2 in Anorina (1971)). Let ν = ν_{nk} = min{i :

⟨ξ_{i}, e_{k−na}⟩ = max_{1≤j≤n} ⟨ξ_{j}, e_{k−na}⟩}.

Corollary 2. Under the conditions of Corollary 1 for any A being a set of continuity, with respect to Lebesque measure, and any ϵ > 0

P (n^{−1/2}(ξ_{ν} − k + na) ∈ A | S_{n} = k) = ∫_{A} ϕ_{B}(u)du(1 + o(1))

uniformly in k, |k − na| ≥ n^{1/2} ln n, e_{k−na} ∈ E_{ϵ}.

The next statement related to regularly sparse supports is, undoubtedly, of interest.

Theorem 2. Let the conditions of Theorem 1 be fulfilled. If D is regularly sparse then (11) is valid for all k indicated but instead of (10)

p_{n}(k) = cnr(|k − na|)h(e_{k−na})(1 + o(1)).

Note that the statement of Theorem 2 is new even in the case d = 1.

Theorem 2 has corollaries which are similar to Corollaries 1 and 2.

3. Proof of Theorem 1 and Corollaries

Throghout this section we suppose, without specifying this, that |k − na| ≥ n^{1/2} ln n and the function w(t) → 0 as t → ∞. In particular, saying ”uniformly in k” we mean ”uniformly in k, |k − na| ≥ n^{1/2} ln n, n → ∞.”

Proof of Theorem 1

Consider A_{0} = {⟨ξ_{j}, e_{k_{n}}⟩ < δ|k_{n}|, j = 1, …, n}, A_{1} = {⟨ξ_{j}, e_{k_{n}}⟩ < δ|k_{n}|, j = 1, …, n − 1, ⟨ξ_{n}, e_{k_{n}}⟩ ≥ δ|k_{n}|}, A_{2} = ⋃_{i,j=1,i≠j}^{n}{⟨ξ_{i}, e_{k_{n}}⟩ ≥ δ|k_{n}|, ⟨ξ_{j}, e_{k_{n}}⟩ ≥ δ|k_{n}|}, where k_{n} = k − na and δ ∈ (0, 1) is fixed.

Obviously, p_{n}(k) can be represented in the form

p_{n}(k) = p_{n0}(k) + np_{n1}(k) + p_{n2}(k),  (12)

where p_{nj}(k) = P (S_{n} = k, A_{j}) , j = 0, 1, 2.

Consider i. i. d. random vectors ξ̃, ξ̃_{1}, ξ̃_{2}, …, such that

P (ξ̃ = l) = P (ξ = l | ⟨ξ, e_{k_{n}}⟩ < δ|k_{n}|).

Denote S̃_{n} = ξ̃_{1} + ⋯ + ξ̃_{n}. Since the sum of vectors a′ = ∑_{l∈L′} lp_{l} and a′′ = ∑_{l∈L′′} lp_{l}, where

L′ = {l ∈ Z^{d} : ⟨l, e_{k_{n}}⟩ < δ|k_{n}|}, L′′ = {l ∈ Z^{d} : ⟨l, e_{k_{n}}⟩ ≥ δ|k_{n}|}, equals to a then

|Eξ̃− a| = O( ∑_{l∈L′′} |l|p_{l}) = O(|k_{n}|^{−1} ∑_{l:|l|≥δ|k_{n}|} |l|^{2}p_{l}) = o (|k_{n}|^{−1}) = o (n^{−1/2}) .

It is evident also that E|ξ̃|^{2} → E|ξ|^{2}. Hence, (2) holds for S̃_{n} as well.

It is of worth to mention the basic properties of the distributions satisfying condition R. Let |k| → ∞. If k − l ∈ D, |l| = o(|k|), then uniformly in e_{k} ∈ S_{ϵ}

p_{k−l} = r(|k|)h(e_{k})(1 + o(1)) ≍ r(|k|);  (13)

if k − l ∈ D, |l| = O(|k|), then uniformly in e_{k} ∈ S_{ϵ}

p_{k−l} = O(r(|k|)).  (14)

In both cases uniformly in e_{k} ∉ S

p_{k−l} = o(r(|k|)).  (15)

Lemma 1. Let the conditions of Theorem 1 be fulfilled. If D is filled then for any ϵ > 0

p_{n1}(k) = r(|k − na|)h(e_{k−na})(1 + o(1))

uniformly in e_{k−na} ∈ S_{ϵ} and

p_{n1}(k) = o(r(|k − na|))  (16)

uniformly in e_{k−na} ∉ S. But if D is sparse then (16) holds uniformly in k.

Proof. By definition,

p_{n1}(k) = P^{n−1} (⟨ξ, e_{k_{n}}⟩ < δ|k_{n}|) ∑_{l∈Z^{d}:⟨l−na,e_{kn}⟩≤(1−δ)|k_{n}|} p_{k−l}P (S̃_{n−1} = l). (17)

Because of Chebyshev’s inequality

P (⟨ξ, e_{k_{n}}⟩ ≥ δ|k_{n}|) = O(P (|ξ| ≥ δ|k_{n}|)) = o(n^{−1}),  (18)

that yields

P^{n} (⟨ξ, e_{k_{n}}⟩ < δ|k_{n}|) = 1 + o(1).  (19)

Let X > 0 be an arbitrarily large fixed number. Partition the integration region in (17) into two the parts J_{1} = {l ∈ Z^{d} : k − l ∈ D, ∥l − na∥ < n^{1/2}X} and J_{2} = {l ∈ Z^{d} : k − l ∈ D, ∥l − na∥ ≥ n^{1/2}X, ⟨l − na, e_{k_{n}}⟩ ≤ (1 − δ)|k_{n}|}, and denote

I_{1} = ∑_{l∈J_{1}} p_{k−l}P (S̃_{n−1} = l), I_{2} = ∑_{l∈J_{2}} p_{k−l}P (S̃_{n−1} = l).

For e_{k_{n}} ∈ S_{ϵ} due to (13)

I_{1} = r(|k_{n}|)h(e_{k_{n}})P (∥S̃_{n−1} − na∥ < n^{1/2}X, k − S̃_{n−1} ∈ D)(1 + o(1)),  (20)

while for e_{k_{n}} ∉ S because of (15)

I_{1} = o(r(|k_{n}|)).  (21)

It remains to study P (∥S̃_{n−1} − na∥ < n^{1/2}X, k − S̃_{n−1} ∈ D).

First, consider the case when the support is filled. From Proposition 1 it follows that

P (∥S̃_{n−1} − na∥ < n^{1/2}X, k − S̃_{n−1} ∈ D) ∼ ∑_{l∈Z^{d}:k−l∈D,∥l−na∥<n^{1/2}X} ϕ_{B} (\frac{l − na}{\sqrt{n}}) \frac{1}{(\sqrt{n})^{d}} .

Partition the cube ∥x∥ < n^{1/2}X into the subcubes of the edge length ϵn^{1/2}. With no loss in generality we can assume that Xϵ^{−1} ∈ Z_{+}^{1}. Let V_{n} be one of such cubes. It is easily understood that under condition (5)

∑_{l∈Z^{d}:k−l∈D,l−na∈V_{n}} ϕ_{B} (\frac{l − na}{\sqrt{n}}) \frac{1}{(\sqrt{n})^{d}} = (ϕ_{B} (\frac{z}{\sqrt{n}}) + θw(ϵ^{−1})) ×

\frac{#(l ∈ Z^{d} : k − l ∈ D, l − na ∈ V_{n})}{(\sqrt{n})^{d}} ∼ ϵ^{d} (ϕ_{B} (\frac{z}{\sqrt{n}}) + θw(ϵ^{−1})) = ϵ^{d}(ϕ_{B}(u)+θw(ϵ^{−1})),

(22)

where z is the center of the cube V_{n} and u is the center of the cube V, which corresponds to V_{n} under partitioning of the cube ∥x∥ < X into the cubes of the edge length ϵ, |θ| < 1.

Summing over all such cubes yields the integral sums for ϕ_{B}. Since ϵ is arbitrary (22) means that

P (∥S̃_{n−1} − na∥ < n^{1/2}X, k − S̃_{n−1} ∈ D) ∼ ∫_{∥u∥<X} ϕ_{B}(u)du = 1 + θw(X) (23)

uniformly in k, where |θ| < 1.

If D is sparse then the similar argument leads to the relation

P (∥S̃_{n−1} − na∥ < n^{1/2}X, k − S̃_{n−1} ∈ D) = o(1)  (24)

uniformly in k and X.

The upper bound for I_{2}, which does not depend on the properties of D, owing to (14) and (23) can be established as follows

I_{2} ≤ P (∥S̃_{n−1} − na∥ ≥ n^{1/2}X) sup_{l∈J_{2}} p_{k−l} ≤ w(X)r(|k_{n}|).  (25)

Since X is arbitrary, (17), (19)–(21), (23)–(25) imply the statements of the lemma. △

Lemma 2. Let the conditions of Theorem 1 be fulfilled. Then

p_{n2}(k) = o(nr(|k − na|))  (26)

uniformly in k.

Proof. Taking into account (14) and (15),

p_{n2}(k) ≤ n^{2} ∑_{l∈Z^{d}:⟨l,e_{kn}⟩≥δ|k_{n}|} ∑_{m∈Z^{d}:⟨m,e_{kn}⟩≥δ|k_{n}|} p_{l}p_{m}P (S_{n−2} = k − l − m) =

O (n^{2}r(|k_{n}|)P (⟨ξ, e_{k_{n}}⟩ ≥ δ|k_{n}|)) .

The desired statement follows now from (18). △ Lemma 3. Let the conditions of Theorem 1 be fulfilled. Then

p_{n0}(k) = o(nr(|k − na|))  (27)

uniformly in k.

Proof. By definition,

p_{n0}(k) = P^{n} (⟨ξ, e_{k_{n}}⟩ < δ|k_{n}|) P (S̃_{n} = k).

Using (19) and Markov’s inequality we obtain

p_{n0}(k) = O (P (⟨ξ̃_{1} − a, e_{k_{n}}⟩ + … + ⟨ξ̃_{n} − a, e_{k_{n}}⟩ ≥ |k_{n}|)) = O (e^{−s|k_{n}|}f^{n}(s)) , (28)

where f(s) = Ee^{s⟨_{ξ}−a,e_{kn}⟩}. Set

s = −|k_{n}|^{−1} ln(n^{1/2}r(|k_{n}|)).  (29)

Under such a choice k_{n} we have s → 0, s = O(n^{−1/2}) and s|k_{n}| → ∞.

Let us show that

f(s) = 1 + O(n^{−1})  (30)

provided 0 < δ < (γ − d − 2)(γ − 1)^{−1}. Then the statement of the lemma will follow from (28) and (29).

Denote η = ⟨ξ̃− a, e_{k_{n}}⟩. For proving (30) let us represent f(s) = Ee^{sη} in the form

f(s) = E (e^{sη}, η ≤ −\frac{β}{s} ) + E (e^{sη}, |η| < \frac{β}{s} ) + E (e^{sη}, \frac{β}{s} ≤ η < δ|k_{n}| − ⟨a, e_{k_{n}}⟩) =

f_{1}(s) + f_{2}(s) + f_{3}(s),

where β > 0 is fixed.

Due to Chebyshev’s inequality

f_{1}(s) = O (P (η ≤ −\frac{β}{s} )) = O (s^{2}) .  (31)

Further,

f_{2}(s) = P (|η| < \frac{β}{s} )+s ∫_{|u|<βs^{−1}} udP (η < u)+O (s^{2}) = 1+O (s^{2})+s ∫_{|u|<βs^{−1}} udP (η < u).

Since

∫ udP (η < u) = ⟨Eξ̃− a, e_{k_{n}}⟩ = o (n^{−1/2}) ,

then

f_{2}(s) = 1 + O (s^{2}) + o (sn^{−1/2}) .  (32)

Finally,

f_{3}(s) = O (E(e^{s⟨ξ,e_{kn}⟩}, \frac{β′}{s} < ⟨ξ, e_{k_{n}}⟩ < δ|k_{n}|))  (33)

for 0 < β′ < β, and

E (e^{s⟨ξ,e_{kn}⟩}, \frac{β′}{s} < ⟨ξ, e_{k_{n}}⟩ < δ|k_{n}|) ≤ \frac{e^{sδ|k_{n}|}}{δ^{β′}|k_{n}|^{β′}} E (⟨ξ, e_{k_{n}}⟩^{β′}, \frac{β′}{s} < ⟨ξ, e_{k_{n}}⟩ < δ|k_{n}|) .

(34)

For any ϵ > 0

E (⟨ξ, e_{k_{n}}⟩^{β′}, \frac{β′}{s} < ⟨ξ, e_{k_{n}}⟩ < δ|k_{n}|) = ∑_{l:β′s^{−1}<⟨l,e_{kn}⟩<δ|k_{n}|}⟨l, e_{k_{n}}⟩^{β′}p_{l} =

O (  ∑_{l:β′s^{−1}<⟨l,e_{kn}⟩<δ|k_{n}|} |l|^{β′−γ+ϵ}) = O (∫_{β′s^{−1}<⟨u,e_{kn}⟩<δ|k_{n}|} |u|^{β′−γ+ϵ}du) = O (s^{1−ϵ}) , (35)

if we set β′ = γ − d − 1 > 1. Since 0 < δ < (γ − d − 2)(γ − 1)^{−1} then from (33)–(35)

f_{3}(s) = O (\frac{e^{sδ|k_{n}|}s^{1−ϵ}}{|k_{n}|^{γ−d−1}} ) = O ((\frac{1}{n^{1/2}r(|k_{n}|)})^{δ} \frac{1}{|k_{n}|^{γ−d−1}n^{(1−ϵ)/2}} ) = o (n^{−1}) . (36)

Together (31), (32) and (36) lead to (30). △

Combining (12) and Lemmas 1–3 we obtain all the statements of the theorem. △

Proof of Corollary 1

Let X > 0 be an arbitrarily large fixed number. Partition the set {l ∈ Z^{d} : l − na ∉ tA_{ρ}} into two the parts J_{1} = {l ∈ Z^{d} : |l − na| < Xt, l − na ∉ tA_{ρ}} and J_{2} = {l ∈ Z^{d} : |l − na| ≥ Xt}, and denote

I_{1} = ∑_{l∈J_{1}} p_{n}(l), I_{2} = ∑_{l∈J_{2}} p_{n}(l).

Then

P (S_{n} − na ∉ tA_{ρ}) = I_{1} + I_{2}.  (37)

Let us represent I_{1} as follows

I_{1} = I_{11} + I_{12} + I_{13},  (38)

where we sum over the sets {l ∈ J_{1} : h(e_{l−na}) ≥ ϵ}, {l ∈ J_{1} : h(e_{l−na}) = 0} and {l ∈ J_{1} : 0 < h(e_{l−na}) < ϵ}, respectively.

Let r(t) = t^{−γ}l(t), where l(t) is slowly varying at infinity. In view of (10)

I_{11} ∼ n ∑_{l∈J_{1},e_{l−na}∈S_{ϵ}} r(|l − na|)h(e_{l−na}) ∼ nl(t) ∑_{l∈J_{1},e_{l−na}∈S_{ϵ}} g(l − na) =

nr_{1}(t) ∑_{l∈J_{1},e_{l−na}∈S_{ϵ}} g (\frac{l − na}{t}) \frac{1}{t^{d}} ∼ nr_{1}(t) ∫_{ρ(e_{u})≤|u|<X,e_{u}∈S_{ϵ}} g(u)du =

nr_{1}(t)(∫_{|u|≥ρ(e_{u})} g(u)du + θ_{1}w(X) + θ_{2}w(ϵ^{−1})),  (39)

where g(y) = h(e_{y})|y|^{−γ}, |θ_{1}| < 1, |θ_{2}| < 1. Further, in accordance with (11),

I_{12} = ∑_{l∈J_{1},h(e_{l−na})=0} o(nr(|l − na|)) = o(n ∑_{l∈J_{1},h(e_{l−na})=0} r(|l − na|)).  (40)

If r(t) monotonically decreases for all sufficiently large t then

∑_{l∈J_{1},h(e_{l−na})=0} r(|l − na|) = O(∫_{tρ(e_{u})≤|u|<Xt} r(|u|)du)= O(r_{1}(t) ∫_{ρ(e_{u})≤|u|<X} |u|^{−γ}du).

(41)

Note that there always exists a slowly varying at infinity function l̃(t) ∼ l(t) with the property l̃′(t) = o(t^{−1}l̃(t)), that means monotonical decreasing of r̃(t) = t^{−γ}l̃(t). Evidently, the function

l_{1}(t) = ∫_{1}^{e} \frac{l(tu)}{u}du ∼ l(t)

is continuous and

l̃(t) = ∫_{1}^{e} \frac{l_{1}(tu)}{u}du ∼ l_{1}(t) ∼ l(t)

is differentiable while

l̃′(t) = t^{−1}(l_{1}(te) − l_{1}(t)) = o (t^{−1}l_{1}(t)) = o (t^{−1}l̃(t)) .

In such a case with no loss in generality we can assume that all regularly varying at infinity functions, which we are dealing with, monotonically decrease.

If l ∈ J_{1}, 0 < h(e_{l−na}) < ϵ, then

p_{n}(l) = O(ϵnr(|l − na|))

whence, according to (41),

I_{13} = ∑_{l∈J_{1},0<h(e_{l−na})<ϵ} O(ϵnr(|l −na|)) = O(ϵn ∑_{l∈J_{1},0<h(e_{l−na})<ϵ} r(|l −na|)) = O(ϵnr_{1}(t)).

(42)

Combining (38)–(42) yields

I_{1} = nr_{1}(t)(∫_{|u|≥ρ(e_{u})} g(u)du + o(1))  (43)

since ϵ and X are arbitrary.

As to I_{2}, the rough estimate

I_{2} = O(n ∑_{l∈J_{2}} r(|k − na|)),

in view of (41), yields

I_{2} = O(nr_{1}(t)w(X)).  (44)

From the other side,

P (ξ ∉ tA_{ρ}) = ∑_{k∈D:|k|≥tρ(e_{k})} r(|k|)(h(e_{k}) + w(|k|)) ∼ \frac{l(t)}{t^{γ−d}} ∑_{k∈D,|k|≥tρ(e_{k})} g (\frac{k}{t} ) \frac{1}{t^{d}} .

Recall that (5) ensures ”the sufficient number of points” which is necessary for proper integral sums for g (see the proof of Lemma 1). In such a case,

P (ξ ∉ tA_{ρ}) ∼ r_{1}(t) ∫_{|u|≥ρ(e_{u})} g(u)du.  (45)

The statement of the corollary follows from (37), (43)–(45). △

Proof of Corollary 2

Using (10), (12), (26) and (27), we have

P (n^{−1/2}(ξ_{ν} − k_{n}) ∈ A | S_{n} = k) = (p_{n}(k))^{−1}(P (S_{n} = k, n^{−1/2}(ξ_{ν} − k_{n}) ∈ A, A_{0}) + nP (S_{n} = k, n^{−1/2}(ξ_{n} − k_{n}) ∈ A, A_{1}) + P (S_{n} = k, n^{−1/2}(ξ_{ν} − k_{n}) ∈ A, A_{2})) ∼

(p_{n}(k))^{−1}nP (S_{n} = k, n^{−1/2}(ξ_{n} − k_{n}) ∈ A, A_{1}) ∼ (r(|k_{n}|)h(e_{k_{n}}))^{−1}×

∑_{l∈Z^{d}:k−l∈D,n^{−1/2}(na−l)∈A} p_{k−l}P (S̃_{n−1} = l) ∼ P (n^{−1/2}(na − S̃_{n−1}) ∈ A, k − S̃_{n−1} ∈ D) .

It remains to show that if D is filled then

P (n^{−1/2}(na − S̃_{n−1}) ∈ A, k − S̃_{n−1} ∈ D) ∼ ∫_{A} ϕ_{B}(u)du.  (46)

Note that the relation (46), in essence, generalizes (23). In order to prove it we partition R^{d} into the cubes V of the edge length ϵ. Define

A_{ϵ}^{−} = ∪_{V ⊂A} V, A_{ϵ}^{+} = ∪_{V ∩A≠∅} V.

Then

P (n^{−1/2}(na−S̃_{n−1}) ∈ A_{ϵ}^{−}, k−S̃_{n−1} ∈ D) ≤ P (n^{−1/2}(na−S̃_{n−1}) ∈ A, k−S̃_{n−1} ∈ D) ≤

P (n^{−1/2}(na − S̃_{n−1}) ∈ A_{ϵ}^{+}, k − S̃_{n−1} ∈ D).

For each particular cube V with the center u the formulae (22) holds. Summing over all such cubes we obtain, since ϵ is arbitrary, (46) (cf (23)).

4. Concluding Remarks

On the Proof of Theorem 2

The proof of Theorem 2, in fact, repeats that of Theorem 1. From the proof of Theorem 1 it is clear that the main contribution to the asymptotics of P (S_{n} = k) is done by P (∥S̃_{n−1} − na∥ < n^{1/2}X, k − S̃_{n−1} ∈ D). Evidently, due to (12), (17), (20), (25)–(27) as |k_{n}| ≥ n^{1/2} ln n, e_{k_{n}} ∈ S_{ϵ}, n → ∞,

\frac{p_{n}(k)}{nr(|k_{n}|)h(e_{k_{n}})} = \frac{p_{n0}(k) + np_{n1}(k) + p_{n2}(k)}{nr(|k_{n}|)h(e_{k_{n}})} = \frac{p_{n1}(k)}{r(|k_{n}|)h(e_{k_{n}})} + o(1) =

P (∥S̃_{n−1} − na∥ < n^{1/2}X, k − S̃_{n−1} ∈ D) + θw(X),

where |θ| < 1. Repeating the argument, utilized in the proof of Lemma 1, which led to the asymptotics of P (∥S̃_{n−1} − na∥ < n^{1/2}X, k − S̃_{n−1} ∈ D) , one should note that for regularly sparse supports (cf (22))

\frac{#(l ∈ Z^{d} : k − l ∈ D, l − na ∈ V_{n})}{(\sqrt{n})^{d}} ∼ cϵ^{d}.

Hence, instead of (23) we get

P (∥S̃_{n−1} − na∥ < n^{1/2}X, k − S̃_{n−1} ∈ D) ∼ c ∫_{∥u∥<X} ϕ_{B}(u)du = c + θw(X)

uniformly in k, where |θ| < 1, that leads to the first part of the statement of Theorem 2. The second follows from (21), (25)–(27), which, obviously, hold in the case.

The same remarks can be made about the analogues of Corollaries 1 and 2.

About the Density of Supports at Infinity

From Theorems 1 and 2 it follows that (10) takes place provided that D is either filled or regularly sparse. Simple examples show that p_{n}(k) may have no definite asymptotic expression unless we impose restrictions on D.

Let for simplicity d = 1 and a = 0. Suppose that the distribution is such that p_{k} = r(|k|)(1 + o(1)), k ∈ D ⊂ Z, |k| → ∞, where r(t) is regularly varying as t → ∞ with exponent −γ, γ > 3, and D is such that its part, which is contained in [1, ∞), has the form ∪_{l∈N}[l^{2} − l/2, l^{2} + l/2] ∩ N. Obviously, such a distribution satisfies the condition R, but has a support which is neither filled or sparse, nor of regular sparse. Indeed, for the sequence k′_{n} = n^{2} and L_{n} = n/2 we have

#{l ∈ Z : |l| < L_{n}, k′_{n} − l ∉ D} = 0,

but for k′′_{n} = n^{2} + n and L_{n}

#{l ∈ Z : |l| < L_{n}, k′′_{n} − l ∈ D} = 0.

It makes no difficulty to show that in this case

0 = lim inf_{k∈K_{n}} \frac{p_{n}(k)}{nr(|k|)} < lim sup_{k∈K_{n}} \frac{p_{n}(k)}{nr(|k|)} = 1,

where K_{n} = {k : |k| ≥ n^{1/2} ln n, n → ∞}, furthermore lim inf is attained at the sequence k′′_{n} and lim sup at k′_{n}. Moreover, the set of the limit points of the ratio p_{n}(k)/nr(|k|) is dense in the whole interval (0, 1), i. e., there is no definite asymptotics of p_{n}(k).

References

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