Multivariate Large Deviations with Stable Limit Laws

Alexander Zaigraev

Nicholas Copernicus University, Toruń

The large deviation problem for sums of i. i. d. random vectors is considered. It is assumed that the underlying distribution is absolutely continuous and its density is of regular variation. An asymptotic expression for the probability of large deviations is established in the case of a nonnormal stable limit law. The role of the maximal summand is also emphasized. Key words and phrases: sums of i. i. d. random vectors, local limit theorem, regular variation, maximal summand. AMS Classification Numbers: Primary 60F10; Secondary 60G50.

1. Introduction

Let ξ, ξ_{1}, ξ_{2}, … be i. i. d. random vectors assuming values in R^{d}. Denote by F the common distribution and consider the partial sums S_{n} = ξ_{1} + ⋯ + ξ_{n}, n = 1, 2 …. Throughout the paper we assume that there exist sequences a_{n} ∈ R^{d}, b_{n} ∈ (0, ∞) and a non-degenerate distribution G such that for any G-continuity set A

P (b_{n}^{−1}(S_{n} − a_{n}) ∈ A) ⟶ G(A), n → ∞  (1)

or, in words, F is attracted by G. The set of distributions attracted by G is called the domain of attraction of G. It is known that the domain of attraction of G is not empty iff G is stable in the sense that for all positive numbers b_{1}, b_{2} there exist a ∈ R^{d} and a positive number b such that b_{1}X_{1} + b_{2}X_{2} and bX + a are identically distributed, where X_{1}, X_{2}, X have distribution G, and X_{1} and X_{2} are independent. Up to a shift, each stable distribution is uniquely determined by a number α ∈ (0, 2] and a probability measure μ defined on the σ-algebra S of the Borel subsets on the unit sphere S^{d−1} (see e.g. Samorodnitsky and Taqqu (1994)).

Domains of attraction for d > 1 were first studied by Rvačeva (1954, 1962) who gave, in particular, necessary and sufficient conditions for F to be attracted by G as follows : F is attracted by G = G_{α,μ} iff for any c > 0

lim_{t→∞} \frac{P (|ξ| > t)}{P (|ξ| > ct)} = c^{α}

while for any Q_{1}, Q_{2} ∈ S

lim_{t→∞} \frac{P (|ξ| > t, |ξ|^{−1}ξ ∈ Q_{1})}{P (|ξ| > t, |ξ|^{−1}ξ ∈ Q_{2})} = \frac{μ(Q_{1})}{μ(Q_{2})}

provided μ(Q_{2}) ≠ 0.

Let us denote by e_{x} the unit vector of the same direction as x that is e_{x} = |x|^{−1}x. Sometimes it is more convenient to utilize the following form of necessary and

sufficient conditions (see e.g. Kalinauskaite (1974)) : F is attracted by G_{α,μ} iff for any c > 0 and any Q ∈ S

P (|ξ| > t, e_{ξ} ∈ Q) = t^{−α}l(t)(μ(Q) + m(Q, t))  (2)

where l(t) varies slowly at infinity and, given t, m(Q, t) is σ-additive while, given Q, m(Q, t) vanishes as t → ∞.

Let (2) be fulfilled. If, furthermore, S_{n_{0}} for some n_{0} ≥ 1 has a uniformly bounded density then the local limit theorem takes place that is

sup_{x∈R^{d}} |b_{n}^{d}p_{n}(x) − g(b_{n}^{−1}(x − a_{n}))| = o(1), n → ∞,  (3)

where p_{n}(x) and g(x) are the densities of S_{n} and the stable limit law, respectively (see Rvačeva (1962)).

A value x = x_{n} in the range of S_{n} is called a large deviation if b_{n}^{−1}|x − a_{n}| → ∞,

n → ∞. That is why P (S_{n} ∈ A) with A = A_{n}, b_{n}^{−1} inf_{x∈A} |x − a_{n}| → ∞, n → ∞,

is called a large deviation probability. In view of (1), a large deviation probability converges to zero. From (3) it follows that, as n → ∞, b_{n}^{d}p_{n}(x) = g(b_{n}^{−1}(x−a_{n}))+o(1) if |x − a_{n}| = O(b_{n}), while b_{n}^{d}p_{n}(x) → 0 if x enjoys a large deviation. The basic problem of large deviation theory is to establish a precise asymptotic expression for P (S_{n} ∈ A) (or p_{n}(x)) when A (or x) lies in the region of large deviations.

In this paper we are going to deal with so-called precise asymptotic of P (S_{n} ∈ A). Another class of problems within the large deviations theory unites those related to the asymptotic expression of ln P (S_{n} ∈ A) which is called a rough asymptotic expression. It is worth noting that such expressions can be established under rather general conditions, e.g. for random variables taking values in various infinite-dimensional spaces, for dependent random variables, etc. The present state of the work in this direction can be found in Deuschel and Stroock (1989), Dembo and Zeitouni (1998). Characterization of stable laws and their domains of attraction in Banach spaces are given e.g. in Jurek and Urbanik (1978), Araujo and Gine (1980).

Let

A^{(n)} = {A ∈ B^{d} : inf_{x∈A} |x| ≥ c_{n}b_{n}}  (4)

where B^{d} is the σ-algebra of the Borel subsets of R^{d} and lim_{n→∞} c_{n} = ∞.

We say that the large deviation problem is solved within A_{0}^{(n)} ⊂ A^{(n)} if it is established that

lim_{n→∞} sup_{A∈A_{0}^{(n)}} |\frac{P (S_{n} − a_{n} ∈ A)}{P (n, A)} − 1| = 0

where P (n, A) is an asymptotic expression that involves certain characteristics of both the underlying distribution and A.

So, to each A_{0}^{(n)} ⊂ A^{(n)} there corresponds a setting of the large deviation problem. The classic theory of large deviations deals with the case when d = 1 and A_{0}^{(n)} is the class of the half-lines (x_{n}, ∞) or (−∞, −x_{n}) with b_{n}^{−1}x_{n} → ∞. An overview of this rather advanced theory can be found e.g. in S. Nagaev (1979). Among those who contributed to the theory studying the case of an infinite variance limit stable law one should mention Zolotarev (1962), Aleskjavičene (1962), Heyde (1968), Tkačuk (1973), Kim and A. Nagaev (1975), A. Nagaev (1975), Rozovskii (1997). Of course, this list is far from being complete.

The first question which arises when we move to higher dimensions is how to choose multidimensional analogues of those half-lines. The simplest way is to define A_{0}^{(n)} as follows

A_{0}^{(n)} = {A ∈ B^{d} : A = tA_{0}^{c}, t ≥ c_{n}b_{n}}, c_{n} → ∞,  (5)

where A_{0} contains the origin, A_{0}^{c} = R^{d} \ A_{0}. In words, A_{0}^{(n)} is the parametric family of the sets homothetic to A_{0}^{c}.

In Tkačuk (1974) it was shown that under such a setting the large deviation problem can be solved provided the boundary of the generating set A_{0} satisfies minimal regularity conditions. Before we present the mentioned result let us assume without loss of generality that Eξ = 0 when the expectation exists.

Proposition 1. Let in (5)

A_{0} = {x ∈ R^{d} : |x| < ρ(e_{x})}

where ρ(e) is a positive continuous function on S^{d−1}. If F belongs to the domain of attraction of a stable distribution with α ∈ (0, 1) ∪ (1, 2) then

lim_{n→∞} sup_{A∈A_{0}^{(n)}} |\frac{P (S_{n} ∈ A)}{nP (ξ ∈ A)} − 1| = 0.

It is rather easy to show that A_{0}^{(n)} in Proposition 1 can be made broader. For instance, for A_{0}^{(n)} we may choose

A_{0}^{(n)} = {A ∈ B^{d} : A = tB^{c}, B ∈ A_{0}, t ≥ c_{n}b_{n}}

where

A_{0} = {A ∈ B^{d} : t_{−}S^{d−1} ⊂ A ⊂ t_{+}S^{d−1}}

for some fixed t_{−}, t_{+} such that 0 < t_{−} < t_{+} < ∞. Apparently, this case can be treated as in Pinelis (1981).

However, for richer families we have to impose more restrictions either on the underlying distribution or on the order of the considered deviations. For instance, we can deal with

Ā^{(n)} = {A ∈ B^{d} : c_{n}^{−}b_{n} ≤ inf_{x∈A} |x| ≤ c_{n}^{+}b_{n}}

where c_{n}^{−} → ∞, c_{n}^{+} → ∞, c_{n}^{−} = o(c_{n}^{+}).

In order to have an impression on the variety of the, so far considered, settings see e.g. Borovkov and Rogozin (1965), Vilkauskas (1965), A. Nagaev (1966), von Bahr (1967), Anorina and A. Nagaev (1971), A. Nagaev and Sakojan (1972), Osipov (1981), Rozovskii (1985).

The simplest way to extend the statement of Proposition 1 to A^{(n)} is to assume that the distribution is absolutely continuous and its density is, asymptotically as |x| → ∞, sufficiently regular. It is worth recalling an assertion proved in Tkačuk (1973).

Proposition 2. Let d = 1. Suppose that F belongs to the domain of normal attraction of a stable distribution G with α ∈ (0, 1) ∪ (1, 2) and its density is such that

p(x) = cx^{−(1+α)}(1 + o(1)), x → ∞, c > 0.  (6)

Then as xn^{−1/α} → ∞, n → ∞,

p_{n}(x) = np(x)(1 + o(1)) = n^{−1/α}g(xn^{−1/α})(1 + o(1)).

Recall A^{(n)} as defined in (4) and let d = 1, b_{n} = n^{1/α}. Then from Proposition 2 it follows that

lim_{n→∞} sup_{A∈A^{(n)}} |\frac{P (S_{n} ∈ A)}{nP (ξ ∈ A)} − 1| = 0.

The conditions of Proposition 2 are rather close to the necessary ones. This means that the local limit theorem provides an approach which enables us to solve the large deviation problem for extremely rich classes of sets. In particular, such an approach was adopted in Borovkov and Rogozin (1965) while studying the case of light-tailed distributions.

The basic goal of the paper is to carry over the statement of Proposition 2 to the multidimensional case. The multidimensional analogue of condition (6) may be chosen as follows. Let non-negative functions r_{γ} and h be defined on [0, ∞) and S^{d−1}, respectively. We say that a density p ∈ P_{γ,h} if it is uniformly bounded and admits the representation

p(x) = r_{γ}(|x|)(h(e_{x}) + w(x))

where r_{γ}(t) varies regularly at infinity with exponent −γ, γ > d, h(e) ≥ 0 is continuous, w(x) → 0 as |x| → ∞ uniformly in e_{x} (cf. Resnick (1987, Section 5.4.2)).

In particular, if p ∈ P_{γ,h} then for any Q ∈ S

P (|ξ| > t, e_{ξ} ∈ Q) = (μ(Q) + w_{Q}(t))r_{γ−d}(t),  (7)

μ(Q) = (∫_{S^{d−1}} h(e)λ(de))^{−1} ∫_{Q} h(e)λ(de), where λ is the surface measure on S and

sup_{Q∈S} |w_{Q}(t)| → 0, t → ∞.

One of the simplest examples of p ∈ P_{γ,h} is given by the function

p(x) = \{ \begin{matrix}0 & if |x| < t_{0} \\ c_{αβ}(|x_{1}|^{α} + ⋯ + |x_{d}|^{α})^{−β} & otherwise \end{matrix} .

It is easily seen that r_{γ}(t) = t^{−γ}, γ = αβ, h(e) = c_{αβ}(|e_{1}|^{α} + ⋯ + |e_{d}|^{α})^{−β} and w(x) = 0, |x| ≥ t_{0}.

It is worth noting that within P_{γ,h} the variables t^{−1}|ξ| and e_{ξ} are asymptotically independent, as t → ∞, given |ξ| > t. More precisely, from (7) it follows that

P (t^{−1}|ξ| > z, e_{ξ} ∈ Q | |ξ| > t) ⟶ z^{d−γ}μ(Q), t → ∞, z ≥ 1.

Obviously, (7) implies (2). Therefore, for γ ∈ (d, d + 2) the underlying distribution belongs to the domain of attraction of a stable law with exponent α = γ−d and measure μ of the mentioned form. If α = 2 it belongs to the domain of attraction of a normal law (see Rvačeva (1962)). Here we confine ourselves to the case α ∈ (0, 1) ∪ (1, 2). The threshold cases α = 1 and α = 2 with γ = d + 2 require additional arguments. They will be studied elsewhere. It should be noted that the present paper complements that of A. Nagaev and Zaigraev (1998) devoted to the case γ > d + 2 when the limit law is normal.

The paper is organized as follows. The main results are formulated in Section 2 while Section 3 contains their proofs.

2. Main Results

It is worth recalling that we confine ourselves to p ∈ P_{γ,h} and α ∈ (0, 1) ∪ (1, 2). Furthermore, when α ∈ (1, 2) we assume that Eξ = 0. It means that in (1) we may choose a_{n} = 0 while b_{n} = inf{t : nP (|ξ| ≥ t) ≤ 1} (see e.g. Rvačeva (1962)). If (7) holds then b_{n} is determined by the equation P (|ξ| ≥ b_{n}) = n^{−1} provided n is sufficiently large.

Let c_{n} be any sequence such that lim_{n→∞} c_{n} = ∞.

Theorem 1. For any ε > 0

lim_{n→∞} sup_{|x|≥c_{n}b_{n}} sup_{e_{x}∈E_{ε}} |\frac{p_{n}(x)}{np(x)} − 1| = 0

where E_{ε} = {e ∈ S^{d−1} : h(e) ≥ ε} while

lim_{n→∞} sup_{|x|≥c_{n}b_{n}} sup_{e_{x}∉E} \frac{p_{n}(x)}{nr_{γ}(|x|)} = 0

where E = {e ∈ S^{d−1} : h(e) > 0}.

As in the case d = 1 mentioned in Section 1 we easily come to the following statement.

Corollary 1. Under the conditions of Theorem 1,

lim_{n→∞} sup_{A∈A^{(n)}} |\frac{P (S_{n} ∈ A)}{nP (ξ ∈ A)} − 1| = 0.

The assertion of Theorem 1 can be rewritten in the following form

lim_{n→∞} sup_{|x|≥c_{n}b_{n}} sup_{e_{x}∈E_{ε}} |\frac{p_{n}(x)}{nr_{γ}(|x|)} − h(e_{x})| = 0.

Simple examples show that the continuity of h(e) is essential. Let, for instance, d = 2 and h(e) is continuous except for e = e^{(0)}. Taking into account the representation e = e_{ϕ} = (cos ϕ, sin ϕ), 0 ≤ ϕ < 2π, assume that inf_{e∈S^{d−1}} h(e) > 0, i. e. E_{ε} = S^{d−1}, and

there exist

h^{−} = lim_{ϕ↑ϕ_{0}} h(e_{ϕ}), h^{+} = lim_{ϕ↓ϕ_{0}} h(e_{ϕ})

such that 0 < h^{−} < h^{+} < ∞. It can be shown that for e_{x} = e^{(0)}

p_{n}(x) = h_{0}nr_{γ}(|x|)(1 + o(1))

with h_{0} = (h^{−} + h^{+})/2 while for any fixed e_{x} ≠ e^{(0)}

p_{n}(x) = h(e_{x})nr_{γ}(|x|)(1 + o(1)).

It is of interest to highlight the role played by the maximal summand when a large deviation of the sum takes place. Let j_{n} be the extreme left point of the set arg max_{1≤j≤n} |ξ_{j}|

and M_{n} = ξ_{j_{n}}.

It is worth noting that, for any α > 0, (2) also presents a nesessary and sufficient condition for the asymptotic relation

P (|M_{n}| ≤ tb_{n}) = e^{−t^{−α}} + o(1)

as well as for

P (|M_{n}| ≤ tb_{n}, e_{M_{n}} ∈ Q) = e^{−t^{−α}}μ(Q) + o(1)

where Q ∈ S (see Resnick (1987, Section 5.4.1)).

Moreover, for α ∈ (0, 1) ∪ (1, 2) there exists the limit distribution for |M_{n}|^{−1}S_{n} (see e. g. Kalinauskaite (1973)). It means that |M_{n}| and |S_{n}| are of the same order. Note that in the case of a normal limit distribution |M_{n}|/|S_{n}| → 0 in probability.

The next statement shows that the role of M_{n} even increases when S_{n} abnormally deviates from the origin.

Theorem 2. Let G be the stable limit law in (1). Then for any ε > 0 and any Gcontinuity set A

lim_{n→∞} sup_{|x|≥c_{n}b_{n}} sup_{e_{x}∈E_{ε}} |\frac{P (b_{n}^{−1}(S_{n} − M_{n}) ∈ A | S_{n} = x)}{G(A)} − 1| = 0.

As a corollary we obtain the limit law for the conditional distribution of the projection of M_{n} onto a deviation direction given the large deviation of the sum. From now on, ⟨·, ·⟩ denotes the ordinary inner product.

Corollary 2. Let g(x) be the density of the stable limit law. Then for any ε > 0 and any t ∈ R^{1}

lim_{n→∞} sup_{|x|≥c_{n}b_{n}} sup_{e_{x}∈E_{ε}} |\frac{P (|S_{n}| − ⟨M_{n}, e_{x}⟩ < tb_{n} | S_{n} = x)}{∫_{⟨u,e_{x}⟩<t} g(u)du} − 1| = 0.

It is useful to compare Theorem 2 and Corollary 2 with the corresponding assertions in A. Nagaev and Zaigraev (1998) (see their Theorem 2 and Corollary 4).

3. Proofs

In what follows it is supposed that n → ∞, |x| ≥ c_{n}b_{n}, c_{n} → ∞. In particular, saying ”uniformly in x, e_{x} ∈ A” we also mean that |x| ≥ c_{n}b_{n}, n → ∞.

Proof of Theorem 1 It should be noted that the proof follows the general scheme given in A. Nagaev and Zaigraev (1998) (see also A. Nagaev (1969)).

It is worth beginning with two relations playing a crucial role in the proof. Let p ∈ P_{γ,h} and |u| = o(|x|) as |x| → ∞. Then

p(x − u) ∼ p(x) ≍ r_{γ}(|x|)  (8)

uniformly in x, e_{x} ∈ E_{ϵ} while

p(x − u) = o(r_{γ}(|x|)), p(x) = o(r_{γ}(|x|))  (9)

uniformly in x, e_{x} ∉ E. Here f(x) ∼ g(x) and f(x) ≍ g(x) mean, respectively, that

f(x)/g(x) ⟶ 1 as |x| → ∞ and 0 < lim inf_{|x|→∞} f(x)/g(x) ≤ lim sup_{|x|→∞} f(x)/g(x) < ∞.

Let y_{n} → ∞ as n → ∞. Consider the events

A_{0} = {⟨ξ_{j}, e_{x}⟩ < y_{n}, j = 1, …, n},

A_{1} = {⟨ξ_{j}, e_{x}⟩ < y_{n}, j = 1, …, n − 1, ⟨ξ_{n}, e_{x}⟩ ≥ y_{n}},

A_{2} = ⋃_{k,j=1,k≠j}^{n}{⟨ξ_{k}, e_{x}⟩ ≥ y_{n}, ⟨ξ_{j}, e_{x}⟩ ≥ y_{n}}.

We choose y_{n} so that b_{n} = o(y_{n}) while y_{n} = o(|x|). It is easily seen that y_{n} = |x|(b_{n}/|x|)^{q} with q ∈ (0, 1) satisfies this requirements. In contrast to the mentioned papers here we have to be more careful when choosing the truncation level.

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

p_{n}(x) = p_{n0}(x) + np_{n1}(x) + p_{n2}(x)  (10)

where

p_{nj}(x) = lim_{|Δ|→0} |Δ|^{−1}P (x ⪯ S_{n} ≺ x + Δ, A_{j}) , j = 0, 1, 2,

and ≺ (⪯) means the componentwise ordering. It should be noted that p_{n1}(x) and p_{n2}(x) are estimated, in essence, as in A. Nagaev and Zaigraev (1998) while all the alterations concern the estimation of p_{n0}(x).

Let ξ̃ be a random vector having density

p̃(u) = \{ \begin{matrix}(P (⟨ξ, e_{x}⟩ < y_{n}))^{−1}p(u) & if ⟨u, e_{x}⟩ < y_{n} \\ 0 & otherwise \end{matrix} .  (11)

Note that due to (7) and the choice of y_{n} we have

nP (⟨ξ, e_{x}⟩ ≥ y_{n}) = O(nr_{α}(y_{n})) = o(1).  (12)

Lemma 1. Under the conditions of Theorem 1 for any k ≥ 1

|E⟨ξ̃, e⟩^{k}| = O(y_{n}^{k−α}l(y_{n}))

uniformly in e ∈ S^{d−1}, where l(t) varies slowly at infinity.

Proof. First, consider the case k = 1. For α ∈ (1, 2)

∫_{⟨u,e⟩<y_{n}}⟨u, e⟩p(u)du = − ∫_{⟨u,e⟩≥y_{n}}⟨u, e⟩p(u)du

yields

|E⟨ξ̃, e⟩| = O(∫_{⟨u,e⟩≥y_{n}}⟨u, e⟩p(u)du).

After performing a change of variables u = T v, where T is an orthogonal transformation such that v_{1} = ⟨u, e⟩, we get

∫_{⟨u,e⟩≥y_{n}}⟨u, e⟩p(u)du = ∫_{y_{n}}^{∞} v_{1}dv_{1} ∫ … ∫ r_{γ}(|v|)(h(e_{T v}) + w(T v))dv_{2} … dv_{d}.

The next change of variables v_{2} = v_{1}y_{1}, …, v_{d} = v_{1}y_{d−1}, fixed v_{1}, implies

∫_{⟨u,e⟩≥y_{n}}⟨u, e⟩p(u)du = ∫_{y_{n}}^{∞} \frac{dv_{1}}{v_{1}^{α}} ∫_{R^{d−1}} \frac{l(v_{1}|ỹ|)}{|ỹ|^{γ}}(h(e_{T _{y}}) + w(v_{1}T ỹ))dy

where ỹ = (1, y)′, y ∈ R^{d−1}, and the function l(t) varies slowly at infinity. Therefore,

∫_{⟨u,e⟩≥y_{n}}⟨u, e⟩p(u)du = O(∫_{y_{n}}^{∞} r_{α}(t)dt)

and

|E⟨ξ̃, e⟩| = O(∫_{y_{n}}^{∞} r_{α}(t)dt) = O(y_{n}^{1−α}l(y_{n}))

uniformly in e ∈ S^{d−1}.

The same relation holds in the case α ∈ (0, 1). Indeed, in a similar manner we obtain

|∫_{⟨u,e⟩<y_{n}}⟨u, e⟩p(u)du| = O(∫_{0}^{y_{n}} r_{α}(t)dt) = O(y_{n}^{1−α}l(y_{n})).

For k > 1 in both cases

|∫_{⟨u,e⟩<y_{n}}⟨u, e⟩^{k}p(u)du| = O(∫_{0}^{y_{n}} r_{α+1−k}(t)dt) = O(y_{n}^{k−α}l(y_{n})).

The lemma is proved.

Keeping in mind representation (10), we divide the proof of Theorem 1 into three parts.

Lemma 2. Under the conditions of Theorem 1 for any ε > 0

lim_{n→∞} sup_{|x|≥c_{n}b_{n}} sup_{e_{x}∈E_{ε}} |\frac{p_{n1}(x)}{p(x)} − 1| = 0

and

lim_{n→∞} sup_{|x|≥c_{n}b_{n}} sup_{e_{x}∉E} \frac{p_{n1}(x)}{r_{γ}(|x|)} = 0.

Proof. Let ξ̃_{1}, ξ̃_{2}, … be i. i. d. random vectors having density (11) and denote S̃_{n} = ξ̃_{1} + ⋯ + ξ̃_{n}. Let p̃_{n}(x) be the density of S̃_{n}. By definition,

p_{n1}(x) = (P (⟨ξ, e_{x}⟩ < y_{n}))^{n−1} ∫_{B} p̃_{n−1}(u)p(x − u)du,  (13)

where B = {u ∈ R^{d} : ⟨u, e_{x}⟩ ≤ |x| − y_{n}}. In view of (12),

(P (⟨ξ, e_{x}⟩ < y_{n}))^{n} = 1 + o(1).  (14)

If n is sufficiently large then we can partition B into

B_{1} = {u ∈ B : |u| < y_{n}}, B_{2} = {u ∈ B : |u| ≥ y_{n}}.

For e_{x} ∈ E_{ε}, due to (8),

∫_{B_{1}} p̃_{n−1}(u)p(x − u)du = P (|S̃_{n−1}| < y_{n})p(x)(1 + o(1)),  (15)

while for e_{x} ∉ E in view of (9)

∫_{B_{1}} p̃_{n−1}(u)p(x − u)du = o(r_{γ}(|x|)).  (16)

On the other hand,

∫_{B_{2}} p̃_{n−1}(u)p(x − u)du = O(P (|S̃_{n−1}| ≥ y_{n})).  (17)

It remains to establish the relation

P (|S̃_{n}| ≥ y_{n}) = o(1)  (18)

which follows from

P (|S_{n}| ≥ y_{n}) = (P (⟨ξ, e_{x}⟩ < y_{n}))^{n}P (|S̃_{n}| ≥ y_{n}) + O(nP (⟨ξ, e_{x}⟩ ≥ y_{n})) due to (12), (14) and the fact that P (|S_{n}| ≥ y_{n}) → 0 since b_{n} = o(y_{n}).

Then the assertion of the lemma follows from (13)–(18). Lemma 3. Under the conditions of Theorem 1,

p_{n2}(x) = o(nr(|x|))

uniformly in x. Proof. By definition,

p_{n2}(x) ≤ n^{2} ∫_{⟨u,e_{x}⟩≥y_{n}} p(u)du ∫_{⟨v,e_{x}⟩≥y_{n}} p_{n−2}(x − u − v)p(v)dv.

Clearly,

∫_{⟨v,e_{x}⟩≥y_{n}} p_{n−2}(x − u − v)p(v)dv = O(r_{γ}(y_{n})).

Therefore, in view of (7)

p_{n2}(x) = O(n^{2}r_{γ}(y_{n})P (⟨ξ, e_{x}⟩ ≥ y_{n})) = O(n^{2}r_{γ}(y_{n})r_{α}(y_{n})) = o(nr_{γ}(|x|)) if we choose q ∈ (0, (γ + α)^{−1}α). The lemma is proved.

Lemma 4. Under the conditions of Theorem 1,

p_{n0}(x) = o(nr_{γ}(|x|))

uniformly in x.

Proof. Consider the moment generating function

f(s) = ∫_{⟨u,e_{x}⟩<y_{n}} e^{s⟨u,e_{x}⟩}p(u)du.

Let ξ(s), ξ_{1}(s), ξ_{2}(s), … be i. i. d. random vectors having density

p_{s}(u) = \{ \begin{matrix}(f(s))^{−1}e^{s⟨u,e_{x}⟩}p(u) & if ⟨u, e_{x}⟩ < y_{n} \\ 0 & otherwise \end{matrix} .

Denote by p_{s}^{(n)}(x) the density of ξ_{1}(s) + ⋯ + ξ_{n}(s). It is easily seen that

p_{n0}(u) = e^{−s⟨u,e_{x}⟩}f^{n}(s)p_{s}^{(n)}(u)

and, in particular,

p_{n0}(x) = e^{−s|x|}f^{n}(s)p_{s}^{(n)}(x).

Set s = y_{n}^{−1}. Obviously, sb_{n} → 0 and s|x| → ∞.

We will prove that

f(s) = 1 + o(n^{−1})  (19)

and

sup_{u} p_{s}^{(n)}(u) = O(b_{n}^{−d}).  (20)

If (19) and (20) hold then the lemma follows since

p_{n0}(x) = O(b_{n}^{−d}e^{y_{n}^{−1}|x|}) = o(nr(|x|))

uniformly in x.

Let us represent

f(s) = ∫_{⟨u,e_{x}⟩<y_{n}} p(u)du + ∫_{⟨u,e_{x}⟩<y_{n}}(e^{s⟨u,e_{x}⟩} − 1)p(u)du = f_{1}(s) + f_{2}(s).

Due to (7), f_{1}(s) = 1 + o(n^{−1}). Furthermore, in view of Lemma 1

|f_{2}(s)| ≤ ∫_{⟨u,e_{x}⟩<y_{n}} |e^{s⟨u,e_{x}⟩} − 1|p(u)du ≤ ∑_{k=1}^{∞} \frac{s^{k}}{k!} ∫_{⟨u,e_{x}⟩<y_{n}} |⟨u, e_{x}⟩^{k}|p(u)du = o(n^{−1}) (21)

and (19) follows.

It remains to prove (20). Let ϕ_{s}(t) and ϕ(t) be the characteristic functions of ξ(s) and ξ, respectively. From (12) and (21) it follows that

|ϕ_{s}(t) − ϕ(t)| = |(f(s))^{−1} ∫_{⟨u,e_{x}⟩<y_{n}} e^{i⟨u,t⟩}e^{s⟨u,e_{x}⟩}p(u)du − ∫_{R^{d}} e^{i⟨u,t⟩}p(u)du|
= O(|∫_{⟨u,e_{x}⟩<y_{n}} e^{i⟨u,t⟩}(e^{s⟨u,e_{x}⟩} − 1)p(u)du − ∫_{⟨u,e_{x}⟩≥y_{n}} e^{i⟨u,t⟩}p(u)du|)
= O(∫_{⟨u,e_{x}⟩<y_{n}} |e^{s⟨u,e_{x}⟩} − 1|p(u)du + ∫_{⟨u,e_{x}⟩≥y_{n}} p(u)du) = o(n^{−1})

uniformly in t ∈ R^{d}. Therefore,

|ϕ_{s}(t)|^{n} ≤ |ϕ(t)|^{n−1}(|ϕ(t)| + w_{n}(t))  (22)

uniformly in t ∈ R^{d}, where sup_{t} |w_{n}(t)| → 0, n → ∞.

By definition sup_{u} p(u) < ∞. Hence ∫_{R^{d}} |ϕ(t)|^{n}dt < ∞, n ≥ 2. Moreover, under the

above choice of s we have sup_{u} p_{s}(u) < ∞ that is ∫_{R^{d}} |ϕ_{s}(t)|^{n}dt < ∞, n ≥ 2. Due to

(22) and the inversion formula,

sup_{u} p_{s}^{(n)}(u) = O( ∫_{R^{d}} |ϕ_{s}(t)|^{n}dt)= O(∫_{R^{d}} |ϕ(t)|^{n}dt) = O(b_{n}^{−d} ∫_{R^{d}} |ϕ(b_{n}^{−1}t)|^{n}dt).

Let g^{∗}(u) be the symmetrization of g(u) (see (3)) that is g^{∗}(u) = ∫_{R^{d}} g(u + v)g(v)dv. By Plancherel’s equality and the classical local limit theorem,

\frac{1}{2π} ∫_{R^{d}} |ϕ(b_{2n}^{−1}t)|^{2n}dt = g^{∗}(0)(1 + o(1)).

Since b_{n}^{−1}b_{2n} → 2^{1/α} the lemma follows.

The assertions of Theorem 1 follow immediately from (10) and Lemmas 2–4.

Proof of Theorem 2 Let A_{0}, A_{1} and A_{2} be as above. In view of Lemmas 3 and 4 for any event A′

lim_{|Δ|→0} \frac{P (x ⪯ S_{n} ≺ x + Δ; A_{j}; A′)}{P (x ⪯ S_{n} ≺ x + Δ)} ≤ \frac{p_{nj}(x)}{p_{n}(x)}, j = 0, 2.  (23)

On the other hand, from the proof of Lemma 2 it follows that

P (b_{n}^{−1}(S_{n} − M_{n}) ∈ A | S_{n} = x) ∼ (p(x))^{−1} ∫_{(b_{n}A)∩B} p̃_{n−1}(u)p(x − u)du

∼ P (b_{n−1}^{−1}S̃_{n−1} ∈ A) ∼ G(A)  (24)

uniformly in x, e_{x} ∈ E_{ϵ}. The last relation utilizes arguments similar to those used for establishing (18). Theorem 2 easily follows from (23) and (24).

Acknowledgements. The author would like to thank A. V. Nagaev, who initiated the research as well as T. Mikosch and the referee for careful reading of the paper and valuable comments.

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Alexander Zaigraev

Faculty of Mathematics and Informatics Nicholas Copernicus University Chopina 12/18, 87–100 Toruń Poland e-mail: alzaig@mat.uni.torun.pl