Multidimensional limit theorems allowing large deviations for densities of regular variation.

Alexander V. Nagaev, Alexander Yu. Zaigraev^{∗}

Nicholas Copernicus University

Large deviations

Alexander Yu. Zaigraev

Faculty of Mathematics and Informatics, Nicholas Copernicus University, ul.Chopina 12/18, 87-100 Toruń, Poland

e-mail: alzaig@mat.uni.torun.pl ABSTRACT: The sums of i.i.d. random vectors are considered. It is assumed that the underlying distribution is absolutely continuous and its density possesses the property which can be referred to as regular variation. The asymptotic expressions for the probability of large deviations are established in the case of a normal limiting law. Furthermore, the role of the maximal summand is emphasized.

∗supported by the KBN Grant PB 591/P03/95/08

Key Words and Phrases: sums of i.i.d. random variables, maximal summand, local theorem for densities, marginal density.

AMS Classification Numbers: Primary 60F10; Secondary 60G50.

1. Introduction

Let ξ, ξ_{1}, ξ_{2} … be i.i.d. random variables taking values in R^{d}. Suppose that Eξ = 0, E|ξ|^{2} < ∞ and write B for the covariance matrix of ξ. Consider the partial sums

S_{n} = ξ_{1} + ⋯ + ξ_{n}, n = 1, 2 ….

Let A ⊂ R^{d} be a set of continuity with respect to Lebesgue measure. From the central limit theorem (c.l.t.) it follows that, as n → ∞,

P (n^{−1/2}S_{n} ∈ A) → ∫_{A} ϕ_{B}(u)du,

where ϕ_{B}(u) is the density of a mean–zero Gaussian vector with covariance matrix B.

We say that a value x = x_{n}, taken by S_{n}, is a large deviation (from the origin) if n^{−1/2}|x| → ∞. Similarly, P_{n}(A) = P (S_{n} ∈ A) is called a probability of large deviation if A depends on n and r_{n} = n^{−1/2} inf_{x∈A} |x| → ∞. It is convenient to refer to r_{n} as the order of the large deviation. The c.l.t. implies that a large deviation probability always converges to zero as n → ∞. It is the asymptotic behaviour of P_{n}(A) that is in the focus of the large deviations theory.

One should distinguish between three classes of problems within this theory. The first class unites those related to the asymptotic expression of ln P_{n}(A) which is called a rough (or coarse) asymptotic expression. It is worth noting that such expressions can be established under rather general conditions, e.g. for variables taking values in various infinite–dimensional spaces, for dependent variables etc. The present state of the work in this direction can be found in [4]–[6], etc.

The problems, which constitute the second class, deal with the precise asymptotic of P_{n}(A). Unfortunately, we can recommend no overview of the works devoted to it. It remains only to mention a number of those who contributed to the development of this part of the theory. Among them are Borovkov and Rogozin [2] (cf. also [11]), Borovkov and Mogulskii [3], von Bahr [1], Osipov [12], Rozovskii [15]. Of course, this list is far from being complete.

The third direction, in essence, goes back to Chebyshev’s inequality. Here the interest is focussed on upper bounds for P_{n}(A). The survey of relevant work is in [10].

The problems considered in the present paper are related to the second class under the above classification. It should be noted that the case of R^{d} with d > 1 allows for a large number of possible formulations of the large deviation problem. The fact is that the sets A_{n} of interest in R^{1} are intervals, finite or infinite, while the variety of sets in R^{d}, which look quite natural from the viewpoint of further applications, is incomparably richer.

The scope of the paper covers the case in which the underlying distribution belongs to the domain of attraction of a normal law, but has the tails which are heavier than it is allowed by the so-called Cramer condition. The case of an infinite variance stable limiting distribution will be investigated elsewhere.

We would like to emphasize that the simplest way here is to derive integral theorems as direct corollaries of the local ones. The latter ones are much more definite. At first sight, such an approach, which was first used in [2] (see also [9]), seems very particular. However, within the context of large deviations of arbitrarily high order all other approaches are hardly tractable.

It is well known, see e.g. [13], [17], that the direct asymptotic analysis, which proved to be very efficient in the case d = 1, remains a powerful tool for higher dimensions. The present paper confirms it once again.

It is also well known that in the case of heavy tails large deviations are due to one single ”abnormally” large summand. Therefore it is important to fix this rather vague statement in a mathematically rigorous form. We do this through the conditional joint distribution of the maximal summand and the maximum of the remaining ones given S_{n} enjoys a large deviation. The notion maximal summand is understood in the sense that it has the largest projection in the direction of x, that is on e_{x} = |x|^{−1}x (cf. [8]).

The paper is organized as follows. In Section 2 we collect some properties of the densities which can be referred to as those of regular variation. Our main results are formulated in Section 3. In Sections 4 and 5 we continue with their proofs. In Section 6 we give some concluding remarks.

2. Densities of regular variation in R^{d}

First, we are going to outline a certain class of absolutely continuous distributions having heavy tails.

Let non-negative functions r and h be defined, respectively, on [0, ∞) and the unit sphere S^{d−1}. Consider the density of the form

p(x) = r(|x|)h(e_{x}), e_{x} = |x|^{−1}x, x ∈ R^{d}.  (1)

In what follows x^{(i)} denotes the ith component of a vector x while the subscripts aim to distinguish between vectors.

Denote by E ⊂ S^{d−1} the support of h. From (1) it follows that p is supported by the cone {x ∈ R^{d} : e_{x} ∈ E}. When h(e) ≡ const we come to the so-called spherically contoured distributions. If ξ has a density (1) then |ξ| and e_{ξ} are independent.

Definition (cf 5.4.2 in [14]). We say that p ∈ P = P_{γh} if p is uniformly bounded in R^{d} and admits the representation

p(x) = r(|x|)(h(e_{x}) + w(x))

where

a) r(t) is of regular variation as t → ∞, with exponent −γ, γ > d;

b) h(e) is continuous, h(e) ≢ 0;

c) w(x) → 0 as |x| → ∞, uniformly in e_{x}.

It is easily seen that within P the events (|ξ| > t) and (e_{ξ} ∈ A) are independent only asymptotically as t → ∞. More precisely, if p ∈ P then for any measurable set A ⊂ S^{d−1}

P (|ξ| > t, e_{ξ} ∈ A) = r_{1}(t) (∫_{A} h(e)λ(de) + w_{A}(t))  (2)

where r_{1}(t) is of regular variation as t → ∞, with exponent d − γ, λ is the uniform measure on the σ-algebra A of measurable subsets of S^{d−1}, and

sup_{A∈A} |w_{A}(t)| → 0 as t → ∞.

If we replace in (2) the absolutely continuous measure ∫_{A} h(e)λ(de) with an arbitrary measure H on A, then for d < γ < d + 2 we come to the necessary and sufficient conditions that the underlying distribution belongs to the domain of attraction of the limiting law referred to as the stable law in the sense of Feldheim-Levy with a parameter α = γ − d, while for γ ≥ d + 2 the underlying distribution belongs to the domain of attraction of the normal law (see e.g. [16]).

Let τ = arg max_{1≤j≤n} |ξ_{j}|, b_{n} = inf{t : nP (|ξ| > t) ≤ 1}. From (2) it follows,

in particular, that

P (|ξ_{τ}| < tb_{n}, e_{ξ_{τ}} ∈ A) ⟶ μ(A)e^{−t^{d−γ}}, n → ∞,  (3)

where μ(A) = (∫_{S^{d−1}} h(e)λ(de))^{−1} ∫_{A} h(e)λ(de) (more about extreme values see in [14]).

The simplest example of p ∈ P is the following

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

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

It is of interest to write up the density of one-dimensional marginal distribution or, more generally, that of ξ_{e} = ⟨e, ξ⟩, e ∈ S^{d−1}. Since now on ⟨·, ·⟩ denotes the inner product. If p ∈ P then p_{e}(t), the density of ξ_{e}, has the form

p_{e}(t) = r_{2}(t)(C_{e}(h) + w_{e}(t))

where r_{2}(t) is of regular variation as t → ∞ with exponent d−γ−1, w_{e}(t) → 0 as t → ∞, uniformly in e ∈ S^{d−1}. Here

C_{e}(h) = ∫_{R^{d−1}} h(e_{e+Qy})(1 + |y|^{2})^{−γ/2}dy  (4)

where Q is a d × (d − 1) matrix such that e^{T} Q = 0 and Q^{T} Q is the unit matrix.

We need also the following elementary property of a density belonging to P. Let A_{ρ} = {x ∈ R^{d} : |x| < ρ(e_{x})} and ρ be a positive continuous function on S^{d−1}, which is separated from zero, that is, inf_{e∈S^{d−1}} ρ(e) > 0. If p ∈ P then

P (t^{−1}ξ ∉ A_{ρ}) = r_{1}(t)(C(ρ, h) + w_{ρ}(t))

where w_{ρ}(t) → 0 as t → ∞, uniformly in ρ. Here

C(ρ, h) = (γ − d)^{−1} ∫_{S^{d−1}} ρ(e)^{d−γ}h(e)|J_{e}|de

where r^{d−1}J_{e} is the Jakobian corresponding to the transformation (x^{(1)}, …, x^{(d)}) = r(e^{(1)}, …, e^{(d)}).

3. Main Results

Let p_{n}(x) be the density of S_{n} = ξ_{1} + ξ_{2} + ⋯ + ξ_{n}. Theorem 1. If p ∈ P with γ > d + 2 then for any ϵ > 0, as n → ∞,

p_{n}(x) = np(x)(1 + o(1))

uniformly in x, |x| ≥ n^{1/2} ln n, e_{x} ∈ E_{ϵ} = {e ∈ S^{d−1} : h(e) ≥ ϵ}. Furthermore,

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

uniformly in x, |x| ≥ n^{1/2} ln n, e_{x} ∉ E. The approach based on local theorems allows great variety of settings. We start with that suggested by Yu.Prokhorov (cf. [17]). Corollary 1. Let A_{ρ} be as in Section 2. Under the conditions of Theorem 1, as n → ∞,

P (t^{−1}S_{n} ∉ A_{ρ}) = nP (t^{−1}ξ ∉ A_{ρ})(1 + o(1))

provided t ≥ n^{1/2} ln n.

The next corollary presents another setting of the problem. Let mes(A) stands for the Lebesgue measure of a Borel set A. Corollary 2. Let A_{n} be the class of the Borel sets defined as

A_{n} = {A ⊂ R^{d} : sup_{u∈A} |u| ≤ a_{n}, a_{n} → ∞}.

Under the conditions of Theorem 1 for any ϵ > 0, as n → ∞,

sup_{A∈A_{n}} | \frac{P (S_{n} − x ∈ A)}{np(x)} − mes(A) |⟶ 0

uniformly in x, |x| ≥ n^{1/2} ln n, a_{n} = o(|x|), e_{x} ∈ E_{ϵ}.

If a_{n} ≡ const then Corollary 2 may be regarded as a local theorem. But if n^{−1/2}a_{n} → ∞ then it is a version of an integral theorem allowing large deviations.

It is of interest to establish the limiting law for the conditional distribution of ⟨e_{2}, S_{n}⟩ given a large deviation of ⟨e_{1}, S_{n}⟩, where e_{1}, e_{2} ∈ E, |⟨e_{1}, e_{2}⟩| < 1. Let P be the projector of R^{d} onto the subspace {x ∈ R^{d} : ⟨e_{1}, x⟩ = 0}. We set e_{1,2} = |P e_{2}|^{−1}P e_{2}. Denote by Q_{1} a d × (d − 2) matrix such that (cf. (4)) e_{1}^{T} Q_{1} = e_{12}^{T}Q_{1} = 0 and Q_{1}^{T} Q_{1} is the unit matrix.

Corollary 3. Under the conditions of Theorem 1, as n → ∞,

P (⟨e_{2}, S_{n}⟩ − t⟨e_{1}, e_{2}⟩ < tz|P e_{2}| | ⟨e_{1}, S_{n}⟩ = t) ⟶ ∫_{−∞}^{z} q(u)du

provided t ≥ n^{1/2} ln n, uniformly in z ∈ R^{1}, where

q(u) = \frac{(1 + u^{2})^{−(γ−d+2)/2} ∫_{R^{d−2}} h(e_{e(u)+Q_{1}v})(1 + |v|^{2})^{−γ/2}dv}{∫_{R^{d−1}} h(e_{e_{1}+Qy})(1 + |y|^{2})^{−γ/2}dy}

and e(u) = (1 + u^{2})^{−1/2}(e_{1} + ue_{1,2}) (cf. (4)).

Let ν = ν_{nx} = min {k : ⟨e_{x}, ξ_{k}⟩ = max_{1≤j≤n} ⟨e_{x}, ξ_{j}⟩}, θ = arg max_{1≤j≤n,j≠ν} |ξ_{j}|.

The following theorem highlights the role played by the maximal summand ξ_{ν} under the large deviation of S_{n} (cf. (3)).

Theorem 2. If p ∈ P with γ > d + 2 then for any Borel sets A, A′ and any ϵ > 0, as n → ∞,

sup_{t} | P (|ξ_{θ}| < tb_{n}; e_{ξ_{θ}} ∈ A; n^{−1/2}(ξ_{ν} − x) ∈ A′ | S_{n} = x)−μ(A)e^{−t^{d−γ}} ∫_{A′} ϕ_{B}(u)du |= o(1)

uniformly in x, |x| ≥ n^{1/2} ln n, e_{x} ∈ E_{ϵ}. Here b_{n} = inf{t : nP (|ξ| > t) ≤ 1}.

As a corollary we obtain the limiting law for the conditional distribution of the maximal projection of summands onto the deviation direction given a large deviation of the sum.

Corollary 4. Under the conditions of Theorem 2 for any ϵ > 0, as n → ∞,

P (⟨e_{x}, ξ_{ν}⟩ − |x| < tn^{1/2}⟨Be_{x}, e_{x}⟩^{1/2} | S_{n} = x) ⟶ (2π)^{−1/2} ∫_{−∞}^{t} e^{−u^{2}/2}du, n → ∞,

uniformly in t ∈ R^{1} and x, |x| ≥ n^{1/2} ln n, e_{x} ∈ E_{ϵ}.

4. Proof of Theorem 1

First, notice that for |u| = o(|x|) we have as |x| → ∞

p(x − u) ∼ p(x) ≍ r(|x|)  (5)

uniformly in e_{x} ∈ E_{ϵ}, and

p(x − u) = o(r(|x|)), p(x) = o(r(|x|))  (6)

uniformly in 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) <

∞.

Consider the events

A_{0} = {⟨e_{x}, ξ_{j}⟩ < δ|x|, j = 1, …, n}, A_{1} = {⟨e_{x}, ξ_{j}⟩ < δ|x|, j = 1, …, n−1, ⟨e_{x}, ξ_{n}⟩ ≥ δ|x|},

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

where δ ∈ (0, 1) is fixed.

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

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

where

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

and ≺ (⪯) means the componentwise ordering.

Consider i.i.d. random vectors {ξ̃, {ξ̃_{i}}_{i≥1}} taking values in {u ∈ R^{d} : ⟨e_{x}, u⟩ < δ|x|} with the density

p̃(u) = (P (⟨e_{x}, ξ⟩ < δ|x|))^{−1}p(u)

and denote S̃_{n} = ξ̃_{1}+⋯+ξ̃_{n}. It is easily seen that for |x| ≥ n^{1/2} ln n, n → ∞, we have E|ξ̃|^{2} → E|ξ|^{2}. Furthermore, since

∫_{⟨e_{x},u⟩<δ|x|} u^{(i)}p(u)du = − ∫_{⟨e_{x},u⟩≥δ|x|} u^{(i)}p(u)du, i = 1, …, d,

we have for all sufficiently large n

|Eξ̃| = O (∫_{|u|≥δ|x|} |u|p(u)du) = O (|x|^{−1} ∫_{|u|≥δ|x|} |u|^{2}p(u)du) = o (|x|^{−1}) = o (n^{−1/2}) .

Therefore,

P (|S̃_{n}| < z) ⟶ 1  (7)

provided n^{−1/2}z → ∞.

Lemma 1. If p ∈ P with γ > d + 2 then for any ϵ > 0 and |x| ≥ n^{1/2} ln n

p_{n1}(x) = p(x)(1 + o(1))

uniformly in e_{x} ∈ E_{ϵ}, while

p_{n1}(x) = o(r(|x|))

uniformly in e_{x} ∉ E.

Proof. By definition,

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

where p̃_{n}(x) denotes the density of S̃_{n}. Applying Chebyshev’s inequality yields

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

uniformly in e_{x} ∈ S^{d−1}.

Let I_{1} = {u ∈ R^{d} : |u| < n^{1/2}r_{n}}, I_{2} = {u ∈ R^{d} : |u| ≥ n^{1/2}r_{n}, ⟨e_{x}, u⟩ ≤ (1 − δ)|x|} where r_{n} → ∞, r_{n} = o(ln n). For u ∈ I_{1} and |x| ≥ n^{1/2} ln n we have due to (5)–(7)

∫_{I_{1}} p̃_{n−1}(u)p(x − u)du ∼ \{ \begin{matrix}p(x) & if e_{x} ∈ E_{ϵ} \\ o(r(|x|)) & if e_{x} ∉ E. \end{matrix}

At the same time, for u ∈ I_{2}

∫_{I_{2}} p̃_{n−1}(u)p(x − u)du = O(r(|x|)) ∫_{I_{2}} p̃_{n−1}(u)du = o(r(|x|))

as |x| ≥ n^{1/2} ln n. The lemma is proved.

Lemma 2. If p ∈ P with γ > d + 2 then for |x| ≥ n^{1/2} ln n

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

uniformly in e_{x} ∈ S^{d−1}. Proof. Obviously,

p_{n2}(x) ≤ n^{2} ∫_{⟨e_{x},u⟩≥δ|x|} p(u)I(u, x)du,

where I(u, x) = ∫_{⟨e_{x},v⟩≥δ|x|} p_{n−2}(x−u−v)p(v)dv. Due to (5) and (6), I(u, x) = O(r(|x|)) uniformly in u, ⟨e_{x}, u⟩ ≥ δ|x|. Since nP (⟨e_{x}, ξ⟩ ≥ δ|x|) = o(1) as |x| ≥ n^{1/2} ln n, the lemma follows. Lemma 3. If p ∈ P with γ > d + 2 then for |x| ≥ n^{1/2} ln n

p_{n0}(x) = o(nr(|x|))

uniformly in e_{x} ∈ S^{d−1}. Proof. Consider the moment generating function

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

Let i.i.d. random vectors {ξ(s), {ξ_{i}(s)}_{i≥1}} take values in {u ∈ R^{d} : ⟨e_{x}, u⟩ < δ|x|} and have the density

p_{s}(u) = f(s)^{−1}e^{s⟨e_{x},u⟩}p(u).

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

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

and, in particular,

p_{n0}(x) = f^{n}(s)e^{−s|x|}p_{s}^{(n)}(x).  (8)

We set

s = −|x|^{−1} ln(n^{1/2}p(x)).  (9)

Obviously, for |x| ≥ n^{1/2} ln n we have s → 0 and s|x| → ∞. It is easily seen that

f(s) = ∫_{−∞}^{δ|x|} e^{sz}p_{e_{x}}(z)dz

where, as before, p_{e_{x}}(z) is the marginal density of ⟨e_{x}, ξ⟩.

Arguing as in [7], we get

f(s) = 1 + O(n^{−1})  (10)

provided 0 < δ < (γ − d − 2)(γ − 1)^{−1}.

Further, for all sufficiently large n

max_{γs^{−1}<⟨e_{x},u⟩<δ|x|} p_{s}(u) ≤ 2 max_{e∈S_{d−1}} h(e) max_{γs^{−1}<⟨e_{x},u⟩<δ|x|} e^{s⟨e_{x},u⟩} | ⟨e_{x}, u⟩ |^{−γ+ϵ_{1}}

for any ϵ_{1} > 0. Since t^{−γ+ϵ_{1}}e^{st} increases within (γs^{−1}, δ|x|), we have

max_{γs^{−1}<⟨e_{x},u⟩<δ|x|} e^{s⟨e_{x},u⟩} | ⟨e_{x}, u⟩ |^{−γ+ϵ_{1}} ≤ δ^{−γ+ϵ_{1}}|x|^{−γ+ϵ_{1}}e^{δs|x|}.

Under our choice

s ≤ |x|^{−1}(γ − ϵ_{1}) ln |x|.

Hence,

|x|^{−γ+ϵ_{1}}e^{δs|x|} ≤ |x|^{−(γ−ϵ_{1})(1−δ)} = o(1).

Obviously,

max_{⟨e_{x},u⟩≤γs^{−1}} p_{s}(u) ≤ 2e^{γ} max_{u∈R^{d}} p(u) < ∞.

Thus, for all sufficiently large n

max_{u} p_{s}(u) ≤ C < ∞

for some C > 0 and, hence,

sup_{u} p_{s}^{(n)}(u) ≤ C.  (11)

It remains to combine (8)–(11). The lemma is proved.

The statements of Theorem 1 follow immediately from Lemmas 1–3.

5. Proof of Theorem 2

Let A_{0}, A_{1} and A_{2} be as in the proof of Theorem 1. It is easily seen that for any event A′′

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

Therefore, taking into account Lemmas 1–3, we have for |x| ≥ n^{1/2} ln n as n → ∞

P (|ξ_{θ}| < tb_{n}; e_{ξ_{θ}} ∈ A; n^{−1/2}(ξ_{ν} − x) ∈ A′ | S_{n} = x) ∼

(p(x))^{−1}P^{n−2}(|ξ| < tb_{n}) ∫_{|v|<tb_{n},e_{v}∈A} p(v) ∫_{n^{−1/2}(u−x)∈A′} p̃_{n−2}(x−u−v)p(u)dudv ∼

P^{n−2}(|ξ| < tb_{n}) ∫_{|v|<tb_{n},e_{v}∈A} p(v) ∫_{n^{−1/2}z∈A′} p̃_{n−2}(z − v)dzdv =
P^{n−2}(|ξ| < tb_{n}) ∫_{|v|<tb_{n},e_{v}∈A} P (n^{−1/2}S̃_{n−2} + n^{−1/2}v ∈ A′)p(v)dv.

In view of (3) and c.l.t.

P^{n−2}(|ξ| < tb_{n}) ∫_{|v|<tb_{n},e_{v}∈A} P (n^{−1/2}S̃_{n−2}+n^{−1/2}v ∈ A′)p(v)dv → e^{−t^{d−γ}}μ(A) ∫_{A′} ϕ_{B}(u)du.

The theorem is proved.

6. Concluding Remarks

1. It should be noted that the assumption on the continuity of h(e) in Theorem 1 is essential. Let d = 2. Then e = e_{ϕ} = (cos ϕ, sin ϕ), 0 ≤ ϕ < 2π, and h(e) = h(e_{ϕ}). Assume that h(e_{ϕ}) is continuous except for ϕ = ϕ_{0}. Assume, further, that there exist

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

Then it is easy to prove that

p_{n}(x) = 2^{−1}(h^{+} + h^{−})nr(|x|) as |x| ≥ n^{1/2} ln n, n → ∞

provided h^{+} + h^{−} > 0, e_{x} = e_{ϕ_{0}}.

2. Let A be a matrix such that det A > 0. It is easy to see that all the results presented above could be restated for AS_{n} as well.

3. It should be noted that the asymptotic properties of the marginal densities of ⟨e_{1}, ξ⟩ and (⟨e_{1}, ξ⟩, ⟨e_{1,2}, ξ⟩) are established by means of straightforward analysis. First, we implement an appropriate orthogonal transformation and then it suffices to utilize the properties of slowly varying functions.

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