Multivariate large deviations for sums of i. i. d. random vectors with compactly supported

distribution

Alexander Zaigraev (Toruń)

Abstract. The sums of i.i.d. random vectors with compactly supported and absolutely continuous distribution are considered. Under some conditions the strong form of the local limit theorem for large deviations is proved. In passing the asymptotic behaviour of the moment generating function as well as possible non-degenerate limit laws for the natural exponential family of distributions are established.

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

Key words and phrases: Abel theorem, Cramér transformation, Fenchel-Legen-dre transformation, Laplace method, local limit theorem, natural exponential family of distributions.

1 Introduction

Let ξ, ξ^{(1)}, …, ξ^{(n)}, … be independent identically distributed random vectors taking values in R^{d}, d ≥ 1. Denote S_{n} = ξ^{(1)} + ⋯ + ξ^{(n)}, n = 1, 2, ….

If E|ξ|^{2} < ∞ then from the central limit theorem it follows that for any set A ⊂ R^{d} being the set of continuity of the Lebesque measure, the relation

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

holds. Here a = Eξ while φ_{B}(u) denotes the density of a mean-zero Gaussian vector with the covariance matrix B = E(ξ − a)(ξ − a)^{T} .

If, furthermore, S_{n_{0}} for some n_{0} ≥ 1 has a uniformly bounded density p_{n_{0}}(x), then the local limit theorem holds, that is

(2)  sup_{x∈R^{d}} |n^{d/2}p_{n}(x) − φ_{B} (n^{−1/2}(x − na))| = o(1),  n → ∞.

A value x = x_{n} in the range of S_{n} is called a large deviation if

n^{−1/2}|x − na| → ∞, n → ∞.

That is why P_{n}(A) = P (S_{n} − na ∈ A), where A = A_{n} is such that r_{n}(A) = n^{−1/2} inf_{x∈A} |x| → ∞, n → ∞, is called a large deviation probability. It is convenient to refer to r_{n}(A) as the order of large deviation. In view of (1), a large deviation probability converges to zero while in view of (2) n^{d/2}p_{n}(x) converges to zero when x enjoys a large deviation. One of the basic problem of the large deviations theory is to establish a precise asymptotic behaviour of P_{n}(A) or p_{n}(x) when x enjoys a large deviation. Another class of problems unites those related to socalled rough asymptotic behaviour of a large deviation probability, that is the asymptotic behaviour of ln P_{n}(A) (see e.g. [6–8]).

Let P be the distribution of ξ. In what follows we assume that it has a bounded convex support X that is an essentially d-dimensional open set containing the origin. Moreover, we suppose that X can be written as

X = {x = te ∈ R^{d} : 0 ≤ t < h(e), e ∈ S^{d−1}},

where h(e) is a positive continuous function on the unit sphere S^{d−1} in R^{d}. Let f(s) be the moment generating function corresponding to P, that is

f(s) = ∫_{X} e^{⟨s,x⟩}P (dx),

where ⟨·, ·⟩ denotes the inner product in R^{d}. It is easily seen that f(s) is finite for all s ∈ R^{d}, that is

S = {s ∈ R^{d} : f(s) < ∞} = R^{d}.

For any s ∈ R^{d} and any Borel set A ⊂ R^{d} we denote

P_{s}(A) = (f(s))^{−1} ∫_{A} e^{⟨s,x⟩}P (dx).

The distribution P_{s} is called the Cramér transformation of the distribution P or the conjugate distribution with respect to P (see e.g. [5, Section 1, §1]). Then {P_{s}, s ∈ R^{d}} is called the family of the conjugate distributions or the natural exponential family of distributions generated by P. From now on we assume that P is absolutely continuous and denote its density by p(x).

The classical method of treating large deviations is based on the conjugate distribution techniques. First general results on the topic can be found in [4]. Among them was a local limit theorem (see also Theorem 1 in [5, Section 1, §3]). Let γ(s) and B(s) be the gradient and the hessian of ln f(s), respectively. As it is known, γ(s) and B(s) are the mean vector and the covariance matrix of the distribution P_{s}, respectively. Moreover, the function γ(·) establishes a one-to-one correspondence between R^{d} and γ(R^{d}) ⊂ R^{d}. Denote by s(x) the inverse function with respect to γ(s). Proposition. Suppose that for n_{0} ≥ 1 there exists the uniformly bounded density p_{n_{0}}(x) of S_{n_{0}}. Then

sup_{x∈γ(F)} |\frac{p_{n}(nx)}{ψ_{n}(x)(ρ(x))^{n}} − 1| = o(1), n → ∞,

where

ψ_{n}(x) = (2πn)^{−d/2}(det B(s(x)))^{−1/2},

ρ(x) = inf_{s∈R^{d}} f(s)e^{−⟨s,x⟩} = f(s(x))e^{−⟨s(x),x⟩}

and F is any closed bounded set in R^{d}.

The function

H(x) = − ln ρ(x) = sup_{s∈R^{d}} (⟨s, x⟩ − ln f(s))

is called the Fenchel-Legendre transformation of ln f(s) (see e.g. [6, pp. 26, 134] or [5, Section 1, §1]). This function is also known as the deviation function being the rate function of the large deviation probabilities. So that

(ρ(x))^{n} = exp{−nH(x)}.

It is worth noting that in the case considered r_{n}(A) = O(n^{1/2}) and the higher order of large deviations does not arise.

The question arises : under what additional conditions the statement of Proposition can be extended to the whole set γ(R^{d})? In other words, under what conditions the relation

sup_{x∈γ(R^{d})} |\frac{p_{n}(nx)}{ψ_{n}(x)(ρ(x))^{n}} − 1| = o(1), n → ∞

holds?

In the case when S is a bounded open set containing the origin, the answer was given in [12,13]. There P was taken to be the gamma-like distribution in R^{d}. Here we consider a distribution with a bounded convex support X. In this case it is expected that γ(R^{d}) = X. It will be confirmed in the next section.

In general, we follow the scheme proved to be effective in [12,13]. We start with establishing the asymptotic behaviour of f(s) when |s| → ∞, that is with establishing the Abel type theorem (see e.g. [9, Section 13, §5]). We make use of the Laplace method for its proving. Possible non-degenerate limit laws for the natural exponential family of distributions are discussed in Section 3. Obtained results are also of interest from the viewpoint of searching for the so-called stable multivariate exponential families (see [2]). Section 4 contains our main result: the strong form of the local limit theorem for large deviations. Some remarks are given in Section 5 while proofs are gathered in Appendix.

2 A theorem of the Abel type

To find an asymptotic expression for f(s) when |s| → ∞, we have need of assumptions on the distribution P. First, we assume a regular behaviour of its density in a neighbourhood of the boundary ∂X.

(A) The density p(x) of the distribution P is bounded in X \ ∂X. In a neighbourhood of ∂X it satisfies the condition

sup_{e∈S^{d−1}} |r_{α−1}(τ^{−1})p((h(e) − τ)e) − λ(e)| = o(1), τ ↓ 0

or

sup_{e∈S^{d−1}} |r_{α−1}((h(e) − |x|)^{−1})p(x) − λ(e)| = o(1), |x| ↑ h(e), e = |x|^{−1}x,

where the function r_{α−1}(u), u > 0, is (α − 1)-regularly varying at infinity (cf. [14, Section 5.4.2]), α > 0, while λ(e) is a positive continuous function on S^{d−1}.

Assumption (A) is not very restrictive. It means some regularity of the underlying distribution in a neighbourhood of the boundary ∂X but allows even tending the density to infinity when α ∈ (0, 1). For instance, a uniform distribution on X satisfies (A) with α = 1, r_{0}(u) ≡ 1 and λ(e) ≡ const.

We also need an assumption on the boundary ∂X. It is formulated in terms of the function a(e), which is the support function for X, that is

(3)  a(e) = sup_{x∈X}⟨x, e⟩ = max_{ε∈S^{d−1}} h(ε)⟨ε, e⟩, e ∈ S^{d−1}.

The relation (3) can be rewritten as (see e.g. [12])

(4)  h(ε) = min_{e∈S^{d−1}:⟨ε,e⟩>0} \frac{a(e)}{⟨ε, e⟩}, ε ∈ S^{d−1}.

For convenience, we partition the assumption onto two parts.

(B) For any e ∈ S^{d−1} the set arg max_{ε∈S^{d−1}} h(ε)⟨ε, e⟩ consists of a single point ε′ = ε′(e).

Since both functions a(e) and h(ε) are positive and continuous, from (3) and (4) it follows that {ε′(e), e ∈ S^{d−1}} = S^{d−1}. Moreover, if (B) holds then each point of the boundary ∂X is regular (see e.g. [10, p. 20]) and X is a strictly convex set.

(C) For any e ∈ S^{d−1} the function h(ε)⟨ε, e⟩, ε ∈ S^{d−1}, in a neighbourhood of ε′(e) admits the representation

h(ε)⟨ε, e⟩ = a(e) − \frac{1}{2}(ε − ε′(e))^{T} Λ_{e}(ε − ε′(e)) + w_{e}(ε),

where Λ_{e} is a non-negative definite symmetric matrix of rank d − 1 which is continuous on e ∈ S^{d−1} and such that Λ_{e}ε′(e) = 0 while

sup_{e∈S^{d−1}} \frac{|w_{e}(ε)|}{|ε − ε′(e)|^{2}} = o(1), ε → ε′(e).

The above representation plays a key role in the proof of the Abel theorem by means of the Laplace method.

Moreover, if (C) holds then X is a rotund body (see Section 10.1 in [2]) with a sufficiently smooth boundary.

Denote Λ_{e}^{(0)} = diag (λ_{1}(e), …, λ_{d−1}(e), 0), Λ̄_{e}^{(0)} = diag (λ_{1}(e), …, λ_{d−1}(e)), where λ_{j}(e), j = 1, …, d − 1, are non-zero eigenvalues of Λ_{e}.

Theorem 1. If (A)–(C) hold then

sup_{e∈S^{d−1}} |r_{β}(t)e^{−ta(e)}f(te) − g_{α}(e)| = o(1), t → ∞,

where β = α+(d−1)/2, g_{α}(e) = (2π)^{(d−1)/2}Γ(α)(h(ε′))^{d−1}λ(ε′)⟨ε′, e⟩^{−α} (det Λ̄_{e}^{(0)})^{−1/2} .

Similarly, one can establish that for s = te as t → ∞

(5)  sup_{e∈S^{d−1}} |r_{β}(t)e^{−ta(e)} \frac{∂f(s)}{∂s_{i}} − g_{α}(e)h(ε′)ε′_{i}| = o(1), i = 1, …, d.

Let p^{(s)}(x) be the density of the conjugate distribution P_{s}, that is

p^{(s)}(x) = \frac{e^{⟨s,x⟩}p(x)}{f(s)}, x ∈ X, s ∈ R^{d}.

Obviously, the distribution P_{s} is supported by the set X. Since γ(s) = grad ln f(s) is the mean vector of P_{s} then γ(s) ∈ X.

From Theorem 1 and (5) it follows that

sup_{e∈S^{d−1}} |γ(te) − h(ε′)ε′| = o(1), t → ∞.

Therefore, if |s| → ∞ then γ(s) ↑ ∂X. That is why γ(R^{d}) = X since both sets are bounded, open and convex with the same boundary.

3 Limit laws for conjugate distributions

Let ξ_{s} be a random vector having the distribution P_{s}. As we know, P_{te} is concentrated in a neighbourhood of h(ε′)ε′ as t → ∞. Therefore, in order to find a non-degenerate limit law for P_{s} when |s| → ∞, one should transform ξ_{s} in a proper way.

One transformation follows immediately from the proof of Theorem 1 (see Appendix). If (A)–(C) hold then the density μ_{s}(x) of the random vector

π_{s}(ξ_{s}) = (t^{1/2}(C_{e}^{T} e_{ξ_{s}})_{1}, …, t^{1/2}(C_{e}^{T} e_{ξ_{s}})_{d−1}, t(h(e_{ξ_{s}}) − |ξ_{s}|)) ,

where C_{e} is the orthogonal matrix reducing Λ_{e} to the diagonal matrix Λ_{e}^{(0)}, satisfies for any δ > 0 and for s = te, t → ∞, the relation

sup_{e∈S^{d−1}} sup_{x∈R^{d}: x_{d}≥δ} |μ_{s}(x_{1}, …, x_{d}) − φ_{(Λ̄_{e}^{(0)})^{−1}}(x_{1}, …, x_{d−1}) · ⟨e, ε′⟩q_{α}(⟨e, ε′⟩x_{d})| = o(1).

Here φ_{(Λ̄_{e}^{(0)})^{−1}}(x_{1}, …, x_{d−1}) stands for the density of a mean-zero (d−1)-dimensional normal distribution with the covariance matrix (Λ̄_{e}^{(0)})^{−1} while q_{α}(z) denotes the density of Γ(α) distribution.

However, the transformation π_{s} is rather useless from the viewpoint of applications since it is non-linear and centering in the last component depends on a random factor. The question arises: does there exist a linear transformation of ξ_{s} possessing a weak non-degenerate limit when t → ∞?

It turns out that the answer to the above question is positive. Let η = (η_{1}, …, η_{d}) be a random vector having the density

π_{d,α}(x; a, B) = φ_{B}(x_{1}, …, x_{d−1}) · aq_{α}(ax_{d} − x̄^{T} B^{−1}x̄/2)

= \frac{a(ax_{d} − x̄^{T} B^{−1}x̄/2)^{α−1}e^{−ax_{d}}}{(2π)^{(d−1)/2}(det B)^{1/2}Γ(α)} ,

where x_{d} > x̄^{T} B^{−1}x̄/(2a) and x̄ = (x_{1}, …, x_{d−1})^{T} . It is easy to see that η̄ = (η_{1}, …, η_{d−1}) has a mean-zero (d − 1)-normal distribution with the covariance matrix B while aη_{d} has Γ(β) distribution. The conditional distribution of aη_{d}, given η̄ = z, is Γ(α) distribution shifted by z^{T} B^{−1}z/(2a).

In what follows, we denote by e^{(i)} the i-th unit vector in R^{d},

A_{δ}(a, B) = {x ∈ R^{d} : x_{d} − x̄^{T} B^{−1}x̄/(2a) ≥ δ}.

Theorem 2. If (A)–(C) hold then the density μ̄_{s}(x) of the random vector

π̄_{s}(ξ_{s}) = D_{s}C_{e}^{T} (ξ_{s} − h(ε′)ε′)

satisfies for any δ > 0 and for s = te, t → ∞, the relation

sup_{e∈S^{d−1}} sup_{x∈A_{δ}(⟨ε′,e⟩,H(e))} |μ̄_{s}(x) − π_{d,α}(x; ⟨ε′, e⟩, H(e))| = o(1),

where H(e) = (h(ε′))^{2}(Λ̄_{e}^{(0)})^{−1} while the rows of the matrix D_{s} are t^{1/2}e^{(1)}, …, t^{1/2}e^{(d−1)}, −(t/⟨ε′, e⟩)C_{e}^{T} e, respectively.

The linear transformation π̄_{s}(ξ_{s}) can also be written as

π̄_{s}(ξ_{s}) = (t^{1/2}(C_{e}^{T} ξ_{s})_{1}, …, t^{1/2}(C_{e}^{T} ξ_{s})_{d−1}, t(h(ε′) − ⟨ξ_{s}, e⟩/⟨ε′, e⟩)) .

Example. Let d = 2 and h(e) ≡ 1. Thus, X is a unit ball centered at the origin. Let P be the uniform distribution on X. Then condition (A) holds with α = 1, r_{0}(u) ≡ 1, λ(e) ≡ π^{−1}. One can easily see that a(e) ≡ 1, conditions (B) and (C) hold and ε′(e) ≡ e.

Let s = te, e = e^{(2)}, t → ∞. As we know, the conjugate distribution P_{s} is concentrated in a neighbourhood of e. The transformations π_{s}(ξ_{s}) and π̄_{s}(ξ_{s}) of the random vector ξ_{s} = (ξ_{s1}, ξ_{s2}) with the distribution P_{s}, look like

π_{s}(ξ_{s}) = (t^{1/2} \frac{ξ_{s1}}{|ξ_{s}|}, t(1 − |ξ_{s}|)) , π̄_{s}(ξ_{s}) = (t^{1/2}ξ_{s1}, t(1 − ξ_{s2}))

while the densities of the limit distributions are

(2π)^{−1/2}e^{−(x_{1}^{2}/2+x_{2})}, x_{1} ∈ R, x_{2} > 0

and

(6)  π_{2,1}(x; 1, 1) = (2π)^{−1/2}e^{−x_{2}},  x_{1} ∈ R, x_{2} > x_{1}^{2}/2,

respectively.

4 Local limit theorems

Our method of proving the strong form of the local limit theorem for large deviations of sums consists in establishing, firstly, the local limit theorem for conjugate distributions.

One can easily calculate the mean vector γ_{e} and the covariance matrix Σ_{e} of the limit distribution from Theorem 2:

γ_{e} = (β/⟨ε′, e⟩)e^{(d)}, Σ_{e} = ( \begin{matrix}H(e) & 0 \\ 0 & β/⟨ε′, e⟩^{2} \end{matrix} )

where, as before, β = α + \frac{d−1}{2}, H(e) = (h(ε′))^{2} \binom{}{Λ̄_{e}^{(0)}}^{−1} , e^{(d)} = (0, …, 0, 1)^{T} .

From Theorem 2 it follows that the density μ̃_{s}(x) of the random vector Σ_{e}^{−1/2}π̄_{s}(ξ_{s}) = Σ_{e}^{−1/2}D_{s}C_{e}^{T} (ξ_{s} − h(ε′)ε′) satisfies for any δ > 0 the relation

lim_{|s|→∞} sup_{x∈A_{δ}(β^{1/2},I)} |μ̃_{s}(x) − π_{d,α}(x; β^{1/2}, I)| = 0,

where I denotes the identity matrix.

So, the random vector Σ_{e}^{−1/2}D_{s}C_{e}^{T} (ξ_{s} − h(ε′)ε′) converges in distribution to a limit vector with the density π_{d,α}(x; β^{1/2}, I). From the next lemma it follows that the random vector B^{−1/2}(s)(ξ_{s} −h(ε′)ε′) also converges in distribution to the same limit vector. Here B(s) denotes the covariance matrix of the distribution P_{s}, as before.

Lemma 1. If (A)–(C) hold then there exists a matrix B^{1/2}(s) with the property B^{1/2}(s)(B^{1/2}(s))^{T} = B(s) such that for any δ > 0

lim_{|s|→∞} sup_{x∈A_{δ}(β^{1/2},I)} |p̄^{(s)}(x) − π_{d,α}(x; β^{1/2}, I)| = 0,

where

p̄^{(s)}(x) = (det B(s))^{1/2}p^{(s)}(B^{1/2}(s)x + h(ε′)ε′)

is the density of B^{−1/2}(s)(ξ_{s} − h(ε′)ε′).

Let ξ_{s}^{(1)}, …, ξ_{s}^{(n)}, … be independent identically distributed random vectors in R^{d} with the distribution P_{s}. By p_{ns}(x) we denote the density of the sum ξ_{s}^{(1)} + ⋯ + ξ_{s}^{(n)}, n = 1, 2, ….

Theorem 3. If (A)–(C) hold then as n → ∞

sup_{s∈R^{d}} sup_{x∈R^{d}} |(2πn)^{d/2}(det B(s))^{1/2}p_{ns}(n^{1/2}B^{1/2}(s)x + nγ(s)) − e^{−|x|^{2}/2}| = o(1),

where the matrix B^{1/2}(s) is that from Lemma 1.

Now we can establish the main result concerning the large deviations of sums.

Theorem 4. If (A)–(C) hold then

sup_{x∈X} |\frac{p_{n}(nx)}{ψ_{n}(x)(ρ(x))^{n}} − 1| = o(1), n → ∞,

where ψ_{n}(x) and ρ(x) as in Proposition.

Due to the Cramér identity we have

p_{n}(nx) = (f(s))^{n}e^{−n⟨s,x⟩}p_{ns}(nx), s ∈ R^{d}.

For x = γ(s) we obtain

p_{n}(nx) = (ρ(x))^{n}p_{ns}(nγ(s)).

The assertion of Theorem 4 follows then from Theorem 3.

5 Some remarks

1. One could note that after the obvious small modifications the main results hold even if the support X of the underlying distribution P is not convex but compact and essentially d-dimensional set. In this case γ(R^{d}) ≠ X but γ(R^{d}) = conv (X).

2. Last time there has been increased an interest in the theory of multivariate exponential families on the one hand and exponential families which are invariant under a group of affine transformations on the other. If there exists a linear transformation of the random vector ξ_{s} with the distribution P_{s}, s ∈ S, such that it converges in distribution to a non-degenerate limit vector ζ when s → ∂S (or |s| → ∞), then the exponential family {P′_{t}, t ∈ T } generated by the distribution of ζ is said to be stable in the sense that all distributions P′_{t} are of the same type. The latter means that for any t ∈ T there exists a linear transformation π_{t} such that π_{t}(ζ) has the distribution P′_{t} (see [2]). In this case it is also said that the distribution of ζ is stable. To the contrary with the univariate case (see [1]), in the multivariate case a complete description of the stable distributions still is not available.

One example of the stable distribution was obtained in [12, 13] as the limit distribution for the exponential family generated by the gamma-like distribution. The density of this stable distribution looks like

ϕ_{x_{d}B_{e}}(x_{1}, …, x_{d−1})q_{β}(x_{d})

for some β > 0 and diagonal positive-definite matrix B_{e}.

Another example is given in Theorem 2, where the density of the stable distribution is of the form π_{d,α}(x; a, B). In fact, the same examples were also given in [2].

It is of interest to remind that in the univariate case any stable distribution is either normal or, possibly shifted, gamma (see [1]).

3. It is of interest to compare the transformations π_{s}(ξ_{s}) and π̄_{s}(ξ_{s}) aiming to answer the question: why does there exist a shift in the last component of the limit distribution after the linear transformation π̄_{s}? Note that

(π̄_{s}(ξ_{s}))_{d} − (π_{s}(ξ_{s}))_{d} = t(h(ε′) − h(e_{ξ_{s}}) + |ξ_{s}| − |ξ_{s}|⟨e_{ξ_{s}}, e⟩/⟨ε′, e⟩).

As we know, if t → ∞ then e_{ξ_{s}} is in a neighbourhood of ε′ and due to (C) and the proof of Theorem 1 we get

⟨ε′, e⟩h(ε′) = h(e_{ξ_{s}})⟨e_{ξ_{s}}, e⟩ + \frac{1}{2}ē^{T} Λ̄_{e}^{(0)}ē + o (t^{−1}) ,

where ē is the vector that consists of the first d − 1 components of C_{e}^{T} e_{ξ_{s}}. Then

h(ε′) − h(e_{ξ_{s}}) + |ξ_{s}|(1 − ⟨e_{ξ_{s}}, e⟩/⟨ε′, e⟩)

= (|ξ_{s}| − h(e_{ξ_{s}}))(1 − ⟨e_{ξ_{s}}, e⟩/⟨ε′, e⟩) + \frac{1}{2⟨ε′, e⟩}ē^{T} Λ̄_{e}^{(0)}ē + o (t^{−1}) .

From the proof of Theorem 1 it follows that

||ξ_{s}| − h(e_{ξ_{s}})| = O (t^{−1}) , |1 − ⟨e_{ξ_{s}}, e⟩/⟨ε′, e⟩| = O (t^{−1/2}) .

Therefore,

t(h(ε′) − h(e_{ξ_{s}}) + |ξ_{s}| − |ξ_{s}|⟨e_{ξ_{s}}, e⟩/⟨ε′, e⟩) = \frac{1}{2⟨ε′, e⟩}(t^{1/2}ē)^{T} Λ̄_{e}^{(0)}(t^{1/2}ē) + o(1).

4. The next example shows what can happen if (B) or (C) does not hold. Let d = 2 and h(e_{1}, e_{2}) = (|e_{1}| + |e_{2}|)^{−1}, (e_{1}, e_{2}) ∈ S^{1}. Thus, X is a square centered at the origin whose vertices are at points (1, 0), (0, 1), (−1, 0), (0, −1). By direct calculation it can be shown that a(e_{1}, e_{2}) = max{|e_{1}|, |e_{2}|}.

Take e = (1/\sqrt{2}, 1/\sqrt{2}). Condition (B) does not hold at this point; any point (e_{1}, e_{2}) with e_{1} ≥ 0, e_{2} ≥ 0 can be taken as ε′(e). Let P be the uniform distribution on X. Then condition (A) holds with α = 1, r_{0}(u) ≡ 1, λ(e) ≡ 1/2.

By direct calculations one can show that f(te) = e^{t/\sqrt{2}}/(\sqrt{2}t)(1 + o(1)) as t → ∞. The conjugate distribution P_{te} is concentrated in a neighbourhood of the boundary ∂X lying in the first quarter-plane (x_{1} > 0, x_{2} > 0).

A linear transformation of ξ_{s} having a non-degenerate limit when s = te, t → ∞, looks like

π̄_{s}(ξ_{s}) = (ξ_{s1}, t(a(e) − ⟨e, ξ_{s}⟩)) = (ξ_{s1}, \frac{t}{\sqrt{2}}(1 − ξ_{s1} − ξ_{s2})) .

The density of the limit distribution is of the form

(7)  g(x_{1}, x_{2}) = e^{−x_{2}},  0 < x_{1} < 1, x_{2} > 0.

The limit distribution has two independent components: the first component has a uniform distribution on (0, 1) while the second one has Γ(1) (exponential) distribution.

Consider also the second case when e = (e_{1}, e_{2}) is such that e_{2} > |e_{1}|. Then (B) holds with ε′(e) = e^{(2)} and a(e) = e_{2}. However, in a neighbourhood of ε′ we have

h(ε)⟨ε, e⟩ = \frac{e_{1}ε_{1} + e_{2}ε_{2}}{|ε_{1}| + ε_{2}} = a(e) − \frac{e_{2}|ε_{1}| − e_{1}ε_{1}}{|ε_{1}| + ε_{2}}

and (C) does not hold.

By direct calculations one can show that

f(te) = \frac{e^{te_{2}}}{t^{2}(e_{2}^{2} − e_{1}^{2})}(1 + o(1)), t → ∞.

The conjugate distribution P_{te} is concentrated in a neighbourhood of e^{(2)}.

A linear transformation of ξ_{s} having a non-degenerate limit when s = te, t → ∞, looks like

π̄_{s}(ξ_{s}) = (te_{2}ξ_{s1}, te_{2}(1 − ⟨e, ξ_{s}⟩/⟨ε′, e⟩)) = (te_{2}ξ_{s1}, t(e_{2} − e_{1}ξ_{s1} − e_{2}ξ_{s2})) .

The density of the limit distribution is of the form

(8) g(x_{1}, x_{2}) = \frac{1 − e_{1}^{2}/e_{2}^{2}}{2} e^{−x_{2}}, −\frac{x_{2}}{1 + e_{1}/e_{2}} < x_{1} < \frac{x_{2}}{1 − e_{1}/e_{2}} , x_{2} > 0.

Again we see that the second component has gamma distribution (it is Γ(2) distribution).

It is of interest to note that in all examples considered we have obtained the densities of the same form but with different supports (see (6)–(8)).

Appendix

From now on, c denotes any positive constant whose concrete value is of no importance. This means that c + c = c, c^{2} = c, etc. As before, C_{e} denotes the orthogonal matrix reducing Λ_{e} to the diagonal matrix Λ_{e}^{(0)}, that is Λ_{e} = C_{e}Λ_{e}^{(0)}C_{e}^{T} . By ω(t) we denote any non-negative function such that lim_{t→∞} ω(t) = 0 while θ varies within [−1, 1]. Proof of Theorem 1. Let e ∈ S^{d−1} be arbitrarily but fixed, s = te, t → ∞. First, we prove the theorem in the case when the coordinate axes are associated with the main axes of the matrix Λ_{e} and the normal direction to the boundary at the point h(ε′)ε′, i.e. we assume that ε′(e) = e^{(d)} and C_{e} is the identity matrix.

Denote

X_{1} = {x = rε : h(ε) − δ ≤ r < h(ε), |ε − ε′| < Mt^{−1/2}},
X_{2} = {x = rε : h(ε) − δ ≤ r < h(ε), Mt^{−1/2} ≤ |ε − ε′| < δ′},
X_{3} = {x = rε : 0 ≤ r < h(ε) − δ, |ε − ε′| < δ′},
X_{4} = {x = rε : 0 ≤ r < h(ε), |ε − ε′| ≥ δ′},

where M is arbitrarily large while δ > 0, δ′ > 0 are arbitrarily small. Obviously,

(9)  f(s) = f_{1}(s) + f_{2}(s) + f_{3}(s) + f_{4}(s),

where

f_{i}(s) = ∫_{X_{i}} e^{⟨s,x⟩}p(x)dx, i = 1, 2, 3, 4.

We estimate f_{i}(s) one after another. Let us start with f_{1}(s) and represent X_{1} as

(10)  X_{1} = X_{11} ∪ X_{12},

where

\begin{matrix}X_{11} = \{x ∈ X_{1} : h(ε) − r < Nt^{−1}\}, \\ X_{12} = \{x ∈ X_{1} : h(ε) − r ≥ Nt^{−1}\} \end{matrix}

and N is arbitrarily large. Denote

(11)  f_{1k}(s) = ∫_{X_{1k}} e^{⟨s,x⟩}p(x)dx,  k = 1, 2.

If x ∈ X_{1} then |ε̄| = O(t^{−1/2}) and 1 − ε_{d} = O(t^{−1}), where ε̄ = (ε_{1}, …, ε_{d−1})^{T} . Changing variables

x_{i} = rε_{i}, i = 1, …, d−1, x_{d} = sign(x_{d})r|ε_{d}| = sign(x_{d})r(1−ε_{1}^{2} −…−ε_{d−1}^{2})^{1/2}

with Jacobian r^{d−1}|ε_{d}|^{−1} yields

f_{11}(s) = ∫_{t^{1/2}|ε̄|<M} |ε_{d}|^{−1} ∫_{t(h(ε)−r)<N} r^{d−1}e^{r⟨s,ε⟩}p(rε)drdε̄.

It is worth noting that x_{d} > 0 in a neighbourhood of ε′ = e^{(d)}.

Next change of variables u = h(ε) − r leads to the relation

f_{11}(s) = ∫_{t^{1/2}|ε̄|<M}|ε_{d}|^{−1} ∫_{tu<N}(h(ε) − u)^{d−1}e^{th(ε)⟨e,ε⟩−t⟨e,ε⟩u}p((h(ε) − u)ε)dudε̄.

Therefore as t → ∞,

f_{11}(s) = e^{ta(e)} ∫_{t^{1/2}|ε̄|<M} |ε_{d}|^{−1}(h(ε))^{d−1}λ(ε)e^{− \frac{t}{2} (ε−ε′)^{T} Λ_{e}(ε−ε′)} ∫_{tu<N}\frac{u^{α−1}e^{−t⟨e,ε⟩u}}{l(u^{−1})} dudε̄(1 + o(1))

due to (A) and (C), where l(t), t > 0, varies slowly at infinity. Since ⟨e, ε⟩ = e_{d} + O(t^{−1/2}) and (ε − ε′)^{T} Λ_{e}^{(0)}(ε − ε′) = ε̄^{T} Λ̄_{e}^{(0)}ε̄, we obtain as t → ∞

f_{11}(s) = \frac{e^{ta(e)}(h(ε′))^{d−1}λ(ε′)}{l(t)t^{α}} ∫_{t^{1/2}|ε̄|<M} e^{− \frac{t}{2} ε̄^{T} Λ̄_{e}^{(0)}ε̄}dε̄(\frac{Γ(α)}{e_{d}^{α}} + ω(N) + o(1))

and finally

f_{11}(s) = \frac{e^{ta(e)}(h(ε′))^{d−1}λ(ε′)}{l(t)t^{α+(d−1)/2}} (\frac{(2π)^{(d−1)/2}}{(det Λ̄_{e}^{(0)})^{1/2}} + ω(M)) (\frac{Γ(α)}{e_{d}^{α}} + ω(N) + o(1))

(12) = \frac{e^{ta(e)}}{r_{β}(t)}(g_{α}(e) + θω(min(M, N)) + o(1)).

Similarly, for f_{12}(s) we get as t → ∞

f_{12}(s) = ∫_{t^{1/2}|ε̄|<M}|ε_{d}|^{−1} ∫_{Nt^{−1}≤u<δ}(h(ε) − u)^{d−1}e^{th(ε)⟨e,ε⟩−t⟨e,ε⟩u}p((h(ε) − u)ε)dudε̄

(13)  ≤ ce^{ta(e)} ∫_{t^{1/2}|ε̄|<M} e^{− \frac{t}{2} ε̄^{T} Λ̄_{e}^{(0)}ε̄} ∫_{tu≥N}e^{−t⟨e,ε⟩u}p((h(ε) − u)ε)dudε̄ ≤ \frac{e^{ta(e)}}{r_{β}(t)}ω(N).

From (10)–(13) we obtain

(14)  f_{1}(s) = \frac{e^{ta(e)}}{r_{β}(t)}(g_{α}(e) + θω(min(M, N)) + o(1)), t → ∞.

Before estimating f_{2}(s) we note that for x ∈ X_{2} we have t^{1/2}|ε̄| ≥ 2M,

(15)  |ε̄| < δ′, 1 − ε_{d} < cδ^{′2}, (ε − ε′)^{T} Λ_{0}(ε − ε′) ≥ c|ε̄|^{2}.

Then due to (A) and (C)

f_{2}(s) ≤ ce^{ta(e)} ∫_{t^{1/2}|ε̄|≥2M} e^{−ct|ε̄|^{2}} ∫_{0}^{δ} \frac{u^{α−1}e^{−t⟨e,ε⟩u}}{l(u^{−1})} dudε̄.

Partitioning the interval (0, δ] as {u ∈ (0, δ] : tu < N} and {u ∈ (0, δ] : tu ≥ N}, and treating both integrals similarly as we did during the investigation of f_{11}(s) and f_{12}(s), we obtain

(16)  f_{2}(s) ≤ \frac{ce^{ta(e)}}{l(t)t^{α}} ∫_{t^{1/2}|ε̄|≥2M} e^{−ct|ε̄|^{2}}dε̄ = \frac{e^{ta(e)}}{r_{β}(t)}ω(M), t → ∞.

Before estimating f_{3}(s) we note that for x ∈ X_{3} again (15) holds and ⟨e, ε⟩ > e_{d} − δ′. Then due to (C)

(17)  f_{3}(s) ≤ ce^{ta(e)} ∫_{|ε̄|<δ′}e^{−ct|ε̄|^{2}−δt⟨e,ε⟩} ∫_{0}^{h(ε)−δ} p(rε)drdε̄ ≤ \frac{ce^{ta(e)−tδ(e_{d}−δ′)}}{t^{(d−1)/2}} .

At last,

(18)  f_{4}(s) ≤ ce^{t(a(e)−c(δ′))} ∫_{X} p(x)dx = ce^{ta(e)−tc(δ′)},

where c(δ′) > 0 is such that

a(e) − max_{ε∈S^{d−1}:|ε−ε′|≥δ′} h(ε)⟨ε, e⟩ ≥ c(δ′) ∀e ∈ S^{d−1}.

Since M, N, δ and δ′ are arbitrary, from (9), (14), (16)-(18) it follows that

(19)  f(s) = \frac{g_{α}(e)e^{ta(e)}}{r_{β}(t)} (1 + o(1)),  t → ∞

uniformly in e ∈ S^{d−1}.

Let us turn to the general case. Clearly, C_{e}e^{(d)} = ε′(e). We can write

f(s) = ∫_{C_{e}^{T} X} e^{⟨s,C_{e}x⟩}p(C_{e}x)dx, s ∈ R^{d}.

If C_{e}ε is in a neighbourhood of ε′(e), then ε is in a neighbourhood of e^{(d)} and due to (C) we obtain

h(C_{e}ε)⟨C_{e}ε, e⟩ − a(e) = −\frac{1}{2}(ε − e^{(d)})^{T} Λ_{e}^{(0)}(ε − e^{(d)}) + w_{e}(C_{e}ε).

The remainder of the proof repeats that given above. ◇

Proof of Theorem 2. In contrast to the proof of Theorem 1, from the very beginning we consider the general case since the particular one gives no special simplifications.

The transformation y = C_{e}D_{s}^{−1}x+h(ε′)ε′ is inverse with respect to D_{s}C_{e}^{T} (x− h(ε′)ε′). Thus,

(20) μ̄_{s}(x) = |det D_{s}^{−1}| p^{(s)} (C_{e}D_{s}^{−1}x + h(ε′)ε′) = \frac{e^{⟨s,y⟩}p(y)}{t^{(d+1)/2}f(s)},  y ∈ X,

where the matrix D_{s}^{−1} is such that its first d−1 rows are t^{−1/2}e^{(1)}, …, t^{−1/2}e^{(d−1)}, while the last row is (−(t^{−1/2}/⟨ε′, e⟩)(C_{e}^{T} e)_{1}, …, −(t^{−1/2}/⟨ε′, e⟩)(C_{e}^{T} e)_{d−1}, −t^{−1}).

Given δ > 0 denote

Z = sup_{e∈S^{d−1}} sup_{x∈A_{δ}(⟨ε′,e⟩,H(e))} |μ̄_{s}(x) − π_{d,α}(x; ⟨ε′, e⟩, H(e))| = sup_{e∈S^{d−1}} max(Z_{1}, Z_{2}),

where

Z_{1} = sup_{x∈Y ∩A_{δ}(⟨ε′,e⟩,H(e))} |t^{−(d+1)/2}p^{(s)} (C_{e}D_{s}^{−1}x + h(ε′)ε′) − π_{d,α}(x; ⟨ε′, e⟩, H(e))| ,

Z_{2} = sup_{x∈Y^{c}∩A_{δ}(⟨ε′,e⟩,H(e))} |t^{−(d+1)/2}p^{(s)} (C_{e}D_{s}^{−1}x + h(ε′)ε′) − π_{d,α}(x; ⟨ε′, e⟩, H(e))| ,

Y = {x ∈ R^{d} : |x̄| < M, x_{d} < N},

Y^{c} is the complement of Y, while M and N are arbitrarily large numbers. We have

(21)  ⟨y, s⟩ − ta(e) = t⟨C_{e}D_{s}^{−1}x, e⟩ = t⟨D_{s}^{−1}x, C_{e}^{T} e⟩ = −⟨ε′, e⟩x_{d},

(22) C_{e}^{T} y =D_{s}^{−1}x + h(ε′)e^{(d)} = (\frac{x_{1}}{t^{1/2}} , …, \frac{x_{d−1}}{t^{1/2}} , h(ε′) − \frac{∑_{i=1}^{d−1} x_{i}(C_{e}^{T} e)_{i}}{t^{1/2}⟨ε′, e⟩} − \frac{x_{d}}{t} )^{T}.

Due to (22) we obtain as t → ∞

|C_{e}^{T} y|=h(ε′) − \frac{∑_{i=1}^{d−1} x_{i}(C_{e}^{T} e)_{i}}{t^{1/2}⟨ε′, e⟩} − \frac{x_{d}}{t} + \frac{(∑_{i=1}^{d−1} x_{i}(C_{e}^{T} e)_{i})^{2}}{2h(ε′)⟨ε′, e⟩^{2}t} + \frac{∑_{i=1}^{d−1} x_{i}^{2}}{2h(ε′)t} +o (t^{−1}) ,

C_{e}^{T} (e_{y} − ε′) = e_{C_{e}^{T} y} − e^{(d)} = (\frac{x_{1}}{t^{1/2}h(ε′)} + O (t^{−1}) , …, \frac{x_{d−1}}{t^{1/2}h(ε′)} + O (t^{−1}) ,

(23)  −\frac{(∑_{i=1}^{d−1} x_{i}(C_{e}^{T} e)_{i})^{2}}{2h^{2}(ε′)⟨ε′, e⟩^{2}t} − \frac{∑_{i=1}^{d−1} x_{i}^{2}}{2h^{2}(ε′)t} + O (t^{−3/2}))^{T} .

From (21), (23) and (C) it follows that, as t → ∞,

0 < h(e_{y}) − |y| = \frac{h(e_{y})⟨e_{y}, e⟩ − ⟨y, e⟩}{⟨e_{y}, e⟩} = \frac{h(e_{y})⟨e_{y}, e⟩ − a(e) − ⟨D_{s}^{−1}x, C_{e}^{T} e⟩}{⟨e_{y}, e⟩}

= \frac{h(e_{y})⟨e_{y}, e⟩ − a(e) + ⟨ε′, e⟩x_{d}/t}{⟨e_{y}, e⟩} = \frac{−\frac{1}{2}(e_{y} − ε′)^{T} Λ_{e}(e_{y} − ε′) + ⟨ε′, e⟩x_{d}/t + o(t^{−1})}{⟨ε′, e⟩ + O(t^{−1/2})}

(24)  = \frac{x_{d} − x̄^{T} H(e)^{−1}x̄/(2⟨ε′, e⟩)}{t}(1 + o(1)).

Therefore, if x ∈ Y ∩ A_{δ}(⟨ε′, e⟩, H(e)) then y ∈ X_{11} (X_{11} is from the proof of Theorem 1).

From (20)-(22), (24) and Theorem 1 it follows that, as t → ∞,

μ̄_{s}(x) = \frac{e^{⟨y,s⟩−ta(e)}t^{α−1}l(t)p(y)}{g_{α}(e)}(1 + o(1))

= \frac{e^{−⟨ε′,e⟩x_{d}}λ(ε′)(x_{d} − x̄^{T} H(e)^{−1}x̄/(2⟨ε′, e⟩))^{α−1}}{g_{α}(e)} (1 + o(1))

= \frac{⟨ε′, e⟩^{α}(x_{d} − x̄^{T} H(e)^{−1}x̄/(2⟨ε′, e⟩))^{α−1}e^{−⟨ε′,e⟩x_{d}}(det Λ̄_{e}^{(0)})^{1/2}}{Γ(α)(2π)^{(d−1)/2}(h(ε′))^{d−1}} (1 + o(1))

and therefore

Z_{1} = o(1), t → ∞

uniformly in e ∈ S^{d−1}.

As to Z_{2}, it is easily seen that Z_{2} ≤ Z_{21} + Z_{22}, where

Z_{21} = sup_{x∈Y^{c}∩A_{δ}(⟨ε′,e⟩,H(e))} π_{d,α}(x; ⟨ε′, e⟩, H(e)),

Z_{22} = sup_{x∈Y^{c}∩A_{δ}(⟨ε′,e⟩,H(e))} t^{−(d+1)/2}p^{(s)} (C_{e}D_{s}^{−1}x + h(ε′)ε′) .

Clearly,

Z_{21} ≤ c sup_{x_{d}≥N} x_{d}^{α−1}e^{−x_{d}} + c sup_{|x̄|≥M} e^{−x̄^{T} H(e)^{−1}x̄/2} = ω(N) + ω(M).

Before estimating Z_{22}, let us define

X′ = {x ∈ R^{d} : |x̄| < M, x_{d} ≥ N},
X′′ = {x ∈ R^{d} : |x̄| ≥ M} ∩ A_{δ}(⟨ε′, e⟩, H(e))

and note that

t^{−(d+1)/2}p^{(s)} (C_{e}D_{s}^{−1}x + h(ε′)ε′) = \frac{e^{−x_{d}}t^{α−1}l(t)p(y)}{g_{α}(e)}(1 + o(1)).

Since y ∉ X_{11} yields x ∉ Y ∩ A_{δ}(⟨ε′, e⟩, H(e)), we have

Z_{22} ≤ c max(Z′, Z′′),

where

Z′ = sup_{x∈X′} e^{−x_{d}}t^{α−1}l(t)p (C_{e}D_{s}^{−1}x + h(ε′)ε′) ,

Z′′ = sup_{x∈X′′} e^{−x_{d}}t^{α−1}l(t)p (C_{e}D_{s}^{−1}x + h(ε′)ε′) .

Due to (24) we obtain

Z′ ≤ c sup_{x∈X′} \frac{e^{−x_{d}}t^{α−1}l(t)(h(e_{y}) − |y|)^{α−1}}{l((h(e_{y}) − |y|)^{−1})} ≤ c sup_{x_{d}≥N} e^{−x_{d}}x_{d}^{α−1} = ω(N)

while

Z′′ ≤ ce^{−cM^{2}} sup_{x∈X′′} \frac{t^{α−1}l(t)(h(e_{y}) − |y|)^{α−1}}{l((h(e_{y}) − |y|)^{−1})} ≤ ce^{−cM^{2}}M^{α−1} = ω(M).

This concludes the proof of the theorem. ◇

Proof of Lemma 1. We start with establishing an auxiliary result.

Lemma 2. If (A)–(C) hold then for s = te

sup_{e∈S^{d−1}} ∥D_{s}C_{e}^{T} B(s)C_{e}D_{s}^{T} − Σ_{e}∥ = o(1), t → ∞,

where ∥ · ∥ stands for any matrix norm.

Proof. In view of Theorem 2 the random vector π̄_{s}(ξ_{s}) = D_{s}C_{e}^{T} (ξ_{s} − h(ε′)ε′) having the covariance matrix D_{s}C_{e}^{T} B(s)C_{e}D_{s}^{T} converges in distribution to a limit vector with the density π_{d,α}(x; ⟨ε′, e⟩, H(e)) which covariance matrix is Σ_{e}. Thus, for proving the lemma it is enough to establish that sup_{s∈R^{d}} E|π̄_{s}(ξ_{s})|^{2} < ∞. It is easy to see that

E|π̄_{s}(ξ_{s})|^{2} = (f(s))^{−1} ∫_{X} |D_{s}C_{e}^{T} x + th(ε′)e^{(d)}|^{2}e^{⟨s,x⟩}p(x)dx

= (f(s))^{−1} ∫_{C_{e}^{T} X}(t|x̄|^{2} + t^{2}(h(ε′) − ⟨x, ẽ⟩/⟨ε′, e⟩)^{2})e^{⟨s,C_{e}x⟩}p(C_{e}x)dx = (f(s))^{−1}f̄(s),

where ẽ = C_{e}^{T} e and x̄ = (x_{1}, …, x_{d−1})^{T} , as before.

Let X_{i}, i = 1, 2, 3, 4, and X_{1k}, k = 1, 2, be the same sets as in the proof of Theorem 1. Denote by f̄_{i}(s), f̄_{1k}(s) the parts of the integral f̄(s) corresponding to those sets. One can show that the ratios f̄_{i}(s)/f(s), i = 2, 3, 4, and f̄_{12}(s)/f(s) can be made arbitrary small while f̄_{11}(s)/f(s) < ∞ when s = te, t → ∞. Note that the integrals f̄_{i}(s), f̄_{1k}(s) are estimated in the same way as f_{i}(s), f_{1k}(s) in the proof of Theorem 1. Let us estimate, for example, f̄_{2}(s). For x = rε, where ε = (ε_{1}, …, ε_{d})^{T} , we have

t|x̄|^{2} + t^{2}(h(ε′) − ⟨x, ẽ⟩/⟨ε′, e⟩)^{2} = tr^{2}|ε̄|^{2} + t^{2}(h(ε′) − r⟨ε, ẽ⟩/⟨ε′, e⟩)^{2}.

Thus (cf. (16))

f̄_{2}(s) ≤ \frac{ce^{ta(e)}}{l(t)t^{α}} ∫_{t^{1/2}|ε̄|≥2M}(t^{1/2}|ε̄|)^{2}e^{−ct|ε̄|^{2}}dε̄

+ce^{ta(e)} ∫_{t^{1/2}|ε̄|≥2M} e^{−ct|ε̄|^{2}} ∫_{0}^{δ} t^{2}u^{2}e^{−tu⟨e,C_{e}ε⟩}p((h(C_{e}ε) − u)C_{e}ε)dudε̄ = \frac{e^{ta(e)}}{r_{β}(t)}ω(M).

From Theorem 1 it follows that

\frac{f̄_{2}(s)}{f(s)} ≤ ω(M).

Similarly, one can estimate all other ratios. ◇

Now we come back to the proof of Lemma 1. Denote R(s) = Σ_{e}^{−1/2}D_{s}C_{e}^{T} . From Lemma 2 it follows that for |s| → ∞

R(s)B(s)R^{T} (s) = ( \begin{matrix}1 + o(1) & o(1) & … & o(1) \\ o(1) & 1 + o(1) & … & o(1) \\ … & … & … & … \\ o(1) & o(1) & … & 1 + o(1) \end{matrix} )

and

(26)  lim_{|s|→∞}(det B(s))^{1/2}det R(s) = 1.

Let B^{1/2}(s) be any matrix with the property

(27)  B^{1/2}(s)(B^{1/2}(s))^{T} = B(s)

and let C be the set of all orthogonal d × d-matrices while C_{0} ⊂ C consists of those matrices whose last column is e^{(d)}. Note that the density π_{d,α}(x; β^{1/2}, I) is invariant with respect to the transformations from C_{0}. Clearly,

lim_{|s|→∞} min_{C∈C} ∥R(s)B^{1/2}(s) − C∥ = 0.

We have a degree of freedom in choosing B^{1/2}(s) since for any C ∈ C the matrix B^{1/2}(s)C also satisfies (27). Therefore, there exists B^{1/2}(s) such that

(28)  lim_{|s|→∞} min_{C∈C_{0}} ∥R(s)B^{1/2}(s) − C∥ = 0.

Finally, due to (26), (28) and Theorem 2 we obtain

lim_{|s|→∞} sup_{x∈A_{δ}(β^{1/2},I)} |p̄^{(s)}(x) − π_{d,α}(x; β^{1/2}, I)|

≤ lim_{|s|→∞} sup_{x∈A_{δ}(β^{1/2},I)} |(det B(s))^{1/2}det R(s)(det R(s))^{−1}p^{(s)}(R^{−1}(s)R(s)B^{1/2}(s)x

+h(ε′)ε′) − (det B(s))^{1/2}det R(s)π_{d,α} (R(s)B^{1/2}(s)x; β^{1/2}, I)|

+ lim_{|s|→∞} sup_{x∈A_{δ}(β^{1/2},I)} |(det B(s))^{1/2}det R(s)π_{d,α} (R(s)B^{1/2}(s)x; β^{1/2}, I)

−π_{d,α}(x; β^{1/2}, I)| = 0. ◇

Proof of Theorem 3. Consider the characteristic function

ψ_{s}(u) = ∫_{R^{d}} e^{i⟨u,x⟩}p̄^{(s)}(x)dx, u ∈ R^{d}

corresponding to the density p̄^{(s)}(x) from Lemma 1. From this lemma it follows that:

(i) for any C > 0

(29)  sup_{s∈R^{d}} sup_{|u|<C} |ψ_{s}^{n} (u/\sqrt{n}) − e^{−|u|^{2}/2}| = o(1),  n → ∞;

(ii) for any sufficiently small δ > 0 there exist c_{δ} > 0 such that

(30)  sup_{s∈R^{d}} |ψ_{s}(u)| ≤ 1 − c_{δ}|u|^{2}, 0 < |u| ≤ δ;

(iii) for any δ > 0

(31)  sup_{s∈R^{d}} sup_{|u|>δ} |ψ_{s}(u)| = ρ_{δ} < 1.

It remains to show that there exists n_{0} > 1 such that |ψ_{s}(u)|^{n_{0}} is integrable uniformly in s ∈ R^{d}. Let s = te. Since

p̄^{(s)}(x) = (det B(s))^{1/2} \frac{e^{ta(e)+t⟨e,B^{1/2}(s)x⟩}p(B^{1/2}(s)x + h(ε′)ε′)}{f(te)},

from (A) it follows that for any C > 0

sup_{e∈S^{d−1}} sup_{t<C} sup_{x∈R^{d}} p̄^{(s)}(x) < ∞.

Hence, for any C > 0

sup_{e∈S^{d−1}} sup_{t<C} ∫_{R^{d}} |ψ_{s}(u)|^{2}du < ∞.

Set

I_{q}(s) = ∫_{X} (e^{⟨s,x⟩}p(x))^{q}dx, q > 1.

Arguing as in the proof of Theorem 1, one can show that

(32) sup_{e∈S^{d−1}} |t^{q(α−1)+(d+1)/2}(l(t))^{q}e^{−ta(e)q}I_{q}(te) − g_{α,q}(e)| = o(1),  t → ∞,

where

g_{α,q}(e) = \frac{(2π)^{(d−1)/2}(h(ε′))^{d−1}(λ(ε′))^{q}Γ(q(α − 1) + 1)}{q^{q(α−1)+(d+1)/2}⟨ε, e⟩^{q(α−1)+1}(det Λ̄_{e}^{(0)})^{1/2}}.

Furthermore, for q ∈ (1, 2)

∫_{R^{d}}(p̄^{(s)}(x))^{q}dx = (det B(s))^{(q−1)/2}(f(s))^{−q}I_{q}(s).

From Theorem 1, (26) and (32) it follows that

lim_{|s|→∞} ∫_{R^{d}}(p̄^{(s)}(x))^{q}dx < ∞.

Therefore (see e.g. [3, §19]),

lim_{|s|→∞} ∫_{R^{d}} |ψ_{s}(u)|^{q/(q−1)}du < ∞.

Thus for n_{0} ≥ q/(q − 1),

(33)  sup_{s∈R^{d}} ∫_{R^{d}} |ψ_{s}(u)|^{n_{0}}du < ∞.

The assertion of the theorem follows from (29)–(31), (33) (see e.g. [10, Section 4, §3]). ◇

References

1. A. Balkema, C. Klüppelberg and S. Resnick, Limit laws for exponential families, Bernoulli 5(6) (1999), 951–968.

2. A. Balkema, C. Klüppelberg and S. Resnick, Stability for multivariate exponential families, J. Math. Sci. 106(2) (2001), 2777–2791.

3. R. N. Bhattacharya and R. Ranga Rao, Normal Approximation and Asymptotic Expansions, John Wiley, New York, 1976.

4. A. A. Borovkov and B. A. Rogozin, On the central limit theorem in multidimensional case, Theory Probab. Appl. 10 (1965), 61-69 [in Russian].

5. A. A. Borovkov and A. A. Mogulskii, Large deviations and testing statistical hypotheses. I. Large deviations of sums of random vectors, Siberian Adv. Math. 2(3) (1992), 52-120.

6. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, 1993.

7. J. D. Deuschel and D. W. Strook, Large Deviations, Academic Press, Boston, 1989.

8. R. J. Ellis, Entropy, Large Deviations and Statistical Mechanics, Springer-Verlag, New York, 1985.

9. W. Feller, An Introduction to Probability Theory and Its Applications, II, Wiley, New York, 1971.

10. H. Groemer, Geometric Applications of Fourier Series and Spherical Harmonics, Cambridge Univ. Press, Cambridge, 1996.

11. I. A. Ibragimov and Yu. V. Linnik, Independent and Stationary Sequences of Random Variables, Wolters-Noordhoff Publishing, Groningen, 1971.

12. A. V. Nagaev and A. Zaigraev, Abelian theorems for a class of probability distributions in R^{d} and their application, J. Math. Sci. 99(4) (2000), 1454–1462.

13. A. V. Nagaev and A. Zaigraev, Abelian theorems, limit properties of conjugate distributions and large deviations for sums of independent random vectors, Theory Probab. Appl. (2002). To appear [in Russian].

14. S. I. Resnick, Extreme Values, Regular Variation, and Point Processes, Springer-Verlag, New York, 1987.

Alexander Zaigraev

Faculty of Mathematics and Computer Science Nicolaus Copernicus University Chopin str. 12/18, 87-100 Toruń Poland

e-mail: alzaig@mat.uni.torun.pl

fax: +48-56-6228979 phone : +48-56-6113441