GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 6, 1997, 579-584

BOUNDS FOR THE CHARACTERISTIC FUNCTIONS OF

THE SYSTEM OF MONOMIALS IN RANDOM VARIABLES AND OF ITS TRIGONOMETRIC ANALOGUE

T. SHERVASHIDZE

Abstract. Using a multidimensional analogue of Vinogradov’s inequality for a trigonometric integral, the upper bounds are constructed for the moduli of the characteristic functions both of the system of monomials in components of a random vector with an absolutely continuous distribution in ℝ^{s} and of the system

(cos j_{1}πξ_{1} ⋯ cos j_{s}πξ_{s}, 0 ≤ j_{1}, …, j_{s} ≤ k, j_{1} + ⋯ + j_{s} ≥ 1), where (ξ_{1}, …, ξ_{s}) is uniformly distributed in [0, 1]^{s}.

Introduction. This note continues the series of studies [1–3] carried out on the initiative of Yu. V. Prokhorov and dealing with estimates of the characteristic functions (c.f.) of degenerate multidimensional distributions of the form

| ∫_{ℝ^{s}} exp{i(t, φ(x))}p(x) dx| ≤ C|t|^{−α} for |t| > t_{0},  (1)

where φ : ℝ^{s} → ℝ^{k}, s < k, t ∈ ℝ^{k}; p(x) is the distribution density; C, α, and t_{0} are positive constants. In what follows we shall use the notation I(t; φ(x), p(x)) for the integral from (1) and omit the condition |t| > t_{0}.

When s = 1, Sadikova has shown in [1] that for φ(x) = φ_{0}(x) = (x, x^{2}, …, x^{k}) one can take α = (1 + 1/k)^{−(k−1)}/k! if the integrals of |p′(x)| and |x|^{k−1}p(x) are finite. Her method is based on van der Corput’s lemma whose generalization allowed Yurinskii to assert that “for decreasing of the c.f. like a negative power of the argument modulus it is, roughly speaking, sufficient that the surface carrying the distribution have no tangencies of an

579 infinitely high order with (k−1)-dimensional hyperplanes, and that the surface density of the distribution be bounded, satisfy the Lipschitz condition in L_{1}-norm, and have several moments” (see [2]).

1991 Mathematics Subject Classification. 60E10.

Key words and phrases. Degenerate multidimensional distributions, bounds for characteristic functions, multidimensional analogue of Vinogradov’s inequality.

1072-947X/97/1100-0579$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1997 Plenum Publishing Corporation

The author’s paper [3] deals with the case s = 1, where φ(x) = φ_{1}(x) = (cos πx, …, cos kπx), p(x) = 11_{[0,1]}(x), where 11_{A}(x) is the indicator of the set A. Using Vinogradov’s famous inequality for a trigonometric integral [4], it is shown in [3] that in that case α = 1/2k (for k = 1 it can be deduced from [2] as well). An order with respect to |t| turned out to be exact (because of the exactness of Vinogradov’s inequality), i.e., in some directions of ℝ^{k} the c.f. behaves like |t|^{−1/2k} for |t| → ∞. Concomitantly, we obtained bounds with α = 1/k for the case φ(x) = φ_{0}(x) assuming that ∫_{ℝ^{1}} max(1, |x|)|p′(x)|dx < ∞ for the density p(x); for analogous estimates see [5].

φ_{0}(x) arises in problems connected with behavior of the joint distribution of sample moments. For φ_{1}(x) the inequality from [3] turns out important in proving the fact that the deviation of the distribution function of ω^{2}-test statistic from the limiting one has order n^{−1} (see [6], cf. [7]).

1. Multidimensional Analogue of Vinogradov’s Inequality. The Results. Let x=(x_{1}, …, x_{s}) ∈ ℝ^{s} and the the multiindex j=(j_{1}, …, j_{s}) ≠ 0 = (0, …, 0) ∈ ℝ^{s} vary in J_{k,s} = {0, 1, …, k}^{s} \ {0}. Consider the system of monomials in s real variables φ_{0}^{(s)}(x) = (x_{1}^{j_{1}} ⋯ x_{s}^{j_{s}}, j ∈ J_{k,s}) and also the corresponding system of cosines products

φ_{1}^{(s)}(x) = ( cos j_{1}πx_{1} ⋯ cos j_{s}πx_{s}, j ∈ J_{k,s}).

Denote τ = τ(t) = max {|t_{j}| : j ∈ J_{k,s}}, t = (t_{j}, j ∈ J_{k,s}) ∈ ℝ^{(k+1)^{s}−1}.

For φ_{j}^{(s}(x), j = 0, 1, inequalities of form (1), which can be used to study distributions of a system of mixed sample moments and other multivariate statistics, can be derived by the following multidimensional analogue of Vinogradov’s inequality [8, p. 39] written for our convenience in the form

|I(t; φ_{0}^{(s)}(x), 11_{[0,1]^{s}}(x)| ≤ 32^{s}(2π)^{1/k}τ^{−1/k} ln^{s−1}(2 + τ/2π);  (2)

note that by [8, p. 41] we have for γ > 1 that

| ∫_{[0,1]^{s}} exp\{iγx_{1}^{k} ⋯ x_{s}^{k}\}dx| ≥ [2πk^{s}(s − 1)!]^{−1}(2π)^{1/k}γ^{−1/k} ln^{s−1} \frac{γ}{2π} .

By virtue of the fact that the inequality τ(t̃) ≥ ∏_{j=1}^{s} [ min(1, |y_{j}|)]^{k}τ holds for y = (y_{1}, …, y_{s}) ∈ ℝ^{s} and for t̃ = (y_{1}^{j_{1}} ⋯ y_{s}^{j_{s}}t_{j}, j ∈ J_{k,s}) and the function on the right-hand side of (2) decreases with respect to τ, we have

|I_{y}(t)| ≤ ∏_{j=1}^{s} | sgn y_{j}| max(1, |y_{j}|)32^{s}(2π)^{1/k}τ^{−1/k} ln^{s−1}(2 + τ/2π) (3)

for

I_{y}(t) = ∫_{0}^{y_{1}} ⋯ ∫_{0}^{y_{s}} exp{i(t, φ_{0}^{(s)}(x))}dx.

Let D = [a_{1}, b_{1}]×⋯×[a_{s}, b_{s}] with positive h_{j} = b_{j} −a_{j}, j = 1, …, s; denote D(j_{1}, …, j_{r}) = [a_{j_{1}}, b_{j_{1}}]×⋯×[a_{j_{r}}, b_{j_{r}}] and D_{c}(j_{1} … j_{r}) = ∏{[a_{j}, b_{j}] : j = 1, …, s, j ≠ j_{1}, …, j_{r}} for 1 ≤ j_{1} < ⋯ < j_{r} ≤ s, 1 < r < s. The notation x(j_{1}, …, j_{s}) and x_{c}(j_{1} … j_{r}) for x ∈ ℝ^{s} is evident. Denote further by V, V_{c}(j_{1} … j_{r}) the sets of vertices of the parallelepipeds D, D_{c}(j_{1} … j_{r}), re-

spectively, and Π(y) = ∏_{j=1}^{r} max(1, |y_{j}|) for y = (y_{1}, …, y_{r}) ∈ ℝ^{r}, 1 ≤ r ≤ s.

Proposition 1. If a random vector ξ has the density

u_{D}(x) = (h_{1} ⋯ h_{s})^{−1}11_{D}(x),

then for the c.f. of φ_{0}^{(s)}(ξ) the inequality

|I(t; φ_{0}^{(s)}(x), u_{D}(x))| ≤ Π(h_{1}^{−1}, …, h_{s}^{−1})32^{s}(2π)^{1/k}τ^{−1/k} ln^{s−1}(2 + τ/2π)

holds.

To proceed to the general case with density p(x) concentrated on D one should use

Proposition 2. If for x ∈ D a function p(x) ≥ 0 is continuous in D and has, in D, continuous partial derivatives p_{i} = p_{x_{i}}, p_{ij} = p_{x_{i}x_{j}}, …, p_{1···s} = p_{x_{1}...x_{s}}, then the estimate

|I(t; φ_{0}^{(s)}(x), 11_{D}(x)p(x))| ≤ C32^{s}(2π)^{1/k}τ^{−1/k} ln^{s−1}(2 + τ/2π)

holds, where C = C_{0} + ⋯ + C_{s} with

C_{r} = ∑_{1≤j_{1}<···<j_{r}≤s} ∑_{x_{c}(j_{1}...j_{r})∈V_{c}(j_{1}...j_{r})} Π(x_{c}(j_{1} … j_{r})) ×

× ∫_{D(j_{1}...j_{r})} Π(x(j_{1} … j_{r}))p_{j_{1}...j_{r}}(x) dx(j_{1} … j_{r}), 1 ≤ r < s,

C_{0} = ∑_{x∈V} Π(x)p(x), C_{s} = ∫_{D} Π(x)p_{1...s}(x) dx.

If the above functions are unbounded or discontinuous in D or D is unbounded, then the estimate will demand obvious changes (conditions of the existence of some limits and integrals).

Since |t| ≤ [(k + 1)^{s} − 1]^{1/2}τ for t ∈ ℝ^{(k+1)^{s}−1}, Propositions 1–2 can be expressed in terms of |t|.

Let now T_{r}(x) = ∑_{j=0}^{r} a_{rj}x^{j}, x ∈ ℝ^{1}, be a Chebyshev polynomial of

the first kind and A_{k} be the matrix with rows (a_{r0}, …, a_{rr}, 0, …, 0), r = 0, …, k.

For a matrix B = (b_{pq})_{p,q=0,...,k} denote ρ(B) = max{∑_{q} |b_{pq}| : p =

0, …, k} and ρ_{k} = ρ((A_{k}^{−1})′) (′ means transposition). Let λ_{k} = λ_{min}(A_{k}A′_{k}) be the least eigenvalue of the matrix A_{k}A′_{k}.

Proposition 3. The following inequalities hold:

(a) |I(t; φ_{1}^{(s)}(x), 11_{[0,1]^{s}}(x))| ≤ 2^{\frac{9ks+1}{k(s+1)}} π^{− \frac{2ks−1}{k(s+1)}} s^{\frac{1}{s+1}} ×

×ρk^{\frac{s}{k(s+1)}} τ^{− \frac{1}{k(s+1)}} ln^{\frac{s−1}{s+1}} (2 + τ/2πρ_{k}^{s});

(b) |I(t; φ_{1}^{(s)}(x), 11_{[0,1]^{s}}(x))| ≤ 2^{\frac{9ks+1}{k(s+1)}} π^{− \frac{2ks−1}{k(s+1)}} s^{\frac{1}{s+1}} [(k + 1)^{s} − 1]^{\frac{1}{2k(s+1)}} ×

×λ_{k}^{− \frac{1}{2k(s+1)}} |t|^{− \frac{1}{k(s+1)}} ln^{\frac{s−1}{s+1}} (2 + \frac{|t|λ_{k}^{s/2}}{2π[(k+1)^{s}−1]^{1/2}} ).

2. Proofs. Proposition 1 is evident. Proof of Proposition 2. If the functions u : ℝ^{s} → ℂ and v : ℝ^{s} → ℂ are continuous in D along with the partial derivatives u_{x_{i}}, v_{x_{i}}, u_{x_{i}x_{j}}, v_{x_{i}x_{j}}, …, u_{x_{1}...x_{s}}, v_{x_{1}...x_{s}}, then

∫_{D} uv_{x_{1}...x_{s}}dx = Δ_{a}^{b}(uv) − ∑_{1≤i≤s} Δ_{a_{c}(i)}^{b_{c}(i)} ∫_{D(i)} vu_{x_{i}}dx(i) +

+ ∑_{1≤i<j≤s} Δ_{a_{c}(ij)}^{b_{c}(ij)} ∫_{D(ij)} vu_{x_{i}x_{j}}dx(ij) − ⋯ + (−1)^{s} ∫_{D} vu_{x_{1}...x_{s}}dx.

Substituting here u = p(x) and v = I_{x}(t), passing to moduli, and applying (3), we complete the proof. Proof of Proposition 3. By the change of variables x_{j} = arccos πy_{j}, j = 1, …, s, we obtain

I(t; φ_{1}^{(s)}(x), 11_{(0,1)^{s}}(x)) = I(t; φ_{0}^{(s)}(y), q(y)),  (4)

where

q(y) = π^{−s} ∏_{j=1}^{s}(1 − y_{j}^{2})^{−1/2}11_{(−1,1)^{s}}(y),

φ_{0}^{(s)}(y) = (T_{j_{1}}(y_{1}) ⋯ T_{j_{s}}(y_{s}), j ∈ J_{k,s}).

Apply Proposition 2 for a cube [−(1 − ε), 1 − ε]^{s} with 0 < ε < 1. Since

(1 − (1 − ε)^{2})^{1/2} < ε^{−1/2} and ∫_{0}^{1−ε}x(1 − x^{2})^{−3/2}dx < ε^{−1/2}, we have

C_{r} < (2/π)^{s} (\begin{matrix}s \\ r \end{matrix}) ε^{−s/2}, 0 ≤ r ≤ s, i.e., C < (4/π)^{s}ε^{−s/2}.

The remaining part of |I(t; φ_{0}^{(s)}(y), q(y)| is less than 1 − (1 − z)^{s} < sz with z = \frac{2}{π} (\frac{π}{2} − arcsin(1 − ε)) ≤ \frac{4}{π} \sqrt{ε}. Finally, we have

|I(t; φ_{0}^{(y)}(s), q(y))| ≤ (\frac{4}{π} )^{s}32^{s}(2π)^{1/k}τ^{−1/k} ln^{s−1}(2 + τ/2π)ε^{−s/2} + \frac{4s}{π} \sqrt{ε},

whence, minimizing with respect to ε, we get

|I(t; φ_{0}^{(s)}(y), q(y))| ≤ 2^{\frac{9ks+1}{k(s+1)}} π^{− \frac{2ks−1}{k(s+1)}} s^{\frac{1}{s+1}} τ^{− \frac{1}{k(s+1)}} ln^{\frac{s−1}{s+1}} (2 + τ/2π). (5)

Further,

(t, φ_{0}^{(s)}(y)) = ∑_{j∈J_{k,s}} t_{j}T_{j_{1}}(y_{1}) ⋯ T_{j_{s}}(y_{s}) = (t^{0})′A_{k}^{(s)}(1, φ_{0}^{(s)}(y))

with A_{k}^{(s)} = \underbrace{A_{k} ⊗ ⋯ ⊗ A_{k}}_{s times}, A_{k}^{(1)} = A_{k}, where ⊗ denotes the Kronecker

product, t^{0} = (0, t) and φ_{0}^{(s)}(y) and t are understood as row vectors with lexicographically arranged components.

From (4) we now obtain

I(t; φ_{1}^{(s)}(x), 11_{(0,1)^{s}}(x)) = I(A_{k}^{(s)′}t^{0}; (1, φ_{0}^{(s)}(y)), q(y)).  (6)

Since ρ(B_{1} ⊗ B_{2}) = ρ(B_{1})ρ(B_{2}) for matrices B_{1} and B_{2}, according to [9, Theorem 6.5.1], τ((A_{k}^{(s)})′t^{0}) ≥ τ/ρ((A_{k}^{(s)′})^{−1}) ≥ τ/ρ_{k}^{s}, which by virtue of (5) and (6) leads to (a).

To prove (b), note that since A_{k}^{(s)}(A_{k}^{(s)})′ = (A_{k}A′_{k})^{(s)} (see [10]) and for two positive definite matrices λ_{min}(B_{1} ⊗ B_{2}) = λ_{min}(B_{1})λ_{min}(B_{2}) (see, e.g., Supplement to [11]), we obtain |A_{k}^{(s)′}t^{0}| = ((t^{0})′A_{k}^{(s)}(A_{k}^{(s)})′(t^{0}))^{1/2} ≥ λ_{k}^{s/2}|t| and τ(A_{k}^{(s)′}t^{0}) ≥ [(k + 1)^{s} − 1]^{−1/2}λ_{k}^{s/2}|t|.

Other applications of (2) to obtain bounds for c.f. can be found in [12]. Inequalities similar to the general ones from [2] can be found in [13].

Based on the theory of singularities and asymptotic expansions of oscillating integrals [14] one can study the exactness properties of bounds (a) and (b) with respect to |t| and τ for the case s > 1, too.

Acknowledgements

The results presented here were reported at the Japan–Russia Symposium on Probability Theory and Mathematical Statistics (Tokyo, 1995). Having received the invitation from the Japanese and Russian Organizing Committees, the author expresses his thanks to both, the first of which funded his travel and participation.

Professors V. V. Yurinskii and S. A. Molchanov and Dr. M. Jibladze have given the author useful references.

References

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11. T. W. Anderson, An introduction to multivariate statistical analysis. Chapman & Hall, New York etc., 1958.

12. F. Götze, Yu. V. Prokhorov, and V. V. Ulyanov, Bounds for characteristic functions of polynomials in asymptotically normal variables. (Russian) Uspekhi Mat. Nauk 51(1996), No. 2(308), 3–26.

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14. V. I. Arnold, A. N. Varchenko, and S. M. Gussein-Zadeh, Singularities of differentiable mappings. 2. Monodromy and asymptotics of the integrals. (Russian) Nauka, Moscow, 1984.

(Received 09.06.1995; revised 10.04.1996)

Authors address: A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia