Tensor products of operators with regular spectral asymptotics and small deviations of Gaussian fields. A.I. Karol’, A.I. Nazarov, Ya.Yu. Nikitin,

Dept. of Mathematics and Mechanics St.Petersburg State University, Russia Contents

1. Introduction

2. Slowly varying functions

3. Spectral asymptotics for tensor products of selfadjoint operators

4. Small deviations and the rate of eigenvalues decrease

5. Fields-products with equal marginal exponents 6. Fields-products with distinct exponents

7. Additional results

1. Introduction

Let X(x), x ∈ Ω ⊂ ℝ^{d}, be a Gaussian random field with zero mean and covariance function

G_{X}(x, y) = EX(x)X(y), x, y ∈ Ω. Let ρ ≥ 0 be a summable weight on Ω. Put

||X||_{ρ} = (∫_{Ω} X^{2}(x)ρ(x)dx)^{1/2}.

Problem: What is the behavior of P{||X||_{ρ} ≤ ε} as ε → 0? Easier problem : To describe as ε → 0

ln P{||X||_{ρ} ≤ ε} (logarithmic asymptotics)

According to Karhunen-Loève expansion, we have equality in distribution

||X||_{ρ}^{2} = ∫_{Ω} X^{2}(x)ρ(x)dx = ∑_{n=1}^{∞} λ_{n}ξ_{n}^{2},

where ξ_{n}, n ∈ ℕ, are independent standard normal r.v.’s, and λ_{n} > 0, n ∈ ℕ, ∑_{n} λ_{n} < ∞ are the

eigenvalues of the integral equation

λf(x) = ∫_{Ω} G_{X}(x, y)\sqrt{ρ(x)ρ(y)}f(y)dy, x ∈ Ω.

Thus we arrive to the problem of studying the asymptotic behavior as ε → 0 of

P {∑_{n=1}^{∞} λ_{n}ξ_{n}^{2} ≤ ε^{2}} or ln P {∑_{n=1}^{∞} λ_{n}ξ_{n}^{2} ≤ ε^{2}} .

We study small deviations of Gaussian fields X having the covariance of ”tensor product” type.

Let X(x), Y (y) be zero mean Gaussian fields with covariances G_{X}(x, u) and G_{Y} (y, v). Consider the zero mean Gaussian function Z(x, y) with the covariance

G_{Z}((x, y), (u, v)) = G_{X}(x, u)G_{Y} (y, v).

We write Z = X ⊗ Y and we call Z the tensor product of X and Y using the term field-product.

Classical examples of such fields are the Brownian sheet (W ⊗ W) , the Brownian pillow ( B ⊗ B) and the Kiefer field ( W ⊗ B).

The generalization to the multivariate case

𝕏(x) = ⊗^{d}_{j=1}X_{j}

is straightforward.

First results on logarithmic L_{2}−small deviations of such Gaussian fields belong to Csáki (1982) and Li (1992).

They showed that the ”marginal” fields can have the pure power spectral asymptotics while the spectral asymptotics for the field-product can contain the logarithmic factor.

We consider more general case. Assume that the marginal eigenvalues λ_{n}^{(j)} have regular behaviour :

λ_{n}^{(j)} ∼ \frac{φ_{j}(n)}{n^{p_{j}}}, n → ∞,  (1)

where p_{j} > 1 and φ_{j} are some slowly varying functions (SVFs), in particular, the powers of logarithms.

This is common in papers on spectral asymptotics, see Birman and Solomyak (2001), Dereich (2003) and Luschgy and Pagès (2002, 2004).

We develop an original approach. It uses new theorems on spectral asymptotics for tensor products of self-adjoint operators in Hilbert space.

Recently Fill and Torcaso (2004) obtained exact asymptotics of L_{2}-small deviations for the Brownian sheet and its integrated counterparts.

Their approach uses the Mellin transform and is entirely different. It is not clear if their approach makes possible obtaining small deviation results for a vast class of Gaussian fields-products studied in the present talk, when the eigenvalues satisfy (1).

Moreover, our results are applicable for the weighted L_{2}-norm in the case of continuous weights on the unit cube in ℝ^{d}. This seems to be beyond the reach of the method using Mellin transform.

2. Slowly varying functions

Recall that a positive function φ(τ), τ > 0, is called a slowly varying function (SVF) at infinity if for any c > 0

φ(cτ)/φ(τ) → 1 as τ → +∞.

We define the Mellin convolution of two SVFs φ and ψ as

(φ∗ψ)(τ) = ∫_{1}^{τ} φ(σ)ψ(τ/σ)\frac{dσ}{σ} = h_{φ,ψ}(τ)+h_{ψ,φ}(τ),

where

h_{φ,ψ}(τ) = ∫_{1}^{\sqrt{τ}}φ(σ)ψ(τ/σ)\frac{dσ}{σ}.

We have proved some auxiliary lemmas concerning the properties of Mellin convolutions. In particular, h_{φ,ψ} is also SVF.

3. Spectral asymptotics for tensor products of compact self-adjoint operators Let T = T^{∗} ≥ 0 be a compact self-adjoint operator in a Hilbert space H; let λ_{n} = λ_{n}(T ) be its positive eigenvalues according to their multiplicity.

In spectral theory it is common to consider the counting function

N (t) = N (t, T ) = #{n : λ_{n}(T ) > t}.

It is easily seen that

1. The relation N (t, T ) = O(t^{−1/p}), t → +0, is equivalent to λ_{n}(T ) = O(n^{−p}), n → ∞.

2. The asymptotics

N (t, T ) ∼ \frac{φ(1/t)}{t^{1/p}}, t → +0,

(here p > 0 while φ is a SVF) is equivalent to the spectral asymptotics

λ_{n}(T ) ∼ \frac{ψ(n)}{n^{p}}, n → ∞,

where ψ is also a SVF.

First suppose that the counting functions have distinct asymptotic exponents near zero.

Theorem 1. Let operators T and T̃ be such compact self-adjoint nonnegative operators in a Hilbert space that as t → +0

N (t) ∼ \frac{φ(1/t)}{t^{1/p}}, Ñ(t) = O(t^{−1/p̃}), p̃ > p.

Then the operator T ⊗ T̃ satisfies the relation

N_{⊗}(t) ∼ \frac{φ(1/t)}{t^{1/p}} ∑_{n} λ_{n}^{1/p}(T̃), t → +0.

In the case of equal asymptotic exponents for counting functions, the result is more refined.

Theorem 2. Let T and T̃ be such that

N (t) ∼ a · \frac{φ(1/t)}{t^{1/p}}, Ñ(t) ∼ ã · \frac{φ̃(1/t)}{t^{1/p}}, t → +0.

1. If

∫_{1}^{∞} φ(τ)\frac{dτ}{τ} = ∫_{1}^{∞} φ̃(τ)\frac{dτ}{τ} = ∞,

then

N_{⊗}(t) ∼ \frac{χ(1/t)}{t^{1/p}}, t → 0, (2)

where

χ(s) = \frac{aã}{p}(φ ∗ φ̃)(s).

2. If

∫_{1}^{∞} φ(τ)\frac{dτ}{τ} < ∞, ∫_{1}^{∞} φ̃(τ)\frac{dτ}{τ} = ∞,

and some mild regularity conditions on φ are fulfilled then (2) holds true with

χ(s) = ãφ̃(s) ∑_{n} λ_{n}^{1/p} + \frac{aã}{p}h_{φ̃,φ}(s).

3. If

∫_{1}^{∞} φ(τ)\frac{dτ}{τ} < ∞, ∫_{1}^{∞} φ̃(τ)\frac{dτ}{τ} < ∞,

and the same regularity conditions on φ, φ̃ are fulfilled, then (2) holds true with

χ(s) = aφ(s) ∑_{n} λ̃_{n}^{1/p} + ãφ̃(s) ∑_{n} λ_{n}^{1/p}.

Example 1. Let, as t → +0,

N (t) ∼ a · \frac{ln^{κ_{1}}(1/t)}{t^{1/p}}, Ñ(t) ∼ ã · \frac{ln^{κ_{2}}(1/t)}{t^{1/p}}.

Then the asymptotics (2) holds true with

χ(s) =

= \{ \begin{matrix}\frac{aã}{p} · B(κ_{1} + 1, κ_{2} + 1)· & ln^{κ_{1}+κ_{2}+1}(s), \\ & κ_{1}, κ_{2} > −1; \\ \frac{aã}{p} · ln(ln(s)) · ln^{κ_{2}}(s), \\ & κ_{1} = −1, κ_{2} > −1; \\ \frac{2aã}{p} · ln(ln(s)) ln^{−1}(s), \\ & κ_{1} = κ_{2} = −1; \\ ã ∑_{n} λ_{n}^{1/p} · ln^{κ_{2}}(s), \\ & κ_{1} < −1, κ_{1} < κ_{2}; \\ (a ∑_{n} λ̃_{n}^{1/p} + ã ∑_{n} λ_{n}^{1/p}) & · ln^{κ_{1}}(s), \\ & κ_{1} = κ_{2} < −1, \end{matrix}

where B is the Euler beta-function. The particular case κ_{1} = 0, κ_{2} = 0 is known from Papageorgiou and Wasilkowski (1990), see also Luschgy and Pagès (2004).

4. Small ball asymptotics and the rate of eigenvalues decrease

Consider the case when

λ_{n} ∼ C \frac{ψ(n)}{n^{p}} , n → ∞,  (3)

where p > 1 while ψ is an SVF.

The case λ_{n} = Cn^{−p}, p > 1, is now classical one and is well-known.

Put Φ(n) = \frac{ψ(n)}{n^{p}} and consider for large u the equation

r = \frac{2^{1/p}π}{2p sin \frac{π}{p}} · \frac{Φ^{−1}(1/u)}{u},

which yields

u(r) ∼ r^{−\frac{p}{p−1}}ζ(1/r), r → 0,  (4)

where ζ is some SVF.

Theorem 3. Let the eigenvalues λ_{n} have the asymptotics (3). Then, as ε → 0,

ln P\{||X||_{ρ} ≤ ε\} ∼ −(p − 1)C^{\frac{1}{p−1}}ε^{−\frac{2}{p−1}}ζ(ε^{−2}), where the SVF ζ is defined in (4).

Example 2. Let the eigenvalues λ_{n} satisfy

λ_{n} ∼ C\frac{ln^{σ}(n + 1)}{n^{p}}, σ ∈ ℝ, p > 1, C > 0.

Then

lim_{ε→0} ε^{\frac{2}{p−1}} ln^{−\frac{σ}{p−1}}(1/ε) · ln P\{||X||_{ρ} ≤ ε\} =

= −C^{\frac{1}{p−1}} (\frac{p − 1}{2})^{1−\frac{σ}{p−1}} (\frac{π}{p sin \frac{π}{p}})^{\frac{p}{p−1}} .

Let examine briefly the special case p = 1. In this case we cannot obtain explicitly the logarithmic asymptotics. We give only one example of weaker (double-logarithmic) asymptotics.

Theorem 4. Suppose that as n → ∞

λ_{n} ∼ \frac{C}{n ln^{σ}(n + 1)} , σ > 1, C > 0.

Then

lim_{ε→0} ε^{\frac{2}{σ−1}} · ln|ln P\{||X||_{ρ} ≤ ε\}| = (\frac{C}{σ − 1})^{\frac{1}{σ−1}} .

5. Fields-products with equal marginal exponents

We deal with the fields of the form

X(x) = ⊗^{d}_{j=1}X_{j}(x_{j}), x = (x_{1}, …, x_{d}) ∈ [0, 1]^{d},

where X_{j} are independent zero mean Gaussian fields with equal marginal exponents p_{j} in the asymptotic representation (1). Also we consider weight functions ρ ∈ C([0, 1]^{d}). Consider the fractional Brownian sheet 𝕎_{d}^{H}(x). Its marginal processes (factors) X_{j} are the fractional Brownian motions W_{j}^{H} (H ∈]0, 1[ is the Hurst parameter) , and the covariance function is

G_{𝕎_{d}^{H}}(x, y) = \frac{1}{2^{d}} ∏_{j=1}^{d}(x_{j}^{2H} + y_{j}^{2H} − |x_{j} − y_{j}|^{2H}).

Theorem 5. Let 0 < H < 1 and let ρ ∈ C([0, 1]^{d}). Denote J_{h} = ∫_{[0,1]^{d}} ρ(x)^{1/h}dx.Then

lim_{ε→0} ε^{1/H} ln^{−(d−1)\frac{2H+1}{2H}}(1/ε)·ln P\{||𝕎_{d}^{H}||_{ρ} ≤ ε\} =

= −\frac{J_{2H+1}H^{2−d}}{(d − 1)!π^{d−1}(2H + 1) sin (\frac{π}{2H+1})} ·

(\frac{J_{2H+1}(Γ(2H + 1) sin(πH))^{d}}{(d − 1)!(πH)^{d−1}(2H + 1) sin (\frac{π}{2H+1})})^{\frac{1}{2H}} .

Remark. This statement in the case ρ ≡ 1 was obtained by Luschgy and Pagès (2004).

Similar result holds for the isotropically integrated fractional Brownian sheet

(𝕎_{d}^{H})_{m}(x) = ⊗^{d}_{j=1}(W^{H})_{m}(x_{j}),

where

(W^{H})_{m}(t) ≡ (W^{H})_{m}^{[β_{1}, ..., β_{m}]}(t) =

= (−1)^{β_{1}+ ... +β_{m}} \underbrace{∫_{β_{m}}^{t} … ∫_{β_{1}}^{t_{1}}}_{m} W^{H}(s) ds dt_{1} …

(here any β_{k} equals either zero or one, t ∈ [0, 1]; for various j the multi-indices [β_{1}, …, β_{m}] can differ), for the fractional Ornstein – Uhlenbeck sheet and for the isotropically integrated fractional OU sheet.

6. Fields-products with distinct marginal exponents

Now let the marginal processes of fields-products have different power rates of eigenvalues decrease. We call X_{j} a slow factor if it has the least exponent p_{j} in (1) and call it a fast factor otherwise.

From Theorem 1 it follows that the rate of spectral asymptotics for the field-product is determined only by slow factors but the constant requires precise information on the fast factors spectra. Theorem 6. Let λ_{n}^{(X)} and λ_{n}^{(Y)} be the eigenvalues corresponding to the zero mean Gaussian processes X and Y on [0, 1]. Suppose that

λ_{n}^{(X)} ∼ C\frac{ψ(n)}{n^{p}}, λ_{n}^{(Y)} = O(n^{−p̃}), n → ∞,

with p̃ > p. Then, as ε → 0,

ln P\{||\sqrt{ρ}X ⊗ Y || ≤ ε\} ∼

∼ (∑_{n}(λ_{n}^{(Y)})^{1/p})^{\frac{p}{p−1}} ln P\{||X||_{ρ} ≤ ε\}. (5)

The result for the tensor product with arbitrary number of fast factors looks similarly. Unfortunately, the trace-type sums in (5) usually cannot be written out explicitly.

( The eigenvalues of Anderson – Darling process , see Anderson and Darling(1952),

B(t)/\sqrt{t(1 − t)}, t ∈ [0, 1],

are λ_{n} = (n^{2} + n)^{−1}, n ≥ 1. Even in this simple example the trace-type sums have no explicit expressions).

Consider some cases when these sums can be expressed in terms of Riemann ζ-function. For any Gaussian process X(t), t ∈ [0, 1], we introduce the centered process \overline{X}(t) = X(t) − ∫_{0}^{1} X(s) ds and the on-line centered process

X̂(t) = X(t) − t^{−1} ∫_{0}^{t} X(s)ds.

Theorem 7. Let the eigenvalues of X satisfy (1). If p ∈]1, 2[, then, as ε → 0

ln P\{||\sqrt{ρ}X ⊗ W || ≤ ε\}

∼ (2^{\frac{2}{p}} − 1)^{\frac{p}{p−1}}π^{−\frac{2}{p−1}}ζ^{\frac{p}{p−1}} (2/p) ln P\{||X||_{ρ} ≤ ε\};

ln P\{||\sqrt{ρ}X ⊗ \overline{W} || ≤ ε\}

∼ π^{−\frac{2}{p−1}}ζ^{\frac{p}{p−1}} (2/p) ln P\{||X||_{ρ} ≤ ε\};

ln P\{||\sqrt{ρ}X ⊗ Ŵ|| ≤ ε\}

∼ π^{−\frac{2}{p−1}}ζ^{\frac{p}{p−1}} (2/p) ln P\{||X||_{ρ} ≤ ε\}.

Similar results hold when replacing W by B and by some integrated processes linked to W and B.

Example 3. Consider the field W^{H} ⊗B⊗(\overline{W})_{1}, 0 < H ≤ 1/2. If H < 1/2, then, as ε → 0,

ε^{1/H} ln P\{||\sqrt{ρ}W^{H} ⊗ B ⊗ (\overline{W})_{1}|| ≤ ε\} ∼

∼ −π^{−\frac{3}{H}}ζ^{\frac{2H+1}{2H}} (\frac{4}{2H+1}) ζ^{\frac{2H+1}{2H}} (\frac{2}{2H+1}) ·

· _{\frac{J_{2H+1}H}{(2H+1) sin(\frac{π}{2H+1})}} (_{\frac{J_{2H+1}Γ(2H+1) sin(πH)}{(2H+1) sin(\frac{π}{2H+1})}})^{\frac{1}{2H}} .

If H = 1/2, then, as ε → 0,

ln P\{||\sqrt{ρ}W ⊗ B ⊗ (\overline{W})_{1}|| ≤ ε\} ∼ ∼ π^{−4}ζ^{2}(2) ln P\{||\sqrt{ρ}W ⊗ B|| ≤ ε\} ∼

∼ −\frac{(∫_{0}^{1} ρ(t)^{1/2}dt)^{2}}{72π^{2}}ε^{−2} ln^{2}(1/ε).

Even for ρ ≡ 1 when our field has the exactly known spectrum

λ_{jkℓ} = π^{−8}(j − 1/2)^{−2}k^{−2}ℓ^{−4}, j, k, ℓ ≥ 1,

the result is new and is not covered by the paper of Li (1992).

7. Additional results

Another type of Gaussian fields we consider is the class of additive processes. For shortness we call it fields-sums. Let X_{j}(x_{j}), 1 ≤ j ≤ d, x_{j} ∈ [0, 1]^{ℓ_{j}}, be independent random functions. The field-sum 𝕏(x) is defined as

𝕏(x) = ∑_{j=1}^{d} X_{j}(x_{j}), x = (x_{1}, ..., x_{d}) ∈ [0, 1]^{D},

where D = ∑_{j=1}^{d} ℓ_{j}, with the covariance

G_{𝕏}(x, y) = ∑_{j=1}^{d} G_{X_{j}}(x_{j}, y_{j}).

Such fields appear, e.g., in the theory of intersections of Brownian processes. Chen and Li (2003) proved some small deviation results with respect to supremum norm. The results in the L_{2}-norm we obtained appear for the first time.

We study the sum of tensor products of compact operators and prove various theorems on the asymptotic behavior of counting function for the sum

N (t, S_{1} + S_{2})

in terms of asymptotic behavior of counting functions N (t, S_{1}) and N (t, S_{2}). Then we pass to the small deviations of fields-sums.

Example 4. Let H ∈ (0, 1/2). Then, as ε → 0

ln P \{∥B^{H} ⊗ W + \overline{B} ⊗ W^{H}∥ ≤ ε\} ∼

∼ C(H) ζ^{\frac{2H+1}{2H}} (\frac{2}{2H+1}) ln P \{∥W^{H}∥ ≤ ε\} .

The results of previous sections can be also generalized to the vector-valued zero mean Gaussian fields X(x) ∈ ℝ^{ℓ}, x ∈ [0, 1]^{d}.

REFERENCES

ANDERSON, T.W. and DARLING, D.A. (1952). Asymptotic theory of certain ”goodness-of-fit” criteria based on stochastic processes. Ann. Math. Statist., 23, 193–212. BIRMAN, M.Sh. and SOLOMYAK, M. (2001). On the negative discrete spectrum of a periodic elliptic operator in a waveguide-type domain, perturbed by a decaying potential. Journ. d’Analyse Math., 83, 337–391. BRONSKI, J.C. (2003). Small ball constants and tight eigenvalue asymptotics for fractional Brownian motions. J. Theor. Probab., 16, 87–100. CHEN, X. and LI, W.V. (2003). Small deviation estimates for some additive processes. Proc. Conf. High Dimensional Probability III, Progress in Probability, 55, Birkhäuser, 225–238. CSÁKI, E. (1982). On small values of the square integral of a multiparameter Wiener process. In: Statistics and Probability. Proc. of the 3rd Pannonian Symp. on Math. Stat. D.Reidel, Boston, 19–26.

DEREICH, S. (2003). Small ball probabilities around random centers of Gaussian measures with applications to quantization. J. Theor. Probab., 16, 427-449. FILL, J.A. and TORCASO, F. (2004). Asymptotic analysis via Mellin transforms for small deviations in L_{2}-norm of integrated Brownian sheets. Probab. Theory and Rel. Fields, 130, 259–288. LI, W.V. (1992). Comparison results for the lower tail of Gaussian seminorms, J. of Theor. Prob., 5, 1–31. LUSCHGY, H. and PAGÈS, G. (2002). Functional quantization of Gaussian processes. J. Funct. Analysis, 196, 486–531. LUSCHGY, H. and PAGÈS, G. (2004). Sharp asymptotics of the functional quantization problem for Gaussian processes, Ann. Probab., 32, 1574– 1599. PAPAGEORGIOU, A. and WASILKOWSKI, G.W. (1990). On the average complexity of multivariate problems. J. of Complexity, 6, 1–23.