On Small Deviation Asymptotics in the L 2 -Norm for Certain Gaussian Processes

: The results obtained allow ﬁnding sharp small deviations in a Hilbert norm for centered Gaussian processes in the case where their covariances have a special form of the eigenvalues and allow us to describe small deviation asymptotics for certain Gaussian processes.


Introduction and Results
Consider a centered Gaussian process X(t), 0 ≤ t ≤ 1, with covariance G(t, s) = EX(t) X(s), t, s ∈ [0, 1] such that 0 < 1 0 G(t, t) dt < ∞. Set ||X(t)|| 2 2 = 1 0 X 2 (t) dt. Our aim is to obtain sharp asymptotics of the probability P(||X 2 (t)|| 2 < ε) as ε → 0. Due to the well-known classical Karunen-Loève expansion (see, for instance, [1]), the following equality in the distribution: λ n ξ 2 n takes place, where ξ n , n ≥ 1, are standard independent Gaussian rv's, while positive summable λ n are the eigenvalues of the integral equation: Hence, the problem examined is equivalent to studying the asymptotic behavior of the probability P( ∑ n≥1 λ n ξ 2 n < ε) as ε → 0 or, in a more general setting, the probability The problem of small deviations for the norms of Gaussian processes (including one of the simplest ones, the norm in L 2 ) has been studied quite intensively (see the bibliography in [2][3][4]). However, due to the fact that explicit formulas for λ n are known for a few concrete processes, sharp asymptotics often cannot be found. Nevertheless, this was done in [2] for an interesting and rather general case where the coefficients λ n behave as quotients of powers of two polynomials.
We present this nice result since it is the starting point for our research. To formulate it, we introduce some notation. For p > 0 and V > 1, set: For example, K(V, 2) = 2 (1−V)/V π/ sin (π/V). Let us define polynomials: where We assume that for n = 1, 2, . . . , the polynomials Q r (n), P q (n) are strictly positive and that real constants µ, ν satisfy the condition: as well ss that the sequence a n = Q ν r (n)/P µ q (n) does not increase. then: as ε → +0, where (see (1), (3), and (4)): and: Proposition 1 was proven in [2], Theorem 1. Note that Condition (4) and the assumption on the sequence a n , which is non-increasing, which is used essentially in the proof, were not explicitly mentioned in the formulation of [2], Theorem 1.
The purpose of the present note is to extend Proposition 1 to a more general class of weights a n and to omit the constraint (4).
Let us formulate the results. For n, m ∈ N, λ j ∈ R, and z j ∈ C (1 ≤ j ≤ m), we set Note that the weights Q ν r (n)/P µ q (n) in Proposition 1 constitute a subcase of the weights a n above (one can derive this by taking λ j to be equal to µ or −ν). Theorem 1. Let the integer k ≥ 0, z j = −1, −2, . . . , and: Then, for every p > 0: as ε → +0, with the same notation as in Proposition 1, except the "new" V and σ.
Note that Equality (7) still holds true if we assume that z j = −k − 1, −k − 2, . . . and replace the numbers a j , 1 ≤ j ≤ k, in (7) with arbitrary positive constants.
Note also that the minimal k satisfying Condition (6) is equal to [max (−σ, 0)], where [x] stands for the integer part of x.

Proofs
Let us prove the statement of Theorem 1 for k = 0. We follow the lines of the proof in [2]. Set b n = (n + σ) −V . Taking into account the equality |n + z j | 2 = n 2 + 2n z j + |z j | 2 , it is easy to verify that: and hence, the product ∞ ∏ j=1 (b n /a n ) converges. To calculate it, we use the well-known representation for the gamma function: where c is the Euler constant. According to this formula, we have: Taking into account that m ∑ j=1 λ j ( z j − σ) = 0, and σ > −1, we get: Hence, Applying the comparison theorem ( [5], Theorem 2.1) (which is possible due to Condition (8)) and [6], Lemma 1, we conclude that Theorem 1 holds for k = 0. Now, let k ≥ 1. Represent the probability on the left-hand side of (7) as P(X 1 + X 2 < ε), where X 1 = k ∑ n=1 a n |ξ n | p , X 2 = ∑ n≥1 a n+k |ξ n+k | p , and let us denote F 1 (·), F 2 (·) the distribution functions of X 1 , X 2 , respectively.
It is easy to see that for any positive a n : P(a n |ξ n | p < ε) ∼ 2 π ε a n 1 p , ε → +0.