On the infinite sums of deflated Gaussian products ∗

In this paper we derive the exact tail asymptotic behaviour of S∞ = ∑∞ i=1 λiXiYi, where λi, i ≥ 1 are non-negative square summable deflators (weights) and Xi, Yi, i ≥ 1, are independent standard Gaussian random variables. Further, we consider the tail asymptotics of S∞;p = ∑∞ i=1 λiXi |Yi| p , p > 1, and also discuss the influence on the asymptotic results when λi’s are independent random variables.


Introduction and Main Result
Let X i , Y i , i ≥ 1 be independent standard (zero mean and unit variance) Gaussian random variables, and let λ i , i ≥ 1 be non-negative constants.Define, for some fixed integer n, the weighted sum Since a one-dimensional projection of a standard Gaussian random vector is distributed as a Gaussian random variable, we have the stochastic representation where d = stands for the equality of the distribution functions.Note in passing that (1.1) holds for general random variables Y 1 , . . ., Y n being independent of Gaussian random variables X 1 , . . ., X n .Clearly, due to the symmetry about 0 of X i , i ≥ 1, the following stochastic representation is also valid, and hence S n is a symmetric (about 0) random variable.In this paper, we will make use of the stochastic representations of S n , following from the fact that X i , i ≥ 1, are Gaussian.For notational simplicity, we consider in the following ordered weights λ i , i ≥ 1 meaning In view of [7] (see also [11] p.168 and [16]), we have (set as x → ∞.The standard notation ∼ stands for asymptotic equivalence as the argument tends to infinity, and Γ(•) denotes the Euler Gamma function.
In the light of Lemma 3.4 (presented in Section 3) we obtain which is shown in the special case λ i = λ, i ≤ n in the first Lemma of [8].
An interesting quantity with application in statistics, insurance and several other applied fields is the random variable where the non-negative weights λ i , i ≥ 1 are square summable, i.e., ∞ i=1 λ 2 i < ∞. ( The random variable S ∞ appears as the distributional limits for various statistics; for instance as shown by [9], S ∞ is essential for the characterisation of continuous, separately exchangeable processes.Several examples of statistics given as infinite weighted sums of Gaussian products appear naturally when dealing with U -statistics or row and column exchangeable processes.The main result of [8] gives an upper bound for the tail probability of the random variable S ∞ , namely with K some unknown constant and S m = λ m i=1 X i Y i .Recall that m is the multiplicity of the largest weight λ = λ 1 , and by the square summability assumption on λ i , i ≥ 1, m is necessarily a finite integer.The main goal of this contribution is to derive, instead of the bound above, the exact tail asymptotic behaviour of |S ∞ |.Our main result below gives such an asymptotic expansion showing further connections with the tail asymptotic behaviours of Z ∞ and S n for any n ≥ m.Theorem 1.1.Let X i , Y i , i ≥ 1 be independent standard Gaussian random variables.For given constants λ : On the infinite sums of deflated Gaussian products holds as x → ∞.Furthermore, as x → ∞, we have and (1.9)

Extensions and Discussions
In this section we discuss two directions which provide natural extensions of the main result presented in Theorem 1.1.Initially, motivated by the stochastic representation (1.2), we consider S n;p , p > 0 defined by The same reasoning as (1.1) yields We show in Lemma 3.1 below that both S ∞;p and Z ∞;p exist almost surely, and moreover Since the tail asymptotic behaviour of Z ∞;p is known (see [3], [10] and [11]) applying Lemma 3.3 for any p > 1 we obtain where we use the same notation and assumptions as in Theorem 1.1.We note in passing that for the cases when p ∈ (0, 1) the asymptotics can not be obtained similarly due to the complexity of the tail asymptotic behaviour of Z ∞;p (e.g., [10,11,12]).
Our second extension concerns the case of random deflators with R i ∈ (0, 1], i ≥ 1 and λ i 's as above.A close inspection of (1.8) indicates that the main contribution in the asymptotics of P {| However, the asymptotic behavior of the latter is, in general, not easy to obtain, and thus in the following we discuss a special case that for some positive constant γ.The above assumption means that the function P {R 1 > 1 − s} is regularly varying at 0, so we have with L(•) a positive slowly varying function at 0, i.e., lim u→0 L(us)/L(u) = 1, ∀s > 0. For more details on the max-domains of attraction and regularly varying functions see [2], [4] or [15].
Our final remark concerns the role of the deterministic weights λ i , i ≥ m.In view of our asymptotic results in the above theorems, the restriction that λ i , i > m, are positive is not necessary since only λ 2 i , i > m, appear.Therefore this condition can be replaced by assuming instead for all i > m.

Further Results and Proofs
We present first some lemmas and then continue with the proofs of Theorem 1.1 and Theorem 2.1.
it follows that P {Z ∞;p < ∞} = 1.Similarly, using again (1.5) establishing thus (3.1).In order to show (3.2), it is sufficient to prove that the characteristic functions of both sides coincide.For any s ∈ R we have ∞;p implying (3.2), and thus the proof is complete.
Lemma 3.2.Let ξ i , i = 1, 2, be two non-negative independent random variables such that, as x → ∞, with some positive constants C i , p i , i = 1, 2, α 1 , α 2 ∈ R, and two positive measurable where ξ * 1 , ξ * 2 are two independent non-negative random variables such that for all x large enough Proof.The proof is similar to that of Lemma 3.2 in [5], and therefore omitted here.
Lemma 3.3.Under the assumptions of Lemma 3.2, if further and L 2 (x) = L 2 ∈ (0, ∞) for all large x, then we have Proof.If L 1 (x) = L 1 > 0 for all x > 0, the claim is established by Lemma 2.1 in [1].In the light of Lemma 3.2, we can restrict our attention to the simpler case that hold for x sufficiently large.As in the proof of Proposition 3.1 of [13], for some 0

By (3.5) we can further write
Since the function ψ(y) = L 1 (Ay) −p1 + L 2 (Ay) p2 attains its minimum in [l 1 , l 2 ] at 1, applying the Laplace approximation we obtain and hence the claim follows.
Lemma 3.4.Let X i , Y i , i ≤ n, be independent standard Gaussian random variables.For given weights λ = Proof.By the representation of S n in (1.1) we need to derive the tail asymptotic of 1 by taking 3) and Lemma 3.3 establish the claim.
The following result, which is restatement of Lemma 2.1 in [14], is crucial for the proof of Theorem 1.1.We present below an alternative proof.Write S m,∞ = ∞ i=m+1 λ i X i Y i , hence S ∞ = S m + S m,∞ .If λ m+1 = 0, then the proof follows by (3.6).We consider therefore below only the case λ m+1 > 0. By the symmetry about 0 of S ∞ and S m for any x > 0 we have In view of (3.6) i Y 1 is in the Gumbel max-domain of attraction with constant auxiliary function a(x) = λ, i.e., Since λ m+1 ∈ (0, λ) (with the multiplicity denoted by m 1 ) we have Furthermore, in view of Lemma 3.1, for any s ∈ (1, λ/λ m+1 ), Consequently, applying Lemma 3.5 we obtain hence the first claim follows from (3.6).Furthermore, Eq. (1.8) follows obviously from (1.7) and (3.6).Finally, in view of (1.3) and (1.7), we establish (1.9), and thus the proof is complete.

2
In the next lemma we present a result of Theorem 3.1 in [6] which will be used in the proof of Theorem 2.1.V n := n i=1 X i Y i for n ≥ 1.We only need to consider the case that λ m+1 > 0 (with the multiplicity denoted by m 2 ≥ 1).With the aid of (1.4), (2.2) and Lemma 3.6, we obtain that, as x → ∞, ∼ exp(y), ∀y ∈ R.