On Small Deviation Asymptotics in the L₂-Norm for Certain Gaussian Processes

Abstract

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.

1. Introduction and Results

Consider a centered Gaussian process $X\left(t\right),0\le t\le 1,$ with covariance $G(t,s)=\mathbf{E}X\left(t\right)X\left(s\right),t,s\in [0,1]$ such that $0<\underset{0}{\overset{1}{\int }}G(t,t)dt<\infty$. Set ${\left|\right|X\left(t\right)\left|\right|}_2^2=\underset{0}{\overset{1}{\int }}{X}^2\left(t\right)dt$. Our aim is to obtain sharp asymptotics of the probability $\mathbf{P}\left(\right|\left|X^2\left(t\right)\right|{|}_2<\epsilon )$ as $\epsilon \to 0$. Due to the well-known classical Karunen–Loève expansion (see, for instance, [1]), the following equality in the distribution:

$${\left|\right|X\left(t\right)\left|\right|}_2^2=\underset{0}{\overset{1}{\int }}{X}^2\left(t\right)dt=\sum _{n\ge 1}{\lambda }_n{\xi }_n^2$$

takes place, where ${\xi }_n,n\ge 1,$ are standard independent Gaussian rv’s, while positive summable ${\lambda }_n$ are the eigenvalues of the integral equation:

$$\lambda f\left(s\right)=\underset{0}{\overset{1}{\int }}G(t,s)f\left(t\right)dt,0\le s\le 1.$$

Hence, the problem examined is equivalent to studying the asymptotic behavior of the probability $\mathbf{P}(\sum _{n\ge 1}{\lambda }_n{\xi }_n^2<\epsilon )$ as $\epsilon \to 0$ or, in a more general setting, the probability $\mathbf{P}\left(\sum _{n\ge 1}{\lambda }_n\right|{\xi }_n{|}^p<\epsilon ),p>0$.

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 ${\lambda }_n$ 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 ${\lambda }_n$ 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:

$$f\left(t\right)=log\mathbf{E}exp\{-t|{\xi }_1{|}^p\},t\ge 0;K(V,p)=-\underset{0}{\overset{\infty }{\int }}{t}^{-1/V}{f}^{\prime }\left(t\right)dt.$$

(1)

For example, $K(V,2)=2^{(1-V)/V}\pi /sin(\pi /V)$.

Let us define polynomials:

$$\begin{gathered} Q_r\left(x\right)=x^r+A_{r-1}{x}^{r-1}+\cdots +A_0=\prod _{i=1}^r(x+{\theta }_i), \\ {P}_q\left(x\right)=x^q+B_{q-1}{x}^{q-1}+\cdots +B_0=\prod _{i=1}^q(x+{\varphi }_i), \end{gathered}$$

(2)

where $r,q\in {\mathbb{Z}}_{+},A_i,B_i\in \mathbb{R},{\varphi }_i,{\theta }_i\in \mathbb{C}$. We assume that for $n=1,2,\dots$, the polynomials $Q_r\left(n\right),P_q\left(n\right)$ are strictly positive and that real constants $\mu ,\nu$ satisfy the condition:

$$V=q\mu -r\nu >1,$$

(3)

as well ss that the sequence $a_n=Q_r^{\nu }\left(n\right)/P_q^{\mu }\left(n\right)$ does not increase.

Proposition 1.

If:

$$\sigma =(\mu B_{q-1}-\nu A_{r-1})/V>-1,$$

(4)

then:

$$\mathbf{P}\left(\sum _{n\ge 1}{a}_n{\left|{\xi }_n\right|}^p<\epsilon \right)\sim {\left(\frac{{\left(\prod _{j=1}^q\Gamma (1+{\theta }_j)\right)}^{\nu }}{{\left(\prod _{j=1}^r\Gamma (1+{\varphi }_j)\right)}^{\mu }}\right)}^{1/p}C{\epsilon }^{A}exp(-B{\epsilon }^{-\frac{1}{V-1}})$$

(5)

as $\epsilon \to +0$, where (see $\left(1\right)$, $\left(3\right)$, and $\left(4\right))$:

$$A=\frac{p-(1+2\sigma )V}{2p(V-1)},B=(V-1){\left(\frac{K}{V}\right)}^{\frac{V}{V-1}},K=K(V,p)$$

and:

$$C=\frac{2^{\frac{-3p+2V-2p\sigma }{4p}}{V}^{\frac{2Vp-V-p-2V\sigma }{2p(V-1)}}{K}^{\frac{V-Vp+2V\sigma }{2p(V-1)}}{\pi }^{\frac{-p+2p\sigma +2V}{4p}}}{\Gamma {(1+\frac{1}{p})}^{\sigma +\frac{1}{2}}\sqrt{V-1}}.$$

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 $n,m\in \mathbb{N}$, ${\lambda }_j\in \mathbb{R},$ and $z_j\in \mathbb{C}$ $(1\le j\le m)$, we set $V={\displaystyle \sum _{j=1}^m}{\lambda }_j>1$, $a_n={\displaystyle \prod _{j=1}^m}{|n+z_j|}^{-{\lambda }_j}$.

Note that the weights $Q_r^{\nu }\left(n\right)/P_q^{\mu }\left(n\right)$ in Proposition 1 constitute a subcase of the weights $a_n$ above (one can derive this by taking ${\lambda }_j$ to be equal to $\mu$ or $-\nu$).

Theorem 1.

Let the integer $k\ge 0$, $z_j\ne -1,-2,\cdots$, and:

$$\sigma :=\frac{1}{V}\sum _{j=1}^m{\lambda }_j\Re z_j>-k-1.$$

(6)

Then, for every $p>0$:

$$\begin{gathered} \mathbf{P}\left(\sum _{n\ge 1}{a}_n{\left|{\xi }_n\right|}^p<\epsilon \right)\sim {\left(\prod _{j=1}^m{|\Gamma (k+1+z_j)|}^{-{\lambda }_j}\right)}^{1/p}{\left(\prod _{j=1}^k{a}_j\right)}^{-\frac{1}{p}}· \\ ·C{\epsilon }^{A}exp(-B{\epsilon }^{-\frac{1}{V-1}}) \end{gathered}$$

(7)

as $\epsilon \to +0$, 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\ne -k-1,$$-k-2,\cdots$ and replace the numbers $a_j,1\le j\le k,$ in (7) with arbitrary positive constants.

Note also that the minimal k satisfying Condition (6) is equal to $[max(-\sigma ,0)]$, where $\left[x\right]$ stands for the integer part of x.

2. Proofs

Let us prove the statement of Theorem 1 for $k=0$. We follow the lines of the proof in [2]. Set $b_n={(n+\sigma )}^{-V}$. Taking into account the equality $|n+z_j{|}^2=n^2+2n\Re z_j+{\left|z_j\right|}^2$, it is easy to verify that:

$$|1-(b_n/a_n)|=O\left(n^{-2}\right),n\to \infty .$$

Therefore,

$$\sum _{n\ge 1}|1-(b_n/a_n)|<\infty$$

(8)

and hence, the product $\prod _{j=1}^{\infty }(b_n/a_n)$ converges. To calculate it, we use the well-known representation for the gamma function:

$$\frac{1}{\Gamma (1+z)}=e^{cz}\prod _{n=1}^{\infty }(1+\frac{z}{n})e^{-z/n},z\in \mathbb{C},$$

where c is the Euler constant. According to this formula, we have:

$$\prod _{n=1}^{\infty }\frac{n+z_j}{n+\sigma }{e}^{-(z_j-\sigma )/n}=e^{-c(z_j-\sigma )}\frac{\Gamma (1+\sigma )}{\Gamma (1+z_j)}.$$

Taking into account that $\sum _{j=1}^m{\lambda }_j(\Re z_j-\sigma )=0$, and $\sigma >-1$, we get:

$$\left|\prod _{n=1}^{\infty }\prod _{j=1}^m{\left(\frac{n+z_j}{n+\sigma }\right)}^{{\lambda }_j}\right|=\left|\frac{{\Gamma }^V(1+\sigma )}{\prod _{j=1}^m{\Gamma }^{{\lambda }_j}(1+z_j)}\right|.$$

Hence,

$$\prod _{j=1}^{\infty }(b_n/a_n)={\Gamma }^V(1+\sigma )\prod _{j=1}^m{|\Gamma (1+z_j)|}^{-{\lambda }_j}.$$

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 $k\ge 1$. Represent the probability on the left-hand side of (7) as $\mathbf{P}(X_1+X_2<\epsilon )$, where $X_1=\sum _{n=1}^k{a}_n{\left|{\xi }_n\right|}^p$, $X_2=\sum _{n\ge 1}{a}_{n+k}{\left|{\xi }_{n+k}\right|}^p$, 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 $a_n$:

$$\mathbf{P}(a_n|{\xi }_n{|}^p<\epsilon )\sim \sqrt{\frac{2}{\pi }}{\left(\frac{\epsilon }{a_n}\right)}^{\frac{1}{p}},\epsilon \to +0.$$

Hence, using [7], Corollary 1 and the just proven statement of Theorem 1 for $k=0$, we obtain:

$$\begin{gathered} F_1\left(\epsilon \right)\sim \frac{{\left(\Gamma (1+\frac{1}{p})\sqrt{\frac{2}{\pi }}{\epsilon }^{\frac{1}{p}}\right)}^k}{\Gamma (1+\frac{k}{p})}{\left(\prod _{j=1}^k{a}_j\right)}^{-\frac{1}{p}}, \\ {F}_2\left(\epsilon \right)\sim {\left(\prod _{j=1}^m{|\Gamma (k+1+z_j)|}^{-{\lambda }_j}\right)}^{1/p}\tilde{C}{\epsilon }^{\tilde{A}}exp(-B{\epsilon }^{-\frac{1}{V-1}}) \end{gathered}$$

(9)

as $\epsilon \to +0$, where $\tilde{C}$ and $\tilde{A}$ are defined similarly to C and A in Proposition 1 with $\sigma$ being replaced by $k+\sigma$.

From [7], Theorem 5, Equation (1.24), and Remark 3 and (9), it follows that:

$$\mathbf{P}(X_1+X_2<\epsilon )\sim \Gamma (1+\alpha )F_2\left(\epsilon \right)F_1(1/|g_2^{\prime }\left(\epsilon \right)\left|\right),\epsilon \to +0,$$

(10)

where $\alpha =k/p$ and $g_2\left(\epsilon \right)=B{\epsilon }^{-\frac{1}{V-1}}$. The observation that the right-hand sides in (10) and (7) coincide finally proves Theorem 1.