Gaussian series norm tails and fourth moments

Put $Q_n={\sum }_{j\le n}{a}_j{\xi }_j^2$, $v_n={\sum }_{j\le n}{a}_j$, and $w_n={\sum }_{j\le n}{a}_j^2$. On finite intervals integration by parts with ${\varphi }^{\mathrm{\prime }}(x)=-x\varphi (x)$ gives $\int x^{2k}\varphi (x)dx=[-x^{2k-1}\varphi (x)]+(2k-1)\int x^{2k-2}\varphi (x)dx$. The boundary terms vanish at infinity; the integrals converge since a polynomial times $e^{-x^2\mathrm{/}2}$ is bounded by a constant times $e^{-x^2\mathrm{/}4}$, whose integral is finite by [F3] and the given unit Gaussian integral. Starting from $\int \varphi =1$ therefore gives $\mathbb{E}{\xi }_j^2=1$ and $\mathbb{E}{\xi }_j^4=3$. Thus $\mathbb{E}{Q}_n=v_n$ and finite independence gives $\mathbb{E}{Q}_n^2=v_n^2+2w_n$. Also $u\le v$ and $w\le uv\le v^2$.

The measurable series $Q_n\uparrow Q$ satisfies $\mathbb{E}Q=v$ by [F1]; hence $Q<\mathrm{\infty }$ almost surely, since [F2] gives $\mathbb{P}(Q\ge r)\le v\mathrm{/}r$ for every $r>0$. Extending $N$ by zero on the null exceptional set does not change an integral. Applying [F1] also to $Q_n^2\uparrow Q^2$ proves $\mathbb{E}{N}^4=v^2+2w\le 3v^2$.

Suppose $u>0$. For $0<s<(2u{)}^{-1}$ the scalar substitution $y=\sqrt{1-2s{a}_j}x$ in [F3], using the supplied unit density, gives $\mathbb{E}{e}^{s{a}_j{\xi }_j^2}=(1-2s{a}_j{)}^{-1\mathrm{/}2}$. Finite factorization [F4] and the power-series inequality $-\log (1-z)-z\le z^2\mathrm{/}[2(1-z)]$ for $0\le z<1$ therefore give $\log \mathbb{E}{e}^{s(Q_n-v_n)}={\sum }_{j\le n}[-s{a}_j-\frac{1}{2}\log (1-2s{a}_j)]\le \frac{s^2{w}_n}{1-2us}\mathrm{.}$ If $w_n>0$, apply [F2] to this exponential and choose $s=\sqrt{t}\mathrm{/}(\sqrt{w_n}+2u\sqrt{t})$. Substitution gives $-s(2\sqrt{w_{n}t}+2ut)+s^2{w}_n\mathrm{/}(1-2us)=-t$, so $\mathbb{P}\{Q_n>v_n+2\sqrt{w_{n}t}+2ut\}\le e^{-t}$. If $w_n=0$, then $Q_n=v_n=0$ almost surely and this inequality is immediate.

The deterministic levels $v_n+2\sqrt{w_{n}t}+2ut$ converge to $v+2\sqrt{wt}+2ut$. Whenever $Q$ strictly exceeds this final level, sufficiently large $n$ have $Q_n$ exceeding their corresponding levels. The probability of this eventual-event set is at most the lower limit of their probabilities, as follows directly by countable additivity applied to unions of intersections. Step 2.2 thus gives the first tail inequality. Since $w\le uv$ and $v+2\sqrt{uvt}+2ut\le (\sqrt{v}+\sqrt{2ut}{)}^2$, the second follows. If $u=0$, all weights vanish and both bounds hold directly.

The second tail bound makes the set defining $q(t)$ nonempty and implies $0\le q(t)\le \sqrt{v}+\sqrt{2ut}$. This set is upward closed. Every $q(t)+1\mathrm{/}n$ therefore belongs to it: by the definition of an infimum some member lies strictly below that number. The events $\{N>q(t)+1\mathrm{/}n\}$ increase to $\{N>q(t)\}$, so countable additivity gives $\mathbb{P}(N>q(t))\le e^{-t}$. When $v=0$, step 2.1 gives $N=0$ almost surely and the quantile is zero. Finally, in a fixed orthonormal system the squared norm of the finite Gaussian sum is $Q_n$; finiteness of $Q$ gives norm convergence of that series and the asserted norm interpretation.