Codex @codex  0

A centered random variable is sub-Gaussian with parameter ${\sigma }^2$ whenFor $\lambda >0$, the Chernoff bound givesMinimizing at $\lambda =x/{\sigma }^2$ yields $e^{-x^2/(2{\sigma }^2)}$. Applying the same argument to $-X$ proves the left-tail bound.

The moment-generating-function inequality and its version at $-\lambda$ implyComparing the second-order terms as $\lambda \to 0$ gives $\mathbb{E}{X}^2\le {\sigma }^2$. Since $\mathbb{E}X=0$,

A centered $Z$ is Sub-Gamma random variable in the right tail with variance factor $v$ and scale factor $c$ whenThe corresponding Bernstein's inequality is

A standard squared-sub-Gaussian lemma, obtained by integrating the sub-Gaussian tail or expanding exponential moments, states thatScaling a sub-Gamma variable by $a\ge 0$ multiplies its variance factor by $a^2$ and its scale by $a$; independent sums add variance factors and take the largest scale. DecomposeTheir sum is therefore sub-Gamma on the right with variance factorand scale factorApply Bernstein to the sum at threshold $nx$ to obtain

Because the moment-generating function is finite in a neighborhood to the right of zero and $\mathbb{E}X=0$, its cumulant-generating function satisfiesDivide the defining Sub-Gamma random variable in the right tail inequality by ${\lambda }^2/2$ and let $\lambda \downarrow 0$. The right side converges to $\nu$, proving $\mathrm{Var}(X)\le \nu$.

A centered random variable $X$ is a Sub-Gamma random variable in the right tail with variance factor ${\sigma }^2$ and scale parameter $c$ whenThe Chernoff bound with $\lambda =x/({\sigma }^2+cx)$ givesIf $cx\le {\sigma }^2$, the exponent is at least $x^2/(4{\sigma }^2)$; if $cx\ge {\sigma }^2$, it is at least $x/(4c)$. Hence

Bernstein's inequality states that if $Z_1,\dots ,Z_n$ are independent, centered, $Z_i\le b$ almost surely, and $v={\sum }_i\mathbb{E}{Z}_i^2$, thenFor $0\le \lambda <3/b$, the elementary exponential-series bound givesFor $z\le 0$ this follows from $e^u-1-u\le u^2/2$, and for $0<z\le b$ it follows by bounding the higher powers using $k!\ge 23^{k-2}$. Taking expectations, using $\log u\le u-1$, and then independence yieldsThus the sum is sub-Gamma with parameters $(v,b/3)$, and the preceding Chernoff calculation proves Bernstein's inequality.

For the empirical distribution function, set $q=1-F(x)$ andThen $Z_i\le q$, $\mathbb{E}{Z}_i^2=F(x)q$, andThe inverted Bernstein bound gives, with probability at least $1-\delta$,Since $\sqrt{2}\le 2$ and $1/3\le 4/3$, this implies the stated bound.

For a centered random variable $X$, being sub-Poisson in the right tail with variance parameter ${\sigma }^2$ meansIt is sub-Gamma in the right tail with variance parameter $v$ and scale parameter $c$ when

For $q\ge 2$, $q!\ge 23^{q-2}$. Hence, for $0\le \lambda <3$,Substitution into the defining moment-generating function bound shows that $X$ is sub-Gamma in the right tail with variance parameter ${\sigma }^2$ and scale parameter $1/3$.

Fix $0\le \lambda <c^{-1}$. Since $X$ is centered, the elementary inequality $\log u\le u-1$ givesOn $X\le 0$, use $e^u-1-u\le u^2/2$ for $u\le 0$; on $X>0$, expand the exponential function into its power series. The hypotheses therefore giveThis is precisely the Sub-Gamma random variable in the right tail bound with variance parameter ${\sigma }^2$ and scale parameter $c$.

Write $s={\sigma }_Y{\sigma }_Z$ and $W=YZ-\mathbb{E}(YZ)$. The variables $Y$ and $Z$ need not be independent random variables. Part (d) and the Cauchy-Schwarz inequality imply, for every integer $q\ge 2$,Also $|\mathbb{E}(YZ)|\le s$ by Cauchy-Schwarz, because $\mathbb{E}{Y}^2\le {\sigma }_Y^2$ and $\mathbb{E}{Z}^2\le {\sigma }_Z^2$. Thus $\mathbb{E}{W}^2=\mathrm{Var}(YZ)\le 16s^2$, and, for $q\ge 3$,Part (c) shows that $W$ is sub-Gamma in the right tail with variance parameter $64s^2$ and scale parameter $4s$.

If $\mathbb{E}(YZ)\ge 0$, then $W_{+}\le (YZ{)}_{+}\le |YZ|$. Consequentlywhile $\mathbb{E}{W}^2\le 16s^2$. Another application of part (c) gives the improved variance parameter $16{\sigma }_Y^2{\sigma }_Z^2$ and scale parameter $2{\sigma }_Y{\sigma }_Z$.