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A real random variable is a sub-Gaussian random variable with exponent $b\ge 0$ when its moment-generating function satisfiesDifferentiating at $t=0$ from both sides gives $\mathbb{E}X=0$. Exponential integrability near zero justifies this differentiation. Conversely, if $|X|\le M$ and $\mathbb{E}X=0$, convexity givesThe last inequality follows by comparing power series using $(2j)!\ge 2^{j}j!$. The case $M=0$ is immediate. Thus a bounded random variable is sub-Gaussian exactly when it is centered; the bound $M$ is one possible exponent.

For $b>0$, the Markov inequality applied to $e^{tX}$ gives $\mathbb{P}(X>R)\le \exp (b^2{t}^2/2-tR)$. Choose $t=R/b^2$ and repeat with $-X$. The union bound yieldsIf $b=0$, the same exponential estimate with arbitrarily large $t$ shows $X=0$ almost surely, so all subsequent inequalities are trivial.

Integrating this tail probability estimate gives, for every real $k\ge 2$,We need $2\Gamma (k+1)\le k^k$. For integer $k$ this follows inductively from $2\cdot 2!=2^2$ and $(k+1)k^k\le (k+1{)}^{k+1}$. To cover real $k$ as well, set $H(k)=k\log k-\log \Gamma (k+1)-\log 2$. Differentiating the Gamma function integral gives $(\log \Gamma {)}^{\prime }(k+1)=\mathbb{E}\log Y$ for a Gamma distribution with shape $k+1$ and rate $1$. The Jensen inequality bounds this by $\log (k+1)$. Hence $H^{\prime }(k)\ge 1-\log (1+1/k)>0$ and $H(2)=0$. Taking the $2k$th root now provesHere $\Vert X{\Vert }_p=(\mathbb{E}|X{|}^p{)}^{1/p}$ is the Lp norm.

For the final assertion we prove the required Littlewood interpolation inequality, rather than assuming it. Apply Holder inequality with conjugate exponents $3/2$ and $3$:Raising to the power $3/2$ gives $\Vert X{\Vert }_2^3\le \Vert X{\Vert }_1\Vert X{\Vert }_4^2$. The preceding Lp norm estimate with $k=2$ says $\Vert X{\Vert }_4\le 2b$. Therefore