Orlicz norms are the general framework used for these sorts of results, though much of the Orlicz norm literature is concerned with concentration of sums $\sum_i X_i$ or suprema $\sup_i X_i$, rather than $\ell_2$ norms. If you want less theoretical literature recommendations, some authors have been looking into "sub-weibull" random variables lately. If one defines $\psi_\alpha(x) = \exp(x^\alpha)-1$, and examines the Orlicz norm $\lVert X\rVert_{\psi_\alpha}$ associated with this Young function, then

• $\alpha = 2$ yields precisely the Sub-gaussian parameter of $X$,
• $\alpha = 1$ yields precisely the sub-exponential parameter.

For $\alpha < 1$ one no longer has that $\lVert \cdot\rVert_{\psi_\alpha}$ is a norm (essentially because the function $\psi_\alpha$ is no longer convex). Still, some can try to recover parts of the general theory, which is precisely what the papers on "sub-weibull" random variables try to do.

Decreasing $\alpha$ is of interest because $\lVert X^2\rVert_{\psi_\alpha} = \lVert X\rVert_{\psi_{\alpha/2}}$, i.e. if one starts with sub-gaussian bounds on $X$, then one gets sub-exponential bounds on $\lVert X\rVert_2$.

One can convert Orlicz norm bounds to concentration bounds as follows. A typical definition of an Orlicz norm is for $\Psi:\mathbb{R}^+\to\mathbb{R}^+$ an increasing function that

$$\lVert X\rVert_\Psi = \inf\{c>0\mid \mathbb{E}\left[\Psi(\lVert X\rVert_2/c)\right] \leq 1\}.$$

For this to be a norm generally one assumes $\Psi$ is what is called a Young's function, namely $\Psi(0)= 0$, $\lim_{x\to\infty}\Psi(x) = \infty$, and $\Psi$ is convex and increasing on $[0,\infty)$.

Markov's inequality then gives that $$ \Pr[\lVert X\rVert_2\geq c] = \Pr[1+\Psi(\lVert X\rVert_2/\lVert X\rVert_\Psi) \geq 1+\Psi(c/\lVert X\rVert_\Psi)] \leq \frac{1+\mathbb{E}[\Psi(\lVert X\rVert_2/\lVert X\rVert_\Psi)]}{1+\Psi(c/\lVert X\rVert_\Psi)} \leq \frac{2}{1+\Psi(c/\lVert X\rVert_\Psi)}. $$

For example, for the Young's function $\Psi_2(x) = \exp(x^2)-1$, we get that

$$\Pr[\lVert X\rVert_2\geq c] \leq \frac{2}{\exp((c/\lVert X\rVert_{\Psi_2})^2)}\implies \Pr[\lVert X\rVert_2 \geq c\lVert X\rVert_{\Psi_2}] \leq 2\exp(-c^2),$$

i.e. something akin to a standard concentration bound. Note that the quantity $\lVert X\rVert_{\Psi_2}$ is defined here in terms of $\lVert X\rVert_2$, rather than something like $\langle t, X\rangle$. To pass between the two, I am pretty sure you write $\lVert X\rVert_2 = \sup_{\lVert t\rVert_2 = 1}\langle X,t\rangle$, and then apply an $\epsilon$-net argument to the set $\{t\mid \lVert t\rVert_2 = 1\}$, but I was never very good with $\epsilon$-net arguments, so perhaps shouldn't be the person to discuss their finer details.

Orlicz norms are the general framework used for these sorts of results, though much of the Orlicz norm literature is concerned with concentration of sums $\sum_i X_i$ or suprema $\sup_i X_i$, rather than $\ell_2$ norms. If you want less theoretical literature recommendations, some authors have been looking into "sub-weibull" random variables lately. If one defines $\psi_\alpha(x) = \exp(x^\alpha)-1$, and examines the Orlicz norm $\lVert X\rVert_{\psi_\alpha}$ associated with this Young function, then

• $\alpha = 2$ yields precisely the Sub-gaussian parameter of $X$,
• $\alpha = 1$ yields precisely the sub-exponential parameter.

For $\alpha < 1$ one no longer has that $\lVert \cdot\rVert_{\psi_\alpha}$ is a norm (essentially because the function $\psi_\alpha$ is no longer convex). Still, some can try to recover parts of the general theory, which is precisely what the papers on "sub-weibull" random variables try to do.

Decreasing $\alpha$ is of interest because $\lVert X^2\rVert_{\psi_\alpha} = \lVert X\rVert_{\psi_{\alpha/2}}$, i.e. if one starts with sub-gaussian bounds on $X$, then one gets sub-exponential bounds on $\lVert X\rVert_2$.

One can convert Orlicz norm bounds to concentration bounds as follows. A typical definition of an Orlicz norm is for $\Psi:\mathbb{R}^+\to\mathbb{R}^+$ an increasing function that

$$\lVert X\rVert_\Psi = \inf\{c>0\mid \mathbb{E}\left[\Psi(\lVert X\rVert_2/c)\right] \leq 1\}.$$

For this to be a norm generally one assumes $\Psi$ is what is called a Young's function, namely $\Psi(0)= 0$, $\lim_{x\to\infty}\Psi(x) = \infty$, and $\Psi$ is convex and increasing on $[0,\infty)$.

Markov's inequality then gives that $$ \Pr[\lVert X\rVert_2\geq c] = \Pr[1+\Psi(\lVert X\rVert_2/\lVert X\rVert_\Psi) \geq 1+\Psi(c/\lVert X\rVert_\Psi)] \leq \frac{1+\mathbb{E}[\Psi(\lVert X\rVert_2/\lVert X\rVert_\Psi)]}{1+\Psi(c/\lVert X\rVert_\Psi)} \leq \frac{2}{1+\Psi(c/\lVert X\rVert_\Psi)}. $$

For example, for the Young's function $\Psi_2(x) = \exp(x^2)-1$, we get that

$$\Pr[\lVert X\rVert_2\geq c] \leq \frac{2}{\exp((c/\lVert X\rVert_{\Psi_2})^2)}\implies \Pr[\lVert X\rVert_2 \geq c\lVert X\rVert_{\Psi_2}] \leq 2\exp(-c^2),$$

i.e. something akin to a standard concentration bound. Note that the quantity $\lVert X\rVert_{\Psi_2}$ is defined here in terms of $\lVert X\rVert_2$, rather than something like $\langle t, X\rangle$. To pass between the two, I am pretty sure you write $\lVert X\rVert_2 = \sup_{\lVert t\rVert_2 = 1}\langle X,t\rangle$, and then apply an $\epsilon$-net argument to the set $\{t\mid \lVert t\rVert_2 = 1\}$, but I was never very good with $\epsilon$-net arguments, so perhaps shouldn't be the person to discuss their finer details. I have in mind something like theorem 8.3 of this though.

An alternative way to handle this all is to appeal to the $\lVert X^2\rVert_{\psi_a} = \lVert X\rVert_{\psi_{a/2}}$ equality mentioned before, to note that (in the case of i.i.d. components for simplicity) $\lVert \lVert X\rVert_2^2\rVert_{\Psi_a} \leq n \lVert X_i\rVert_{a/2}$. This simply applies the above equality, and then triangle inequality. As mentioned though, for $a < 1$, $\lVert X\rVert_{\psi_a}$ is not convex (so does not satisfy triangle equality). Various things can be done to try to fix this, for example

• settle for triangle equality up to some multiplicative constant (see section 4 of this), or
• modify $\psi_a$ near zero to be convex, see problem 4 of this.

Orlicz norms are the general framework used for these sorts of results, though much of the Orlicz norm literature is concerned with concentration of sums $\sum_i X_i$ or suprema $\sup_i X_i$, rather than $\ell_2$ norms. If you want less theoretical literature recommendations, some authors have been looking into "sub-weibull" random variables lately. If one defines $\psi_\alpha(x) = \exp(x^\alpha)-1$, and examines the Orlicz norm $\lVert X\rVert_{\psi_\alpha}$ associated with this Young function, then

• $\alpha = 2$ yields precisely the Sub-gaussian parameter of $X$,
• $\alpha = 1$ yields precisely the sub-exponential parameter.

For $\alpha < 1$ one no longer has that $\lVert \cdot\rVert_{\psi_\alpha}$ is a norm (essentially because the function $\psi_\alpha$ is no longer convex). Still, some can try to recover parts of the general theory, which is precisely what the papers on "sub-weibull" random variables try to do.

Decreasing $\alpha$ is of interest because $\lVert X^2\rVert_{\psi_\alpha} = \lVert X\rVert_{\psi_{\alpha/2}}$, i.e. if one starts with sub-gaussian bounds on $X$, then one gets sub-exponential bounds on $\lVert X\rVert_2$.

Orlicz norms are the general framework used for these sorts of results, though much of the Orlicz norm literature is concerned with concentration of sums $\sum_i X_i$ or suprema $\sup_i X_i$, rather than $\ell_2$ norms. If you want less theoretical literature recommendations, some authors have been looking into "sub-weibull" random variables lately. If one defines $\psi_\alpha(x) = \exp(x^\alpha)-1$, and examines the Orlicz norm $\lVert X\rVert_{\psi_\alpha}$ associated with this Young function, then

• $\alpha = 2$ yields precisely the Sub-gaussian parameter of $X$,
• $\alpha = 1$ yields precisely the sub-exponential parameter.

For $\alpha < 1$ one no longer has that $\lVert \cdot\rVert_{\psi_\alpha}$ is a norm (essentially because the function $\psi_\alpha$ is no longer convex). Still, some can try to recover parts of the general theory, which is precisely what the papers on "sub-weibull" random variables try to do.

Decreasing $\alpha$ is of interest because $\lVert X^2\rVert_{\psi_\alpha} = \lVert X\rVert_{\psi_{\alpha/2}}$, i.e. if one starts with sub-gaussian bounds on $X$, then one gets sub-exponential bounds on $\lVert X\rVert_2$.

One can convert Orlicz norm bounds to concentration bounds as follows. A typical definition of an Orlicz norm is for $\Psi:\mathbb{R}^+\to\mathbb{R}^+$ an increasing function that

$$\lVert X\rVert_\Psi = \inf\{c>0\mid \mathbb{E}\left[\Psi(\lVert X\rVert_2/c)\right] \leq 1\}.$$

For this to be a norm generally one assumes $\Psi$ is what is called a Young's function, namely $\Psi(0)= 0$, $\lim_{x\to\infty}\Psi(x) = \infty$, and $\Psi$ is convex and increasing on $[0,\infty)$.

Markov's inequality then gives that $$ \Pr[\lVert X\rVert_2\geq c] = \Pr[1+\Psi(\lVert X\rVert_2/\lVert X\rVert_\Psi) \geq 1+\Psi(c/\lVert X\rVert_\Psi)] \leq \frac{1+\mathbb{E}[\Psi(\lVert X\rVert_2/\lVert X\rVert_\Psi)]}{1+\Psi(c/\lVert X\rVert_\Psi)} \leq \frac{2}{1+\Psi(c/\lVert X\rVert_\Psi)}. $$

For example, for the Young's function $\Psi_2(x) = \exp(x^2)-1$, we get that

$$\Pr[\lVert X\rVert_2\geq c] \leq \frac{2}{\exp((c/\lVert X\rVert_{\Psi_2})^2)}\implies \Pr[\lVert X\rVert_2 \geq c\lVert X\rVert_{\Psi_2}] \leq 2\exp(-c^2),$$

i.e. something akin to a standard concentration bound. Note that the quantity $\lVert X\rVert_{\Psi_2}$ is defined here in terms of $\lVert X\rVert_2$, rather than something like $\langle t, X\rangle$. To pass between the two, I am pretty sure you write $\lVert X\rVert_2 = \sup_{\lVert t\rVert_2 = 1}\langle X,t\rangle$, and then apply an $\epsilon$-net argument to the set $\{t\mid \lVert t\rVert_2 = 1\}$, but I was never very good with $\epsilon$-net arguments, so perhaps shouldn't be the person to discuss their finer details.