1 Sub-Gaussian and Related Random Variables

The sub-Gaussian random variables were explicitly defined by Kahane in [1] (see also [2]). They were further studied by Buldygin and Kozachenko in [3, 4] (see also [5, Chap. 3] and [6]).

A real valued random variable \(\xi \) given on a probability space \((\varOmega , \mathcal A, \mathbb {P})\) is called sub-Gaussian if there exists \(a\ge 0\) such that

$$ \mathbb {E}\,e^{t\xi }\le \,e^{\frac{1}{2}t^2a^2},\quad {\text {for every}} \quad t\in {\mathbb R}\,. $$

To a random variable \(\xi \) let us associate a quantity \(\tau (\xi )\in [0, +\infty ]\) defined by the equality:

$$ \tau (\xi )=\inf \{a\ge 0: \mathbb {E}\,e^{t\xi }\le \,e^{\frac{1}{2}t^2a^2}\quad {\textit{for every}} \quad t\in \mathbb R\,\}, $$

and call it the Gaussian standard of \(\xi \) [3] (it is called the Gaussian deviation (“écart de Gauss”) of \(\xi \) in [1]).

Lemma 1

( [1, 4]; see also, [6, Proposition 2.1 and Corollary 2.1]) For a real valued random variable \(\xi \) the following statements are equivalent:

(i) \(\xi \) is sub-Gaussian.

(ii) \(\tau (\xi )<+\infty \quad \text {and}\quad \mathbb {E}\,\xi =0\,.\)

(iii) There is   \(\lambda >0\)   such that   \({\mathbb E}\,{\exp (\lambda \xi ^2)}<+\infty \)   and   \({\mathbb E}\,\xi =0\).

Moreover, if (i) holds, then

and

$$(\mathbb {E}\,|\xi |^p)^{\frac{1}{p}}\le \beta _p \tau (\xi )\,\quad {for every} \quad p\in ]0,\infty [\,,$$

where \(\beta _p=1\) if \(p\in ]0,2]\) and \(\beta _p=2^{\frac{1}{p}}(\frac{p}{e})^{\frac{1}{2}}\) if \(p\in ]2,\infty [\).

In particular we have

$$ \mathbb {E}\, \xi =0\quad {\text {and}}\quad \mathbb {E}\, \xi ^2\le \tau ^2(\xi )\,.$$

Remark 1

An interesting application of implication \((i)\Longrightarrow (iii)\) of Lemma 1 is the following observation: if \(\xi \) is sub-Gaussian random variable with infinitely divisible distribution, then \(\xi \) is (possibly degenerate) Gaussian. This can be derived e.g. from [7, Theorem 2], or from [8, Theorem 1(a)] or (more directly) from [9, Theorem 2] which asserts in particular that if for a random variable \(\xi \) with infinitely divisible distribution we have

$$ \mathbb {E}\, \exp (\alpha |\xi |\ln (|\xi |+1))<\infty \quad {\text {for every}} \quad \alpha >0\, , $$

then it is Gaussian.

A sub-Gaussian random variable \(\xi \) with \(\tau (\xi )\le 1\) is called in [2, p. 67] subnormal.

For a centered Gaussian random variable \(\xi \) clearly \(\tau ^2(\xi )={\mathbb E}\, \xi ^2.\)

A random variable \(\xi \) is called strictly sub-Gaussian if it is sub-Gaussian and \(\tau ^2(\xi )={\mathbb E}\,\xi ^2\).

Let \(SG(\varOmega )\) be the set of all sub-Gaussian random variables \(\xi :\varOmega \rightarrow \mathbb R\). It is known that \(SG(\varOmega )\) is a vector space with respect to the natural point-wise operations, the functional \(\tau (\cdot )\) is a norm on \(SG(\varOmega )\) (provided the random variables which coincide a.s. are identified) and, moreover, \((SG(\varOmega ), \tau (\cdot ))\) is a Banach space [3, 4]. It follows, that if \(\xi _1\) and \(\xi _2\) are centered Gaussian random variables (not necessarily jointly Gaussian) then the random variable \(\xi _1+\xi _2\) is sub-Gaussian, but in general \(\xi _1+\xi _2\) may not be strictly sub-Gaussian (even if \(\mathbb {E}\,\xi _1\xi _2=0\)) [6, Example 3.7 (d)].

From Lemma 1 we can conclude that for every \(p\in ]0,+\infty [\) we have

$$ SG(\varOmega )\subset L_p(\varOmega ) $$

and the norm of the inclusion mapping \(\le \beta _p\).

2 Sub-Gaussian Random Elements

Below X will be a real normed space with the dual space \(X^*\).

We recall that a mapping \(\eta :\varOmega \rightarrow X\) is a random element (in X) if

$$\langle x^*,\,\eta \rangle :=x^*\circ \eta $$

is a random variable for every \(x^*\in X^*\).

A random element \(\eta :\varOmega \rightarrow X\) is called Gaussian if for every \(x^*\in X^*\) the random variable \(\langle x^*,\,\eta \rangle \) is Gaussian.

Such a definition of a Gaussian random element goes back to Kolmogorov [10] and Fréchet [11]. For a Gaussian random element we have the following important integrability result (Vakhania [12] for \(X = lp, 1\le p < +\infty \); Fernique [13], Landau-Shepp [14], Skorokhod [15] in general; see [16, Corollary 2 of Proposition V.5.5, p. 329–330] for a proof):

Theorem 1

Let \(\eta \) be a separably valued Gaussian random element in a normed space X. Then there is \(\lambda >0\) such that \(\mathbb {E}\,{\exp (\lambda \Vert \eta \Vert ^2)}<+\infty \).

A random element \(\eta :\varOmega \rightarrow X\) is called weakly sub-Gaussian if for every \(x^*\in X^*\) the random variable \(\langle x^*,\,\eta \rangle \) is sub-Gaussian (cf. [6, 17]).

In [17] it was shown that an analogue of Theorem 1 may fail for weakly sub-Gaussian random elements (see also [6, Theorem 4.2 and Remark 4.1]).

Let us call a random element \(\eta :\varOmega \rightarrow X\) strictly sub-Gaussian if for every \(x^*\in X^*\) the random variable \(\langle x^*,\,\eta \rangle \) is strictly sub-Gaussian.

Definition 1

( [18]) A random element \(\eta :\varOmega \rightarrow X\) is called sub-Gaussian, if there is a finite constant \(C_{\eta }\ge 0\) such that

$$ \tau (\langle x^*,\,\eta \rangle )\le C_{\eta } \left( {\mathbb E}\,|\langle x^*,\,\eta \rangle |^2\right) ^{\frac{1}{2}}<+\infty \quad {\text {for every}} \quad x^* \in X^*\,. $$

We call a random element \(\eta :\varOmega \rightarrow X\) satisfying conditions of Definition 1 sub-Gaussian in Fukuda’s sense, or F-sub-Gaussian.

An analogue of Theorem 1 remains true for F-sub-Gaussian random elements with values in \(X=L_p\) with \(1\le p<+\infty \) [18, Theorem 4.3]; however, it may fail for \(X=c_0\) (S. Kwapien, personal communication).

In [18] (motivating by [19, Theorem 15 (p. 120)], where a similar concept is implicitly used) a random element \(\eta :\varOmega \rightarrow X\) is called \(\gamma \)-sub-Gaussian if there exists a centered Gaussian random element \(\zeta \) in X such that

$$ \mathbb {E}\,e^{\langle x^*,\,\eta \rangle }\le \mathbb {E}\,e^{\langle x^*,\,\zeta \rangle } \quad {\text {for every}} \quad x^*\in X^*\,. $$

We call a \(\gamma \)-sub-Gaussian random element sub-Gaussian in Talagrand’s sense or T-sub-Gaussian. In [20, Remark 4] the definition of a \(\gamma \)-sub-Gaussian random element in a Hilbert space is attributed to [19].

An analogue of Theorem 1 remains true for \(\gamma \)-sub-Gaussian random elements in a Banach space [18, Theorem 3.4].

If \(X=\mathbb R\) then the notion of a T-sub-Gaussian, as well as the notion of a F-sub-Gaussian random element coincides with the notion of a sub-Gaussian random variable and the notion of a F-sub-Gaussian random variable \(\xi \) with the constant \(C_{\xi }=1\) coincides with the notion of a strictly sub-Gaussian random variable.

If X is a finite-dimensional Banach space then weakly sub-Gaussian random elements are \(\gamma \)-sub-Gaussian (see [6, Proposition 4.4]). In every infinite-dimensional Banach space there exists a weakly sub-Gaussian random element, which is not \(\gamma \)-sub-Gaussian (see [6, Theorem 4.4]).

In what follows H will denote an infinite-dimensional separable Hilbert space with the inner product \(\langle \cdot ,\cdot \rangle \).

Definition 2

( [20, Definition 2.1]) Let \(\mathbf{e}:=\{e_n, n \in {\mathbb N}\}\) be an orthonormal basis of H. A random element \(\eta \) with values in H is subgaussian with respect to \(\mathbf{e}\) if the following conditions are satisfied:

(1) For every \(x\in H\) the real valued random variable \(\langle x,\eta \rangle \) is sub-Gaussian (i.e. \(\eta \) is weakly sub-Gaussian),

(2) \(\sum _{n=1}^\infty \tau ^2(\langle e_n,\eta \rangle )<\infty \).

Using the terminology of the definition we have obtained (see [21, Theorem 1.6]) the following characterization of weakly sub-Gaussian random elements in a separable Hilbert space which are \(\gamma \)-sub-Gaussian.

Theorem 2

For a random element \(\eta \) with values in H the following statements are equivalent:

(i) \(\eta \) is \(\gamma \)-sub-Gaussian.

(ii) For every orthonormal basis \(\mathbf{e}:=\{e_n, n \in {\mathbb N}\} \) of H the random element \(\eta \) is subgaussian with respect to \(\mathbf{e}\).

For a weakly sub-Gaussian random element \(\eta \) in a Banach space X let

$$T_{\eta }:X^*\rightarrow {SG}(\varOmega )$$

be the induced operator, which sends each \(x^*\in X^*\) to the element \(\langle x^*,\,\eta \rangle \in {SG}(\varOmega )\) (the continuity and other related properties of induced operators can be seen in [6, Proposition 4.2]).

Theorem 2 in [21] is derived from the following general result (the definitions of a 2-summing operator and a type 2 space can be seen e.g.. in [16]):

Theorem 3

For a weakly sub-Gaussian random element \(\eta \) with values in a Banach space X consider the assertions:

(i) \(\eta \) is \(\gamma \)-sub-Gaussian;

(ii) \(T_{\eta }:X^*\rightarrow {SG}(\varOmega )\) is a 2-summing operator.

Then \((i)\Rightarrow (ii)\). The implication \((ii)\Rightarrow (i)\) is true when X is a reflexive type 2 space.

The following statement, which is a refinement of a similar assertion contained in [5, Chap. 3], shows in particular that the implication \((i)\Longrightarrow (ii)\) of Theorem 2 may fail for a bounded symmetrically distributed elementary random element \(\eta \).

Proposition 1

Let \(\mathbf{e}:=\{e_n, n \in {\mathbb N}\}\) be an orthonormal basis of H. Then there exists a symmetric bounded random element \(\eta :\varOmega \rightarrow H\) with a countable range, such that

(a) \(\sum _{i=1}^{\infty }\Vert \langle \eta ,e_i\rangle \Vert _{L_p}^2<\infty \)    for every    \(p\in ]0,\infty [\);

(b) \(\sum _{i=1}^{\infty }\left( \tau (\langle \eta ,e_i\rangle )\right) ^2=\infty \)   and hence   \(\eta \)   is not subgaussian with respect to   \(\mathbf{e}\).

Proof

(a). Denote

$$ I_n=\{2^n-1,\dots ,2^{n+1}-2\},\,\,n=1,2,\dots $$

and

$$ b_n=2^{-n}\sum _{k\in I_n}e_k,\,\,n=1,2,\dots . $$

Observe that

$$ \sum _{k=1}^{\infty }\Vert b_k\Vert ^2=\sum _{n=1}^{\infty }\sum _{k\in I_n}\Vert b_k\Vert ^2=\sum _{n=1}^{\infty }2^{-2n}\cdot 2^{n}=1\,. $$

Thus we can define a probability measure \(\mathbb {P}\) on \(\varOmega :=\mathbb {N}\) and a random element \(\eta :\varOmega \rightarrow H\) by setting:

$$ \mathbb P(\{2n-1\})=\mathbb P(\{2n\})=\frac{1}{2}\Vert b_n\Vert ^2,\,\,n=1,2,\dots $$

and

$$ \eta (2n-1)=-\frac{b_n}{\Vert b_n\Vert },\,\,\eta (2n)=\frac{b_n}{\Vert b_n\Vert },\,\,n=1,2,\dots \,. $$

Fix now \(p\in ]0,\infty [\) and \(i\in \mathbb N\). Clearly,

$$ \mathbb {E}|\langle \eta ,e_i\rangle |^p=\sum _{n=1}^{\infty }\left( \sum _{k\in I_n}\langle e_k, e_i\rangle \right) \frac{1}{2^{n(1+p/2)}}\,. $$

Hence

$$ \mathbb {E}|\langle \eta ,e_i\rangle |^p=\frac{1}{2^{n(1+p/2)}}\,\quad {\text {for every}} \quad i\in I_n,\,\,n=1,2,\dots $$

and so

$$ \sum _{i=1}^{\infty }\Vert T_{\eta }e_i\Vert _{L_p}^2=\left( \mathbb {E}|\langle \eta ,e_i\rangle |^p\right) ^{2/p}=\sum _{n=1}^{\infty }\sum _{k\in I_n}\frac{1}{2^{n(1+2/p)}}= $$
$$ \sum _{n=1}^{\infty }\frac{2^n}{2^{n(1+2/p)}}=\sum _{n=1}^{\infty }\frac{1}{2^{2n/p}}<\infty \,. $$

(b). To a (real-valued) random variable \(\xi \) let us associate a quantity \(\vartheta _2(\xi )\in [0, +\infty ]\) defined by the equality:

$$ \vartheta _2(\xi )=\sup _{m\in \mathbb N} \frac{\left( \mathbb {E}\, |\xi |^{2m}\right) ^{1/2m}}{\sqrt{m}}\,. $$

According to [6, Proposition 2.9(b)] we have:

$$ \vartheta _2(\xi )\le \frac{2}{\sqrt{e}}\tau (\xi )\qquad {\text {for every}} \qquad \xi \in {SG}(\varOmega ). $$

So, it is sufficient to show that

$$ \sum _{i=1}^{\infty }\left( \vartheta _2(T_{\eta }e_i)\right) ^2=\infty \,. (2.1) $$

We have for every \(n\in \mathbb N\) and \(i\in I_n\):

$$ \vartheta _2(\langle \eta ,e_i\rangle )=\sup _m \frac{\left( \mathbb {E}\, |\langle \eta ,e_i\rangle |^{2m}\right) ^{1/2m}}{\sqrt{m}}=\sup _m \frac{1}{2^{n(1/2+1/2m)}\sqrt{m}}\ge $$
$$ \frac{1}{2^{n(1/2+1/2n)}\sqrt{n}}\,. $$

Hence

$$ \sum _{i=1}^{\infty }\vartheta _2^2(\langle \eta ,e_i\rangle )=\sum _{n=1}^{\infty }\sum _{i\in I_n}\vartheta _2^2(\langle \eta ,e_i\rangle )\ge \sum _{n=1}^{\infty }2^n\left( \frac{1}{2^{n(1/2+1/2n)}\sqrt{n}}\right) ^2 = $$
$$ \frac{1}{2}\sum _{n=1}^{\infty }\frac{1}{n}=\infty $$

and (2.1) is proved.

The authors do not know whether the following conjecture related with Proposition 1 is true.

Conjecture 1

There exists a symmetric bounded random element \(\eta :\varOmega \rightarrow H\) such that

(a) \(\sum _{i=1}^{\infty }\Vert \langle \eta ,e_i\rangle \Vert _{L_p}^2<\infty \) for every \(p\in ]0,\infty [\) and for every orthonormal basis \(\mathbf{e}:=\{e_n, n \in {\mathbb N}\}\) of H;

(b) \(\sum _{i=1}^{\infty }\left( \tau (\langle \eta ,e_i\rangle )\right) ^2=\infty \) for some orthonormal basis \(\mathbf{e}:=\{e_n, n \in {\mathbb N}\}\) of H.