Abstract
We give a short survey concerning sub-Gaussian random elements in a Banach space and prove a statement about the induced operator of a bounded random element in a Hilbert space.
Dedicated to 90th birthday anniversary of Nicholas Vakhania (1930–2014).
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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
To a random variable \(\xi \) let us associate a quantity \(\tau (\xi )\in [0, +\infty ]\) defined by the equality:
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
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
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
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
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
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
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
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
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
and
Observe that
Thus we can define a probability measure \(\mathbb {P}\) on \(\varOmega :=\mathbb {N}\) and a random element \(\eta :\varOmega \rightarrow H\) by setting:
and
Fix now \(p\in ]0,\infty [\) and \(i\in \mathbb N\). Clearly,
Hence
and so
(b). To a (real-valued) random variable \(\xi \) let us associate a quantity \(\vartheta _2(\xi )\in [0, +\infty ]\) defined by the equality:
According to [6, Proposition 2.9(b)] we have:
So, it is sufficient to show that
We have for every \(n\in \mathbb N\) and \(i\in I_n\):
Hence
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.
References
Kahane, J.P.: Propriétés locales des fonctions à séries de Fourier aleatoires. Stud. Math. 19, 1–25 (1960)
Kahane, J.P.: Some Random Series of Functions, 2nd edn. Cambridge University Press, Cambridge (1985)
Buldygin, V.V., Kozachenko, YuV.: Subgaussian random variables. Ukrainian Math. J. 32, 723–730 (1980)
Buldygin, V.V., Kozachenko, Yu.V.: Metric Characterization of Random Variables and Random Processes. Translations of Mathematical Monographs, vol. 188. AMS, Providence R.I. (2000)
Kvaratskhelia, V.V.: Unconditional Convergence of Functional Series in Problems of Infinite-Dimensional Probability Theory. Doctoral Dissertation, Tbilisi (2002)
Vakhania, N.N., Kvaratskhelia, V.V., Tarieladze, V.I.: Weakly Sub-Gaussian random elements in Banach spaces. Ukrainian Math. J. 57, 1187–1208 (2005)
Ruegg, A.: A characterization of certain infinitely divisible laws. Ann. Math. Stat. 41, 1354–1356 (1974)
Horn, R.A.: On necessary and sufficient conditions for an infinitely divisible distribution to be normal or degenerate. Z. Wahrscheinlichkeitstheorie verw. Geb. 21, 179–187 (1972)
Kruglov, V.M.: Characterization of a class of infinitely divisible distributions in a Hilbert space. Mat. Zametki 16, 777–782 (1974)
Kolmogorov, A.N.: La transformation de Laplace dans les espaces linéaires. C.R. Acad. Sc. Paris 200, 1717–1718 (1935)
Fréchet, M.: Généralisation de la loi de probabilité de Laplace. Ann. Inst. H. Poincare 12(1), 1–29 (1951)
Vakhania, N.N.: Sur les répartitions de probabilités dans les espaces de suites numériques. C.R. Acad. Sc. Paris 260, 1560–1562 (1965)
Fernique, X.: Intégrabilité des vecteurs gaussiens. C.R. Acad. Sc. Paris, Série A 270, 1698–1699 (1970)
Landau, H.J., Shepp, L.A.: On the supremum of Gaussian process. Sankhya, Série A 32(4), 369–378 (1970)
Skorokhod, A.V.: A note on Gaussian measures in a Banach space. Theory Probab. Appl. 15(3), 508–508 (1970)
Vakhania, N.N., Tarieladze, V.I., Chobanyan, S.A.: Probability distributions on Banach spaces. D. Reidel Publishing Company, Dordrecht (1987)
Vakhania, N.: Subgaussian random vectors in normed spaces. Bull. Georg. Acad. Sci. 163, 8–11 (2001)
Fukuda, R.: Exponential integrability of sub-Gaussian vectors. Probab. Theory Relat. Fields 85, 505–521 (1990)
Talagrand, M.: Regularity of gaussian processes. Acta Math. 159, 99–149 (1987)
Antonini, R.G.: Subgaussian random variables in Hilbert spaces. Rend. Sem. Mat. Univ. Padova 98, 89–99 (1997)
Kvaratskhelia, V., Tarieladze, V., Vakhania, N.: Characterization of \(\gamma \)-subgaussian random elements in a Banach space. J. Math. Sci. 216, 564–568 (2016)
Acknowledgements
1. The authors are grateful to Professor Sergei Chobanyan for useful remarks. 2. The third author was partially supported by Shota Rustaveli National Science Foundation of Georgia (SRNSFG) grant no. DI-18-1429: “Application of probabilistic methods in discrete optimization and scheduling problems”.
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Giorgobiani, G., Kvaratskhelia, V., Tarieladze, V. (2020). Notes on Sub-Gaussian Random Elements. In: Jaiani, G., Natroshvili, D. (eds) Applications of Mathematics and Informatics in Natural Sciences and Engineering. AMINSE 2019. Springer Proceedings in Mathematics & Statistics, vol 334. Springer, Cham. https://doi.org/10.1007/978-3-030-56356-1_11
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