##
**Banach spaces of random variables of sub-Gaussian type.**
*(English.
Russian original)*
Zbl 0626.60026

Theory Probab. Math. Stat. 32, 45-56 (1986); translation from Teor. Veroyatn. Mat. Stat. 32, 42-53 (1985).

In this paper, the authors study the Banach space sub(\(\phi)\) of centered random variables (r.v.) X, with generating moments functions g(s) majorized by exp \(\phi\) (cs), for all s, where c is a constant and \(\phi\) an N-function satisfying some additional regularity conditions.

In the first part, using the functions g and \(\phi\), they define two equivalent norms on sub(\(\phi)\). If \(\phi^*\) is the Young’s conjugate of \(\phi\), they show that the tail of \(X\in sub(\phi)\) is controlled by a function \(\phi^*\), and that sub(\(\phi)\) is the Orlicz space \(L_ u\) with \(u(x)=\exp (\phi^*(x))-1\). They also establish some properties of sums of independent r.v. in sub(\(\phi)\).

The last part of this work concerns separable random processes \(X=\{X(t),t\in T\}\) of sub-Gaussian type. When \(\phi\) is the natural metric induced on T by X, the authors give the following sufficient condition for sample path \(\phi\)-continuity of X, \[ \int_{0}H(x)/\phi^{-1}(H(x))dx<\infty, \] where H is the metric entropy of (T,\(\phi)\). They also prove that sup(X(t),t\(\in T)\) is in \(L_ u\) and evaluate the tail of this r.v. and the Luxemberg u-norm of X’s modulus of continuity. The authors conclude with sufficient conditions of the same type for Central Limit Theorems (CLT) in C(T); they give an example where only the continuity condition holds and they construct a sample path continuous process on [0,1] not satisfying the CLT.

In the first part, using the functions g and \(\phi\), they define two equivalent norms on sub(\(\phi)\). If \(\phi^*\) is the Young’s conjugate of \(\phi\), they show that the tail of \(X\in sub(\phi)\) is controlled by a function \(\phi^*\), and that sub(\(\phi)\) is the Orlicz space \(L_ u\) with \(u(x)=\exp (\phi^*(x))-1\). They also establish some properties of sums of independent r.v. in sub(\(\phi)\).

The last part of this work concerns separable random processes \(X=\{X(t),t\in T\}\) of sub-Gaussian type. When \(\phi\) is the natural metric induced on T by X, the authors give the following sufficient condition for sample path \(\phi\)-continuity of X, \[ \int_{0}H(x)/\phi^{-1}(H(x))dx<\infty, \] where H is the metric entropy of (T,\(\phi)\). They also prove that sup(X(t),t\(\in T)\) is in \(L_ u\) and evaluate the tail of this r.v. and the Luxemberg u-norm of X’s modulus of continuity. The authors conclude with sufficient conditions of the same type for Central Limit Theorems (CLT) in C(T); they give an example where only the continuity condition holds and they construct a sample path continuous process on [0,1] not satisfying the CLT.

Reviewer: Ph.Nobelis

### MSC:

60F05 | Central limit and other weak theorems |

60G15 | Gaussian processes |

60G17 | Sample path properties |

60G60 | Random fields |

46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |