##
**Generalization of a probability limit theorem of Cramér.**
*(English)*
Zbl 0063.01342

From the introduction: Let \(\{X_k\}\), \(k = 1, 2,\ldots\), be mutually independent random variables, that is to say independent real functions in a space in which a probability measure is defined. Let \(V_k(x) = \text{Prob}\, \{X_k\leq x\}\) be the distribution function of \(X_k\). \(\ldots\)

H. Cramér [Actual. Sci. Industr. 736, 5–23 (1938; Zbl 0022.24104; JFM 64.0529.01)] considered the case of “equal components” where all \(V_k(x)\) are identical, \(V_k(x) = V(x)\), and it is moreover supposed that the integral \(\int_{-\infty}^{+\infty} e^{ax}\,dV(x)\) exists for all \(|a| <a_0\neq 0\). Nothing seems to be known in the case of unequal components. The problem as such has been stated by P. Lévy (see p. 289 in [Théorie de l’addition des variables aléatoires. Paris: Gauthier-Villars (1937; Zbl 0016.17003)]).

The purpose of the present paper is to generalize Cramér’s result to the case of unequal components. This generalization seems of interest in itself, but it is also to serve as the basis for the establishment of the general form of the so-called law of the iterated logarithm.

We shall consider only the case where \(| X_n|\) has an upper bound of the form \(o(s_n)\). Since \(s_n\to\infty\), this restriction is not very strong. In some cases it is possible to free oneself from this restriction, provided the “tails” \(\int_{| x|>\varepsilon s_n} dV_n(x)\) are sufficiently small. However, in many applications the principle of equivalent sequences of random variables will automatically reduce the consideration to the case of bounded variables. Our proof is based on a straightforward adaptation of Cramér’s proof, although we shall avoid the use of characteristic functions. Cramér’s proof, in turn, rests on a transformation first used by F. Esscher [Skand. Aktuarie Tidskr. 15, 175–195 (1932; Zbl 0004.36101; JFM 58.1177.05)] in connection with an insurance problem. In the special case of equal components our condition \(| X_n| =o(s_n)\) is more restrictive than Cramér’s condition that the integral (1.7) should exist. On the other hand, our results will be slightly sharper than Cramér’s, since it will be possible for us to free ourselves of the logarithmic term in the remainder without introducing a new hypothesis.

H. Cramér [Actual. Sci. Industr. 736, 5–23 (1938; Zbl 0022.24104; JFM 64.0529.01)] considered the case of “equal components” where all \(V_k(x)\) are identical, \(V_k(x) = V(x)\), and it is moreover supposed that the integral \(\int_{-\infty}^{+\infty} e^{ax}\,dV(x)\) exists for all \(|a| <a_0\neq 0\). Nothing seems to be known in the case of unequal components. The problem as such has been stated by P. Lévy (see p. 289 in [Théorie de l’addition des variables aléatoires. Paris: Gauthier-Villars (1937; Zbl 0016.17003)]).

The purpose of the present paper is to generalize Cramér’s result to the case of unequal components. This generalization seems of interest in itself, but it is also to serve as the basis for the establishment of the general form of the so-called law of the iterated logarithm.

We shall consider only the case where \(| X_n|\) has an upper bound of the form \(o(s_n)\). Since \(s_n\to\infty\), this restriction is not very strong. In some cases it is possible to free oneself from this restriction, provided the “tails” \(\int_{| x|>\varepsilon s_n} dV_n(x)\) are sufficiently small. However, in many applications the principle of equivalent sequences of random variables will automatically reduce the consideration to the case of bounded variables. Our proof is based on a straightforward adaptation of Cramér’s proof, although we shall avoid the use of characteristic functions. Cramér’s proof, in turn, rests on a transformation first used by F. Esscher [Skand. Aktuarie Tidskr. 15, 175–195 (1932; Zbl 0004.36101; JFM 58.1177.05)] in connection with an insurance problem. In the special case of equal components our condition \(| X_n| =o(s_n)\) is more restrictive than Cramér’s condition that the integral (1.7) should exist. On the other hand, our results will be slightly sharper than Cramér’s, since it will be possible for us to free ourselves of the logarithmic term in the remainder without introducing a new hypothesis.

### MSC:

60F05 | Central limit and other weak theorems |