S. V. NAGAEV

AN INTEGRAL LIMIT THEOREM FOR LARGE DEVIATIONS

(Presented by Academician A. N. Kolmogorov on 7 VII 1962)

Let \(\mathfrak{G}\) be the class of distribution functions \(G(x)\), absolutely continuous for \(x>B\) (\(B\) depends on \(G(x)\)), with a density that can be represented in the form
\[ G'(x)=\int_{0}^{\infty} e^{-xu}\varphi(u)\,du, \]
where \(\varphi(u)\geq 0\) is integrable, has a second derivative satisfying the Hölder condition, and \(\varphi'(0)=0\). It is evident that distributions from the class \(\mathfrak{G}\) do not satisfy Cramér’s well-known condition \((^1)\).

Theorem. Let \(\xi_i\) be a sequence of independent identically distributed random variables with distribution function \(F(x)\), where \(M\xi_i=0\), and let \(F_n(x)\) be the distribution function of the sum \(\xi_1+\xi_2+\cdots+\xi_n\). If there exists \(G(x)\in\mathfrak{G}\) such that, for any \(b>a>B\),
\[ \operatorname*{Var}_{a\leq x\leq b}[F(x)-G(x)]\leq \int_{a}^{b}\frac{\psi(x)}{x^2}\,G'(x)\,dx, \tag{1} \]
where \(\psi(x)\) is integrable on \((B,\infty)\), then
\[ 1-F_n(x)=n(1-F(x))(1+O(1))* \tag{2} \]
for such \(x\) that
\[ n u_x^2=O(1), \tag{3} \]
where \(u_x\) is the solution of the equation
\[ u_x e^{x u_x}(1-G(x))=1. \]

In particular, if \(\varphi(u)\sim u^\alpha,\ \alpha>2\), as \(u\to 0**\), then (2) holds for
\[ x>\sqrt{n}\ln n\,\rho(n), \]
where \(\rho(n)\) is an arbitrary function such that
\[ \lim_{n\to\infty}\rho(n)=\infty. \]

Recently V. V. Petrov \((^4)\) obtained, for \(0\leq x\leq n^\alpha/\rho(n)\), where \(\alpha<1/2\), an asymptotic expression for \(1-F_n(x)\) of the same type as in Cramér’s work \((^1)\), under the assumption that
\[ M\exp |\xi_1|^{\frac{4\alpha}{2\alpha+1}}<\infty. \tag{4} \]
If, along with (4), condition (2) is also satisfied, then simple calculations show that the \(x\) satisfying (3) grow faster than \(n^\alpha\).

The question of large deviations for intermediate values of \(x\) remains open.

Romanovskii Institute of Mathematics
Academy of Sciences of the Uzbek SSR

Received
7 VII 1962

REFERENCES

1. H. Cramer, Actuel sci. et ind., No. 736 (1938).
2. S. V. Nagaev, Vestn. LGU, No. 1 (1962).
3. Yu. V. Linnik, Proc. of the Fourth Berkeley Symposium on Mathem. Statistics and Probability, 2, 1961.
4. V. V. Petrov, DAN, 138, No. 4 (1961).

* An analogous representation is obtained in \((^3)\) under the condition that
\[ 1-F(x)=\frac{A_a}{x^a}+\frac{A_{a+1}}{x^{a+1}}+\cdots+\frac{A_{4a+5}}{x^{4a+5}}+ O\left(\frac{1}{x^{4a+5+\varepsilon}}\right), \]
where \(a\geq 3\) is an integer, and \(A_i\) are constants.

** Under the same qualitative condition, local limit theorems were proved in \((^2)\).