UDC 519.214

MATHEMATICS

A. V. NAGAEV

INTEGRAL THEOREMS TAKING LARGE DEVIATIONS INTO ACCOUNT WHEN CRAMÉR’S CONDITION IS VIOLATED

(Presented by Academician Yu. V. Linnik on 6 VII 1967)

Let \(\xi_j,\ j=1,2,\ldots,\) be independent identically distributed random variables, \({\bf M}\xi_j=0,\ {\bf D}\xi_j=1\). Form the sums \(\zeta_n=\xi_1+\cdots+\xi_n\), and let \(P_n(x)={\bf P}\{\zeta_n>x\}\). In this note we describe the behavior of the probability \(P_n(x)\) for \(x>\sqrt n\). The order of growth of \(x\) from above is not restricted. We shall assume that the \(\xi_j\) have a distribution density \(p(x)\), and moreover (cf. \((^{2,5})\))\(^*\)

\[ p(x)\sim \exp[-|x|^{1-\varepsilon}],\qquad |x|\to\infty,\qquad 0<\varepsilon<1. \tag{*} \]

Let \(\rho\) tend to infinity arbitrarily slowly as \(n\to\infty\). The following theorem is a certain refinement of a theorem of V. V. Petrov \((^3)\).

Theorem 1. Let condition \((*)\) be satisfied. Then, for
\[ x<(c_\varepsilon-\delta)(n-1)^{1/(1+\varepsilon)}, \]

\[ P_n(x)=\left[1-\Phi\left(\frac{x}{\sqrt n}\right)\right] \exp\left\{\frac{x^3}{n^2}\lambda^{[k]}\left(\frac{x}{n}\right)\right\}(1+o(1)), \]

where \(\Phi(x)\) is the distribution function of the normal \((0,1)\) law; \(\lambda^{[k]}(z)\) denotes the first \(k\) terms of the Cramér series (\((^1)\), p. 170), \(k>1/\varepsilon-1\), and, finally, \(c_\varepsilon=(1+\varepsilon)(2\varepsilon)^{-\varepsilon/(1+\varepsilon)}\), while \(\delta\) is arbitrarily small.

The following theorem is due to S. V. Nagaev \((^4)\).

Theorem 2. Under our conditions, for \(x>\rho n^{1/2\varepsilon}\),

\[ P_n(x)=nP_1(x)(1+o(1)). \]

The remaining three theorems are stated for the first time.

Theorem 3. Under our conditions, for
\[ (c_\varepsilon+\delta)(n-1)^{1/(1+\varepsilon)}<x<n^{1/2\varepsilon}/\rho \]

\[ P_n(x)= \frac{n(1+\varepsilon)}{((1-\alpha)x)^\varepsilon} \sqrt{\frac{2\pi n}{1-(n-1)(1-\varepsilon)^\varepsilon/|\alpha|^{1+\varepsilon}}}\, P_{n-1}(\alpha x)P_1((1-\alpha)x)(1+o(1)). \]

\(^*\) It should be noted that condition \((*)\) can be weakened considerably. For example, one may assume that:

\[ 1)\quad p(x)\sim \exp[-x^{1-\varepsilon}],\quad x\to\infty,\qquad \int_{-\infty}^{0}|x|^k p(x)\,dx<\infty,\quad k>\frac1\varepsilon-1 \]

or

\[ 2)\quad p(x)\sim x^\alpha \exp[-x^{1-\varepsilon}l(x)],\quad x\to\infty,\qquad \int_{-\infty}^{0}|x|^k p(x)\,dx<\infty,\quad k>\frac1\varepsilon-1, \]

where \(l(x)\) varies slowly and sufficiently regularly, etc.

Here \(\delta\) is any sufficiently small positive number, and \(\alpha\) is the smaller positive root of the equation

\[ \frac{n-1}{x^{1+\varepsilon}}=\frac{\alpha(1-\alpha)^{\varepsilon}}{1-\alpha},\qquad \varepsilon>1/2, \]

and of the equation

\[ \frac{n-1}{x^{1+\varepsilon}}= \frac{\alpha(1-\alpha)^{\varepsilon}}{1-\varepsilon} \left(1+\sum_{j=3}^{k} j c_j \left(\frac{\alpha x}{n-1}\right)^{j-2}\right), \qquad 0<\varepsilon\leqslant 1/2, \]

\(c_j\) are the coefficients of the Cramér series \(\lambda(z)\).

Let us note that \(ax<(c_\varepsilon-\delta)(n-1)^{1/(1+\varepsilon)}\), so that the behavior of the probability \(P_{n-1}(ax)\) is known to us from Theorem 1.

Theorem 3 takes an especially simple form if \(\varepsilon>1/2\) and \(\rho n^{1/(1+\varepsilon)}<x<n^{1/2\varepsilon}/\rho\). In this case

\[ P_n(x)=n\sqrt{2\pi n}\left[1-\Phi\left(\frac{\alpha x}{\sqrt{n-1}}\right)\right] \exp\left[-(1-\alpha)^{1-\varepsilon}x^{1-\varepsilon}\right](1+o(1)). \]

It remains to describe two intermediate cases, when
\((c_\varepsilon-\delta)(n-1)^{1/(1+\varepsilon)}<x<(c_\varepsilon+\delta)(n-1)^{1/(1+\varepsilon)}\) and when
\(\dfrac{n^{1/2\varepsilon}}{\rho}<x<\rho n^{1/2}\).

The two following theorems are devoted to them.

Theorem 4. If \((c_\varepsilon-\delta)(n-1)^{1/(1+\varepsilon)}<x<(c_\varepsilon+\delta)(n-1)^{1/(1+\varepsilon)}\),

\[ \begin{aligned} P_n(x)=& \left[1-\Phi\left(\frac{x}{\sqrt n}\right)\right] \exp\left[\frac{x^3}{n^2}\lambda^{[k]}\left(\frac{x}{n}\right)\right](1+o(1))+\\ &+n\sqrt{\frac{2\pi n}{1-(n-1)(1-\varepsilon)\varepsilon/x^{1+\varepsilon}}} \left[1-\Phi\left(\frac{\alpha x}{\sqrt{n-1}}\right)\right]\times\\ &\times \exp\left\{\frac{\alpha^3x^3}{(n-1)^2}\lambda^{[k]}\left(\frac{\alpha x}{n-1}\right)\right\} \exp\left[-(1-\alpha)^{1-\varepsilon}x^{1-\varepsilon}\right](1+o(1)). \end{aligned} \]

The meaning of the notation is the same as before.

Theorem 5. If \(n^{1/2\varepsilon}/\rho<x<\rho n^{1/2}\),

\[ \begin{aligned} P_n(x)=& \frac{nx^\varepsilon \exp[-x^{1-\varepsilon}]}{\varepsilon e} \exp\left[-\frac{n}{2x^{2\varepsilon}}\right] \int_{0}^{\infty}\int_{0}^{\infty}\exp[-u-v]\times\\ &\times\left[ \Phi\left(\frac{vx^\varepsilon}{\varepsilon\sqrt{n-1}}+ \frac{ux^\varepsilon}{\sqrt{n-1}}- \frac{\sqrt{n-1}}{x^\varepsilon}\right) -\Phi\left(\frac{vx^\varepsilon}{\varepsilon\sqrt{n-1}}- \frac{\sqrt{n-1}}{x^\varepsilon}\right) \right]\,du\,dv\,(1+o(1))+\\ &+nx^\varepsilon\exp[-x^{1-\varepsilon}] \int_{-\infty}^{0} \left[1-\Phi\left(\frac{ux^\varepsilon}{\sqrt{n-1}}\right)\right] \exp[(1-\varepsilon)u]\,du\,(1+o(1)). \end{aligned} \]

Thus, Theorems 1–5 completely describe the behavior of the probability \(P_n(x)\). Let us examine what gives rise to the large deviations when condition (*) is satisfied. Let \(x_m\) have the property that

\[ n\int_{x_m}^{\infty} u^m\exp[-u^{1-\varepsilon}]\,du=o(1). \]

1. In the case of normal deviations \(\sqrt n<x<C\sqrt n\) (\(C\) is an arbitrarily large constant), the event \(\{\xi_n>x\}\) is formed mainly at the expense of realizations \(\xi_1,\ldots,\xi_n\) in which \(|\xi_i|<x_0,\ i=1,\ldots,n\).

1. If \(C\sqrt n < x < (c_\varepsilon-\delta)(n-1)^{1/(1+\varepsilon)}\), then it is necessary to distinguish two cases: \(\varepsilon>1/2\) and \(0<\varepsilon\le 1/2\). In the first case the principal contribution to the event \(\{\zeta_n>x\}\) is made by realizations \(\xi_1,\ldots,\xi_n\) for which \(|\xi_i|<x_2,\ i=1,\ldots,n\). In the second case \(|\xi_i|<x_m,\ i=1,\ldots,n\), where \(m\) is determined from the condition
   \[ n\left(\frac{x}{n}\right)^m=o(1). \]
   We note that, in any case, \(m\le [1/\varepsilon]+1\).

2. If \((c_\varepsilon-\delta)(n-1)^{1/(1+\varepsilon)}<x<n^{1/2\varepsilon}/\rho\) and \(\varepsilon>1/2\), then the event \(\{\zeta_n>x\}\), roughly speaking, is arranged as follows: \(n-1\) random variables lie in the interval \((-x_2,x_2)\), and one lies in \(((1-\alpha)x-\rho\sqrt n,(1-\alpha)x+\rho\sqrt n)\). If, however, \((c_\varepsilon-\delta)(n-1)^{1/(1+\varepsilon)}<x<n^{1/2\varepsilon}/\rho\) and \(0<\varepsilon\le 1/2\), then \(n-1\) random variables lie in the interval \((-x_m,x_m)\), where \(m\) is determined by the requirement \(nx^{-m\varepsilon}=o(1)\), and one falls into the interval \((x,x+\rho\sqrt n)\).

3. If \(x>\rho n^{1/2\varepsilon}\), then the event that interests us consists of realizations in which \(n-1\) random variables lie in the interval \((-x_0,x_0)\), and one lies in \((x,x+\rho\sqrt n)\).

4. If
   \[ \frac{n^{1/2\varepsilon}}{\rho}<x<\rho n^{1/2\varepsilon}, \]
   then \(n-1\) random variables lie in the interval \((-x_2,x_2)\), and one lies either in the interval \((-\rho\sqrt n+x,x)\) (which gives rise to the first term in Theorem 5), or in the interval \((x,x+\rho\sqrt n)\) (which gives rise to the second term).

5. If, however, \((c_\varepsilon-\delta)(n-1)^{1/(1+\varepsilon)}<x<(c_\varepsilon+\delta)(n-1)^{1/(1+\varepsilon)}\), then large deviations are formed partly by realizations characteristic of case 2, and partly by realizations characteristic of case 3.

V. I. Romanovskii Institute of Mathematics
Academy of Sciences of the Uzbek SSR

Received
3 VII 1967

REFERENCES

1. H. Cramér, UMN, 10, 166 (1944).
2. Yu. V. Linnik, Proc. of IV-th Berk. Symp., 1960.
3. V. V. Petrov, DAN, 154, No. 4, 771 (1964).
4. S. V. Nagaev, Izv. AN UzSSR, ser. phys.-math. sciences, No. 6, 37 (1962).
5. A. V. Nagaev, Collection: Limit Theorems of Probability Theory, Tashkent, 1963.