UDC 519.214

MATHEMATICS

A. V. NAGAEV

ON THE ROLE OF THE EXTREME TERMS OF A VARIATION SERIES IN THE FORMATION OF A LARGE DEVIATION OF THE SUM OF INDEPENDENT RANDOM VARIABLES

(Presented by Academician Yu. V. Linnik on 20 I 1970)

Let \(\xi_j,\ j=1,2,\ldots,\) be independent identically distributed random variables, \(M\xi_j=0,\ D\xi_j=\sigma^2<\infty\). Let, further,

\[ \underline{\xi}=\xi_1^* \leqslant \xi_2^* \leqslant \cdots \leqslant \xi_n^*=\bar{\xi} \tag{1} \]

be the variation series constructed from the realization \(\xi_1,\xi_2,\ldots,\xi_n\). We shall be interested in the limiting distribution law of the extreme terms of the series (1) under conditions imposed on the sum \(\zeta_n=\xi_1+\cdots+\xi_n\).

At first we shall assume that the random variables \(\xi_j\) have an absolutely continuous distribution with bounded density \(p(x)\), and that, as \(x\to 0\),

\[ p(x)\sim e^{-x^\alpha},\qquad \alpha>0. \tag{2} \]

Put, as usual,

\[ \Phi(w)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{w} e^{-u^2/2}\,du. \]

Let, further, the sequences \(A_n\) and \(B_n\) satisfy the equalities*

\[ 2n\exp\{-A_n^2/2\}=A_n\sqrt{2\pi},\qquad B_n=A_n^{-1}, \tag{3} \]

and let the sequences \(a_n\) and \(b_n\) be such that

\[ n\exp\{-a_n^\alpha\}=\alpha a_n^{\alpha-1},\qquad b_n=\alpha a_n^{1-\alpha}. \tag{4} \]

Theorem 1. Let in relation (2) \(\alpha>1\). Then

1°. If \(0\leqslant x\leqslant n(\ln n)^{-\gamma}\), \(\gamma>1/\alpha\), then as \(n\to\infty\)

\[ \mathbf{P}\{\bar{\xi}<a_n+b_nz\mid \zeta_n=x\} = \mathbf{P}\{\bar{\xi}<a_n+b_nz\}+o(1) = \exp\{-e^{-z}\}+o(1) \]

uniformly in \(z,\ z\geqslant z_0>-\infty\).

2°. If, however, \(x\geqslant n(\ln n)^{\gamma_1}\), \(\gamma_1>\max(12/\alpha,\ 1/(\alpha-1))\), then as \(n\to\infty\)

\[ \mathbf{P}\left\{ \max_{1\leqslant j\leqslant n}\left|\xi_j-\frac{x}{n}\right| \sqrt{\frac{x^{\alpha-2}}{\alpha(\alpha-1)}}<A_n+B_nz \mid \zeta_n=x \right\} = \]

\[ =\exp\{-e^{-z}\}+o(1) \]

uniformly in \(z,\ z\geqslant z_0>-\infty\).

* It is easy to see that, uniformly in \(z,\ -\infty<z<\infty\),

\[ \mathbf{P}\left\{\max_{1\leqslant j\leqslant n}|\eta_j|<A_n+B_nz\right\} = \exp\{-e^{-z}\}+o(1), \]

where the \(\eta_j\) are independent and normally distributed with parameters \((0,1)\).

Here the sequences \(A_n, B_n, a_n\), and \(b_n\) are defined by equations (3) and (4).

Theorem 2. Suppose that in relation (2) \(0<\alpha<1\). Suppose further that \(M|\xi_1|^k<\infty\), \(k=4+\left[\dfrac{2\alpha-1}{1-\alpha}\right]\) (in particular, for \(0<\alpha<1/2\) we have \(k=3\)). Then:

\(1^\circ\). If
\[ 0\le x\le (c_\alpha-\delta)\sigma^{2/(2-\alpha)}n^{1/(2-\alpha)},\qquad c_\alpha=(2-\alpha)(2-2\alpha)^{(\alpha-1)/(2-\alpha)}, \]
where \(\delta>0\) is an arbitrarily small positive number, then, as \(n\to\infty\),
\[ \mathbf P\{\bar\xi<a_n+b_nz\mid \zeta_n=x\} = \mathbf P\{\bar\xi<a_n+b_nz\}+o(1) = \exp\{-e^{-z}\}+o(1) \]
uniformly in \(z,\ z\ge z_0>-\infty\), and \(w,\ -\infty<w<\infty\).

\(2^\circ\). If, however,
\[ (c_\alpha-\delta)\sigma^{2/(2-\alpha)}n^{1/(2-\alpha)}<x=o\bigl(n^{1/(2-2\alpha)}\bigr), \]
then, as \(n\to\infty\),
\[ \mathbf P\{\xi_{n-1}^*<a_n+b_nz;\ \bar\xi-(1-\beta)x<w\sigma\sqrt n\mid \zeta_n=x\} = \]
\[ =\exp\{-e^{-z}\}\Phi\left(\frac{w}{\sigma_1}\right)+o(1) \]
uniformly in \(z,\ z\ge z_0>-\infty\), and \(w,\ -\infty<w<\infty\).

Here the quantity \(\beta\) is the smaller positive root of the equation
\[ \frac{n\sigma^2}{x^{2-\alpha}}=\frac{\beta(1-\beta)^{1-\alpha}}{\alpha}, \]
if \(0<\alpha<1/2\), and of the equation
\[ \frac{n\sigma^2}{x^{2-\alpha}} = \frac{\beta(1-\beta)^{1-\alpha}}{\alpha} \left[ 1-\sigma^2\sum_{j=0}^{k-4}(j+3)\lambda_j \left(\frac{\beta x}{n}\right)^{j+1} \right], \]
if \(1/2\le \alpha<1\).

Here the sequences \(a_n\) and \(b_n\) are defined by equalities (4), while \(\lambda_j\), \(j=0,\ldots,k-4\), are the first coefficients of the Cramér series (see \((^1)\), equality (19)), and
\[ \sigma_1=\bigl(1-\alpha(1-\alpha)n\sigma^2/(1-\beta)^{1+\varepsilon}x^{1+\varepsilon}\bigr)^{-1/2}. \]

Theorem 3. Suppose that in relation (2) \(0<\alpha<1\). Then:

\(1^\circ\). If \(n^{1/(2-2\alpha)}\ll x\), then, as \(n\to\infty\),
\[ \mathbf P\{\xi_{n-1}^*<a_n+b_nz;\ \bar\xi-x+n\alpha\sigma^2x^{\alpha-1}<w\sigma\sqrt n\mid \zeta_n=x\} = \]
\[ =\exp\{-e^{-z}\}\Phi(w)+o(1) \]
uniformly in \(z,\ z\ge z_0>-\infty\), and \(w,\ -\infty<w<\infty\).

\(2^\circ\). If, however, \(n^{1/(2-2\alpha)}=o(x)\), then, as \(n\to\infty\),
\[ \mathbf P\{\xi_{n-1}^*<a_n+b_nz;\ \bar\xi<x+w\sigma\sqrt n\mid \zeta_n=x\} = \]
\[ =\exp\{-e^{-z}\}\Phi(w)+o(1) \]
uniformly in \(z,\ z\ge z_0>-\infty\), and \(w,\ -\infty<w<\infty\).

The theorems stated above refine the considerations on the nature of large deviations contained in the papers \((^2,^3)\). We now briefly describe the case where the random variables are integer-valued. Suppose that instead of representation (2) the relation
\[ \mathbf P\{\xi_1=k\}\sim e^{-k^\alpha},\qquad \alpha>0. \tag{5} \]
holds.

Under condition (5), the assertions of Theorems 2 and 3 remain valid. If in relation (5) \(1<\alpha<2\), then assertion \(2^\circ\) of Theorem 1 is also valid. Under the conditions of \(1^\circ\) of Theorem 1, there is not even an unconditional limiting distribution for \(\bar\xi\). Concerning the case \(\alpha\ge2\), see paper \((^4)\).

If one does not require the existence of a density of the distribution of the random variables \(\xi_j\) and does not assume that they are integer-valued, then one has to use

to trace the limiting behavior of our order statistics under the condition that $\zeta_n>x$. For example, the following is valid.

Theorem 4. Suppose that instead of representation (2) the relation

\[ 1-F(x)\sim x^{-\gamma}, \qquad \gamma>2. \tag{6} \]

holds. Then, as $n\to\infty$, $x/\ln x\geqslant \sqrt n$

\[ \mathbf P\{\xi_{n-1}^{*}<zn^{1/\gamma};\ \bar\xi>x+wx\mid \zeta_n>x\} = \exp\{-z^{-\gamma}\}(1+w)^{-\gamma}+o(1) \]

uniformly in $z$, $z\geqslant z_0>-\infty$, and $w$, $w\geqslant 0$.

Conditions (2), (5), and (6) are one-sided in character; therefore the theorems given above (except for item $2^\circ$ of Theorem 1) describe the limiting law of distribution only for the extreme right-hand terms of series (1). If the behavior of the probability $\mathbf P\{\xi_1<x\}$ as $x\to-\infty$ is specified, then one can obtain an explicit expression for the limiting distribution of the statistics $\xi$, $\bar\xi$, and $\xi_{n-1}^{*}$. For example, instead of item $2^\circ$ of Theorem 3 the following may be formulated.

Theorem 5. Suppose that in representation (2) $0<\alpha<1$, and as $x\to\infty$

\[ \mathbf P\{\xi_1<-x\}\sim x^{-\gamma},\quad \gamma>4+\left[\frac{2\alpha-1}{1-\alpha}\right]. \]

Then, if $n^{1/(2-2\alpha)}=o(x)$,

\[ \mathbf P\{-yn^{1/\gamma}<\xi;\xi_{n-1}^{*}<a_n+b_nz;\bar\xi<x+w\sigma\sqrt n\mid \zeta_n=x\} = \exp\{-y^{-\gamma}-e^{-z}\}\Phi(w)+o(1) \]

uniformly in $y$, $z$, and $w$, $y\geqslant 0$, $z\geqslant z_0>-\infty$, $-\infty<w<\infty$.

Romanovsky Institute of Mathematics
Academy of Sciences of the Uzbek SSR
Tashkent

Received
26 XII 1970

REFERENCES

1. H. Cramér, UMN, 10, 166 (1944).
2. S. V. Nagaev, Winter School on Probability Theory and Mathematical Statistics, Kiev, 1964, p. 147.
3. A. V. Nagaev, DAN, 180, No. 2, 279 (1968).
4. A. V. Nagaev, in: Limit Theorems and Random Processes, Tashkent, 1967, p. 71.