MATHEMATICS

V. V. PETROV

ON LOCAL THEOREMS FOR LARGE DEVIATIONS

(Presented by Academician V. I. Smirnov on V 16, 1960)

1. Consider a sequence of independent random variables (X_1, X_2, \ldots), having the same distribution with finite variance (\sigma^2 > 0) and mathematical expectation (EX_1) equal to zero. Put

[
Z_n=\frac{X_1+X_2+\cdots+X_n}{\sigma\sqrt n},\qquad
\Phi(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x} e^{-t^2/2}\,dt,\qquad
\varphi(x)=\frac{1}{\sqrt{2\pi}}e^{-x^2/2}.
]

The distribution function of the random variable (Z_n) will be denoted by (F_n(x)), and the corresponding density of the distribution, if it exists, by (p_n(x)). Studying the behavior of (F_n(x)) as (x\to\infty) together with (n), Cramér ({}^{(1)}) obtained a fundamental theorem, which was later generalized by the author ({}^{(2)}) to the case of non-identically distributed variables, with a concomitant strengthening of it for the special case of identical distributions. We give the statement of Cramér’s theorem with this strengthening taken into account.

If, for some (a>0),

[
E e^{a|X_1|}<\infty,
\tag{A}
]

then for (x>1), (x=o(\sqrt n)), and (n\to\infty) we have

[
\frac{1-F_n(x)}{1-\Phi(x)}
=
\exp\left[
\frac{x^3}{\sqrt n}\lambda\left(\frac{x}{\sqrt n}\right)
\right]
\left[
1+O\left(\frac{x}{\sqrt n}\right)
\right],
]

[
\frac{F_n(-x)}{\Phi(-x)}
=
\exp\left[
-\frac{x^3}{\sqrt n}\lambda\left(-\frac{x}{\sqrt n}\right)
\right]
\left[
1+O\left(\frac{x}{\sqrt n}\right)
\right],
]

where

[
\lambda(t)=c_0+c_1t+c_2t^2+\cdots
\tag{1}
]

is a power series converging for all sufficiently small values of (|t|).

Denote by (\gamma_m) the cumulant of order (m) of the random variable (X_1). The coefficient (c_k) in Cramér’s series (1) is expressed only in terms of (\gamma_2=\sigma^2,\gamma_3,\ldots,\gamma_{k+3}). In particular, (c_0=\gamma_3/(6\sigma^3)), (c_1=(\sigma^2\gamma_4-3\gamma_3^2)/(24\sigma^6)).

Analogous results, under the assumption that condition (A) is satisfied, were obtained by V. Richter ({}^{(3)}) for local theorems.

Yu. V. Linnik ({}^{(4,5)}) obtained a number of integral and local limit theorems for the case when Cramér’s condition (A) is not satisfied and therefore the previously used methods are inapplicable. In ({}^{(4,5)}) the problem of normal convergence was studied, i.e., the problem of determining conditions under which, for example, ([1-F_n(x)]/[1-\Phi(x)]\to 1) or (p_n(x)/\varphi(x)\to 1) as (n\to\infty) and (0\le x\le \psi(n)), where (\psi(n)) is a monotone function, (\lim_{n\to\infty}\psi(n)=+\infty).

However, the method proposed by Yu. V. Linnik can also be applied to the solution of other problems. We formulate several results obtained in this way.

If (s) is a nonnegative integer, then by (\lambda^{[s]}(t)) we shall denote the segment of Cramér’s series (1) consisting of its first (s) terms. Thus,

[
\lambda^{[s]}(t)=\sum_{k=0}^{s-1} c_k t^k \quad (s\geqslant 1);
]

in the case (s=0) we put (\lambda^{[0]}(t)\equiv 0).

Theorem 1. Suppose that for some (n_0) there exists a bounded density (p_n(x)) of the distribution of the random variable (Z_n). If the condition

[
\mathbf E \exp |X_1|^{\frac{4\alpha}{2\alpha+1}}<\infty
\tag{2}
]

is satisfied for some (\alpha), (0<\alpha<1/2), then, for (|x|\leqslant n^\alpha/\rho(n)), where (\rho(n)) is any monotone function such that (\lim\limits_{n\to\infty}\rho(n)=+\infty), as (n\to\infty) we have, uniformly with respect to (x),

[
\frac{p_n(x)}{\varphi(x)}
=
\exp\left[
\frac{x^3}{\sqrt n}\,
\lambda^{[s]}\left(\frac{x}{\sqrt n}\right)
\right][1+o(1)],
\tag{3}
]

where (s) is a nonnegative integer determined by the inequalities

[
\frac{s}{2(s+2)}<\alpha\leqslant \frac{s+1}{2(s+3)} .
\tag{4}
]

The following theorem shows that condition (2) is necessary in order that (3) hold as (n\to\infty) uniformly with respect to (x) in the interval (|x|\leqslant n^\alpha \rho(n)).

Theorem 2. If (3) holds as (n\to\infty), for (|x|\leqslant n^\alpha \rho(n)) and some integer (s\geqslant 0), uniformly with respect to (x), where (\rho(n)) is some monotone function and (\alpha) is some number such that (\lim\limits_{n\to\infty}\rho(n)=+\infty), (0<\alpha<1/2), then condition (2) is satisfied for the given (\alpha).

As consequences of Theorems 1 and 2 we obtain certain results of Yu. V. Linnik. Suppose that for some (n_0) there exists a bounded density (p_n(x)). By (\rho(n)) below is denoted a monotone function increasing (arbitrarily slowly) to infinity.

Corollary 1. Let (0<\alpha\leqslant 1/6). Condition (2) is sufficient in order that, for (|x|\leqslant n^\alpha/\rho(n)) and (n\to\infty), the relation

[
\frac{p_n(x)}{\varphi(x)}\to 1
\tag{5}
]

hold uniformly with respect to (x), and is necessary in order that (5) hold for (|x|\leqslant n^\alpha \rho(n)) and (n\to\infty) uniformly with respect to (x).

Corollary 2. Let (\alpha) satisfy condition (4) with (s>0). Conditions (2) and

[
\gamma_m=0 \quad (m=3,\ldots,s+2)
\tag{6}
]

are sufficient in order that (5) hold for (|x|\leqslant n^\alpha/\rho(n)) and (n\to\infty) uniformly with respect to (x), and are necessary in order that (5) hold for (|x|\leqslant n^\alpha \rho(n)) and (n\to\infty) uniformly with respect to (x).

We now turn to the case where the random variables (X_1,X_2,\ldots) have a lattice distribution. Let (X_1,X_2,\ldots) be a sequence of identically distributed independent random variables, (\mathbf E X_1=0), (0<\mathbf D X_1=\sigma^2<\infty). Suppose that (X_1) has only possible values of the form (a+kh) ((k=0,\pm1,\pm2,\ldots)), where (a) is some real number and (h) is the maximal span of the distribution. Introduce the notation

[
p_k=\mathbf P{X_1=a+kh},\quad
P_n(N)=\mathbf P\left{\sum_{j=1}^n X_j=an+Nh\right},\quad
x=x_{nN}=\frac{an+Nh}{\sigma\sqrt n}.
]

Theorem 3. If condition (2) is satisfied for some (\alpha), (0<\alpha<1/2), then for (|x|\le n^\alpha/\rho(n)), where (\rho(n)) is any monotone function such that (\lim_{n\to\infty}\rho(n)=+\infty), as (n\to\infty) we have, uniformly with respect to (x),

[
\frac{\dfrac{\sigma\sqrt n}{h}\,P_n(N)}{\varphi(x)}
=
\exp\left[
\frac{x^3}{\sqrt n}\lambda^{[s]}\left(\frac{x}{\sqrt n}\right)
\right][1+o(1)].
\tag{7}
]

Here (s) is a nonnegative integer determined by means of (4).

Theorem 4. If (7) holds as (n\to\infty) for some integer (s\ge0), uniformly with respect to (x) in the segment (|x|\le n^\alpha\rho(n)), where (\alpha) is some number and (\rho(n)) is some monotone function satisfying the conditions (0<\alpha<1/2), (\lim_{n\to\infty}\rho(n)=+\infty), then condition (2) is satisfied for the given (\alpha).

From Theorems 3 and 4 there follow corollaries analogous to the above-indicated corollaries from Theorems 1 and 2.

1. The results obtained can be generalized to the case of non-identically distributed random variables. We shall confine ourselves to the following assertion.

Let (X_1, X_2,\ldots) be a sequence of independent random variables,

[
\mathbf E X_j=0,\qquad \mathbf D X_j=\sigma_j^2<\infty\quad (j=1,2,\ldots).
]

Put (s_n^2=\sum_{j=1}^n\sigma_j^2),

[
Z_n=\frac1{s_n}\sum_{j=1}^n X_j.
]

By (V_j(x)) denote the distribution function of the random variable (X_j), and by (v_j(t)) its characteristic function. As before, by (F_n(x)) and (p_n(x)) we shall denote respectively the distribution function and the density of the distribution of the random variable (Z_n).

Theorem 5. Suppose that for some (\alpha), (0<\alpha<1/2), the following conditions are satisfied:

I.

[
\mathbf E\exp\left{|X_j|^{\frac{4\alpha}{2\alpha+1}}\right}\le C
\qquad (j=1,2,\ldots),
]

where (C) is a constant.

II.

[
\int_{|t|>\varepsilon}\prod_{j=1}^n |v_j(t)|\,dt
=
O\left(e^{-\delta n^{2\alpha}}\right)
]

for every (\varepsilon>0) and some (\delta>0).

III.

[
\lim_{n\to\infty}\frac{s_n^2}{n}>0.
]

Then, for all sufficiently large (n), there exists a density (p_n(x)), and for (|x|\le n^\alpha/\rho(n)), where (\rho(n)) is an arbitrary monotone function such that (\lim_{n\to\infty}\rho(n)=+\infty), we have

[
\frac{p_n(x)}{\varphi(x)}
=
\exp\left[
\frac{x^3}{\sqrt n}\lambda_n^{[s]}\left(\frac{x}{\sqrt n}\right)
\right][1+o(1)]
]

as (n\to\infty), uniformly with respect to (x). Here (s) is a nonnegative integer determined by means of (4), and (\lambda_n^{[s]}(t)) is the truncation of the series (\lambda_n(t)) consisting of its first (s) terms. The series (\lambda_n(t)) is defined in the work ({}^{(2)}).

We note that the power series (\lambda_n(t)), whose coefficients in the general case depend on (n), in the particular case of identical distributions coincides with Cramér’s series (\lambda(t)).

Theorem 1 follows from Theorem 5.

In conclusion, I express my deep gratitude to Yu. V. Linnik for his attention to the present work and for valuable advice.

Leningrad State University
named after A. A. Zhdanov

Received
26 IV 1960

REFERENCES

1. H. Cramér, Actual. sci. et ind., No. 736, Paris, 1938.
2. V. V. Petrov, UMN, 9, no. 4, 195 (1954).
3. B. Richter, Probability Theory and Its Applications, 2, no. 2, 214 (1957).
4. Yu. V. Linnik, Proc. Fourth Berkeley Symposium on Probability and Mathematical Statistics, 1960.
5. Yu. V. Linnik, DAN, 133, No. 6 (1960).