MATHEMATICS

A. A. BOROVKOV

SOME PROBLEMS ON LARGE DEVIATIONS OF THE MAXIMUM OF SUMS OF INDEPENDENT IDENTICALLY DISTRIBUTED RANDOM VARIABLES

(Presented by Academician A. N. Kolmogorov, 28 II 1958)

Let (\xi_1, \xi_2, \ldots) be a sequence of independent identically distributed random variables. Denote:

[
\mathbf{M}\xi_k=m,\qquad s_n=\sum_{k=1}^{n}\xi_k,\qquad \bar{s}n=\maxs_\nu .
]

It is required to study the asymptotic behavior of the probability (\mathbf{P}(\bar{s}_n0). A complete solution of this question, when the jump size (\xi_k) can take only the two values (\pm 1) with probabilities (p) and (1-p), is contained, for example, in Feller’s book ((^1)). The method used there for finding the explicit form of the generating function of the time of first passage does not extend to a more general type of distribution of the jumps. Therefore the behavior of the generating function near its first singular points was studied; with the use of the saddle-point method ((^2,\,^3)), this gives the required asymptotic expansion for the probabilities. Below we give the results for the case in which the (\xi_k) are bounded and have a lattice distribution.

We shall assume that (\xi_k) can take only the integer values (l_1,l_2,\ldots,l_q) ((l_1=-r<0;\ l_{i+1}>l_i,\ i=1,2,\ldots,q-1;\ l_q=s>0)) with probabilities (p_{l_i}=\mathbf{P}(\xi_k=l_i)), and that the greatest common divisor of the numbers (l_1,l_2,\ldots,l_q) is equal to (1). Let (\Delta_i=l_{i+1}-l_i,\ i=1,2,\ldots,q-1;\ d=(\Delta_1,\Delta_2,\ldots,\Delta_{q-1})), and let (l) be the residue of any of the numbers (l_i) modulo (d) (((l,d)=1;\ 0\le l\le d-1)). We shall further denote by (w(\lambda)) the “generating” function of the jump

[
w(\lambda)=\sum_{i=-r}^{s}p_i\lambda^i
]

and by (\lambda_1) the point at which (w(\lambda)) attains its minimum on the ray (\lambda>0).

Consider the equation (w(\lambda)-w(\lambda_1)=0). Using Pellet’s theorem ((^4)), one can show that this equation contains in the circle (|\lambda|<\lambda_1) exactly (r-1) roots (\lambda_2,\lambda_3,\ldots,\lambda_r), counted with their multiplicities. For simplicity, we shall assume that among the roots (\lambda_2,\lambda_3,\ldots,\lambda_r) there are no multiple roots. Then one may put

[
\sigma_i=\sum_{\substack{k=1\ k\ne i}}^{r}\frac{1}{\lambda_i-\lambda_k},\qquad
\pi_i=\prod_{\substack{k=1\ k\ne i}}^{r}(\lambda_i-\lambda_k),\qquad
\varepsilon_i=\prod_{\substack{k=1\ k\ne i}}^{r}(1-\lambda_k).
]

Moreover, let (f_{ij}(p)), (f_i(p)) be the elementary symmetric polynomials of order (p) in all the roots (\lambda_1,\lambda_2,\ldots,\lambda_r), except respectively (\lambda_i,\lambda_j).

and (\lambda_i). We shall assume that (f_{ij}(p)=1,\ f_i(p)=1), if (p=0), and (f_{ij}(p)=0), if (p<0). Then, for integral (x), the following theorem is true.

Theorem. If (u_{x,n}) is the probability of absorption at the (n)-th step by the screen at the point (0) of a particle starting from the point (x>0), then:

A. For (x) independent of (n),

[
u_{x,n}=\frac{[w(\lambda_1)]^{n+1}}{n^{3/2}}\,
\frac{d}{\sqrt{2\pi w(\lambda_1)w''(\lambda_1)}}\,b_0(x,n)
\left{1+\frac{b_1(x,n)}{n}+\frac{b_2(x,n)}{n^2}+\cdots+O!\left(\frac1{n^k}\right)\right},
]

where (b_0(x,n)) is a function periodic in (n) with period (d), equal to

[
\begin{aligned}
b_0(x,n)=&\frac{\lambda_1^{r+x-1}}{\pi_1}
\left{\frac{r+x+1}{\lambda_1}-\sigma_1\right}
\sum_{k=0}^{\left[\frac{r-1-c(n)}{d}\right]}
(-1)^{c(n)+kd}f_1(c(n)+kd)
\
&+\sum_{i=2}^{r}\frac{\lambda_i^{r+x-1}}{\pi_i}
\sum_{k=0}^{\left[\frac{r-1-c(n)}{d}\right]}
(-1)^{c(n)+kd}
\left{
f_{i1}(c(n)+kd-i)+
\frac{f_i(c(n)+kd)}{\lambda_i-\lambda_1}
\right}.
\end{aligned}
]

Here (c(n)\equiv ln-x\pmod d,\ 0\le c(n)\le d-1), and ([y]) denotes the integer part of (y); (b_i(x,n)) also have period (d) in (n), are bounded, and can be calculated; (k) is any natural number.

B(^*). For (x(n)\to\infty,\ n\to\infty) and (x(n)=o(\sqrt n)),

[
u_{x,n}=
\frac{[w(\lambda_1)]^{n+1}}{n^{3/2}}\,
x\lambda_1^{x+r-2}
\frac{d}{\pi_1\sqrt{2\pi w(\lambda_1)w''(\lambda_1)}}
\left{
\sum_{k=0}^{\left[\frac{r-1-c(n)}{d}\right]}
(-1)^{c(n)+kd}f_1(c(n)+kd)
\right}
]

[
{}\times
\left(1+O!\left(\frac1x\right)+O!\left(\frac{x^2}{n}\right)\right).
]

C. For (\dfrac{\sqrt n}{x(n)}=O(1)) and (x(n)=o(n)),

[
u_{x,n}=
\frac{[w(\lambda_1)]^{n+1}}{n^{3/2}}\,
x\lambda_1^{x+r-2}e^{nh(x/n)}
\frac{d}{\pi_1\sqrt{2\pi w(\lambda_1)w''(\lambda_1)}}
\times
]

[
{}\times
\left{
\sum_{k=0}^{\left[\frac{r-1-c(n)}{d}\right]}
(-1)^{c(n)+kd}f_1(c(n)+kd)
\right}
\left(1+O!\left(\frac{x}{n}\right)\right).
]

Here (h(t)) is a power series convergent for sufficiently small values of (|t|):

[
h(t)=-\frac{w(\lambda_1)}{w''(\lambda_1)}
\frac{t^2}{2\lambda_1^2}
-\left(\frac{w(\lambda_1)}{w''(\lambda_1)}\right)^2
\left[
\frac{w'''(\lambda_1)}{w''(\lambda_1)}+\frac{3}{\lambda_1}
\right]
\frac{2t^3}{3\lambda_1^3}
+\cdots .
]

If among (\lambda_2,\ldots,\lambda_r) there are multiple roots, then only the expressions for (b_i(x,n)) in item A change.

For (d=1) the formulas simplify: (b_i(x,n)\equiv b_i(x)), and (b_0(x)) can be written in the form

[
b_0(x)=\lambda_1^{r+x-1}\frac{\varepsilon_1}{\pi_1}
\left(\frac{r+x-1}{\lambda_1}-\sigma_1\right)
+\sum_{i=2}^{r}\lambda_i^{r+x-1}
\frac{\varepsilon_i}{\pi_i}
\left(
\frac1{\lambda_i-\lambda_1}-\frac1{1-\lambda_1}
\right),
]

and the expression in braces in the formulas of items B and C is replaced by (\varepsilon_1/\pi_1).

For (x(n)\sim cn) with (c<r) and (c=r), analogous theorems can also be formulated.

* In items B, C we restrict ourselves only to the first terms of the asymptotic expansion. The notation (c(n)) will have the same meaning as in item A.

Next, let us consider (p(x))—the probability that the wandering point will at some time be absorbed by the screen at the point (0). (p(x)) differs from 1 only when the mean jump is directed away from the absorbing screen. If we set

[
\Pi_i=\prod_{\substack{k=1\ k\ne i}}^{r}(\mu_i-\mu_k),\qquad
E_i=\prod_{\substack{k=1\ k\ne i}}^{r}(1-\mu_k),
]

then

[
p(x)=\sum_{i=1}^{r}\mu_i^{x+r-1}\frac{E_i}{\Pi_i},
]

where (\mu_1,\mu_2,\ldots,\mu_r) are the roots of the equation (w(\mu)-1=0) such that (|\mu_i|<1) (in view of (2) there will be exactly (r) such roots; among them there is always a positive one, maximal in modulus). For simplicity we again assume that none of the roots (\mu_1,\mu_2,\ldots,\mu_r) is multiple.

Now put (\xi'k=-\xi_k) and denote by (u'), (p'(x)) the absorption probabilities corresponding to the random variable (\xi'_k); then

[
u'{x,n}= \mathbf{P}(\bar s<x,\bar s_n\ge x)
]

and

[
1-p'(x)=\mathbf{P}(\bar s_i<x,\ i=1,2,\ldots).
]

For (m\ge0), the desired probability (\mathbf{P}(\bar s_n<x(n))) can be computed as

[
\sum_{k=n+1}^{\infty}u'_{x(n),k}.
]

For (m<0),

[
\mathbf{P}(\bar s_n<x(n))=1-p'(x(n))+\sum_{k=n+1}^{\infty}u'_{x(n),k}.
]

If a second absorbing screen is introduced at the point (a>0) and one again considers (u_{x,n})—the probability of absorption at the (n)-th step by the screen at the point (0) of a particle starting from a point (0<x<a)—then the asymptotic behavior of (u_{x,n}) will be different. It turns out that in this case the generating function

[
U_x(z)=\sum_{n=0}^{\infty}u_{x,n}z^n
]

is a rational function (U_x(z)=P_x(z)/Q(z)), analytic in a disk of radius (\rho>1/w(\lambda_1)). The polynomials (P_x(z)) and (Q(z)) can be found.

Some conclusions concerning the probability (u_{x,n}) with one absorbing screen can also be drawn for other types of distribution functions of the random variables (\xi_k). We shall assume that Cramér’s condition ({}^{(5)}) is satisfied, i.e., there exists an interval ([\alpha,\beta]), (\alpha<1), (\beta>1), on which (w(\lambda)) exists, and, moreover, the point (\lambda_1) at which (w(\lambda)) attains its minimum is an interior point of ([\alpha,\beta]). Then the use of the results of works ({}^{(5-7)}) and Lemma 5 of work ({}^{(8)}) permits one to assert that, for a sufficiently broad class of distribution functions of the jump and (x=o(n)),

[
\lim_{n\to\infty}\frac{\log u_{x,n}}{n}=\log w(\lambda_1).
]

The author expresses gratitude to B. A. Savostyanov for a number of valuable comments.

Received
24 II 1958

CITED LITERATURE

({}^{1}) A. Feller, An Introduction to Probability Theory and Its Applications, IL, 1952.
({}^{2}) O. Perron, Sitzungsber. Kais. Bayr. Akad. Wiss. München, Phys.-Math. Kl., 1919 (1917).
({}^{3}) B. A. Fuks, V. I. Levin. Functions of a Complex Variable and Some of Their Applications, Moscow–Leningrad, 1951.
({}^{4}) Pellet, Bull. Sci. Math., (2), 5 (1881).
({}^{5}) H. Cramér, Uspekhi Mat. Nauk, vol. 10 (1944).
({}^{6}) H. E. Daniels, Ann. Math. Stat., 25, 631 (1954).
({}^{7}) I. N. Sanov, Proceedings of the 3rd All-Union Mathematical Congress, 1, 1956.
({}^{8}) A. N. Kolmogorov, Math. Ann., 101, 484 (1929).