Mathematics

A. A. BOROVKOV

SOME RESULTS OF THE ANALYSIS OF LARGE DEVIATIONS IN BOUNDARY PROBLEMS

(Presented by Academician A. N. Kolmogorov, January 29, 1963)

Let \(s_0 = 0,\ s_1 = \xi_1,\ s_2 = \xi_1 + \xi_2,\ldots\) be a sequence of sums of independent identically distributed random variables \(\xi_1,\xi_2,\ldots\), for which \(M\xi_k = 0\). Let \(x = x(n)\) be an increasing function of \(n\), and let \(g_1(t) > 0 > g_2(t)\) be two functions defined on the interval \([0,1]\). In the well-known work of A. N. Kolmogorov \((^1)\), and in a number of later works, the problem was considered of finding the asymptotic behavior, as \(n \to \infty\), of the probability

\[ \mathbf{P}\left(g_1\left(\frac{k}{n}\right) > \frac{s_k}{x} > g_2\left(\frac{k}{n}\right),\ k = 1,2,\ldots,n\right). \tag{1} \]

The limiting value of (1) and estimates of the rate of convergence were found in the case when \(x=\sqrt{n}\), and under very broad assumptions concerning the distribution of \(\xi_k\) (for bibliography see, for example, \((^2)\)).

The present note is devoted to probabilities of large deviations in problem (1) and related problems, when \(x/\sqrt{n}\to\infty\). In this case the case of two boundaries \(g_1\) and \(g_2\) is easily reduced to the case of one boundary \(g>0\), while certain additional restrictions must be imposed on the random variables \(\xi_k\). We shall consider the most meaningful “extreme” case, when

\[ x = n(1+\varkappa), \qquad \varkappa = o(1), \]

and assume that \(\mathbf{P}(\xi_k < t)\) converges to its limits as \(t\to \pm\infty\) according to exponential laws and contains a nonzero absolutely continuous component (the latter condition may be replaced by the lattice property of \(\xi_k\)). Introduce the necessary notation. Let \(\eta_g\) be the time of the first passage by the polygonal trajectory with vertices at the points \((k/n,\ s_k/x)\), \(k=0,1,\ldots,n\), of the boundary \(g\): \(\eta_g = k/n\), if \(s_j/x < g(j/n)\), \(j<k\), \(s_k/x \geq g(k/n)\); let \(\chi_g = s_{n\eta_g} - xg(\eta_g)\) be the unnormalized amount of the first overshoot over \(g\). Further, let \(\varphi(\lambda)=M e^{-\lambda \xi_k}\), \(\lambda_-=\inf\{\lambda:\varphi(\lambda)<\infty\}\), \(\lambda_+=\sup\{\lambda:\varphi(\lambda)<\infty\}\),

\[ \mathfrak{m}(a)=\min_{\lambda\in[\lambda_-,\lambda_+]} e^{\lambda a}\varphi(\lambda), \]

let \(\lambda(a)\) be the point of the interval \([\lambda_-,\lambda_+]\) at which this minimum is attained (it is unique), and

\[ a_{\pm}=-\lim_{\lambda \uparrow \downarrow \lambda_{\pm}} \frac{\varphi'(\lambda)}{\varphi(\lambda)}. \]

The functions \(\mathfrak{m}(a)\), \(\lambda(a)\) are regular for \(a\in(a_-,a_+)\), and \(\lambda(a)=\lambda_{\pm}\) for \(a\gtreqless a_{\pm}\).

We shall call problems of types I and II, respectively, the problems of finding the asymptotic behavior of the probabilities

\[ \mathbf{P}\left(\eta_g=\frac{k}{n},\ \chi_g<y\right),\quad 1\leq k\leq n; \qquad \mathbf{P}\left(\eta_g\leq 1,\ \chi_g<y\right), \]

\[ \mathbf{P}\left(\eta_g=\frac{k}{n},\ \chi_g<y,\ s_n<\rho x\right),\quad 1\leq k\leq n; \qquad \mathbf{P}\left(\eta_g\leq 1,\ \chi_g<y,\ s_n<\rho x\right). \]

The basic concept in the study of large deviations in boundary problems of types I and II is the notion of a level line.

The level line \(a(t)>0\) of type I (for problems of type I) is defined as a line along which the function \(t\ln \mathfrak m\left(\dfrac{a}{t}\right)\), \(0<t\leqslant 1\), retains a constant value equal to \(\ln \mathfrak m(\tau)\) (\(\tau\) is a parameter). However, if \(\alpha_-<\infty,\lambda_-=-\infty\) (\(\xi_k\) are bounded above by the value \(\alpha_-\)) and \(\mathfrak m(\alpha_-)=\mathbf P(\xi_k=\alpha_-)>0\), then \(\mathfrak m(\alpha)\) has a discontinuity at the point \(\alpha_-\) (in this case \(\mathfrak m(\alpha)=0\) for \(\alpha>\alpha_-\)), and the function \(a_\tau(t)\) is not defined for all \(t\). For such \(t\), by definition, we put \(a_\tau(t)=\alpha_-t\). The line \(a_\tau(t)\), \(0<t\leqslant 1,\ 0<\tau<\infty\), is thus uniquely defined, increasing, convex, and is obtained by pasting together, generally speaking, two pieces of regular functions. The pasting occurs at the intersection of the curve \(a_\tau(t)\) with the straight line \(a/\alpha_-=t\). For values of \(t\) smaller than the abscissa of the pasting point, \(a_\tau(t)\) is the continuation of the tangent to \(a_\tau(t)\) at the pasting point.

The functions \(a_\tau(t)\), where they differ from linear functions, can also be defined as the solution \(a_\tau(t)=a(\tau,t)\) of the Cauchy problem for the absolute equation

\[ a_{\tau t}''\left(\frac{a}{t}-a_t'\right)+a_{tt}''a_\tau'=0 \]

under the conditions \(a(\tau,1)=\tau,\ a_t'(\tau,1)=-\ln \varphi(\lambda(\tau))/\lambda(\tau)\).

Next, we shall say that a function \(R(t)\), defined in a neighborhood \(\delta\) of the point \(t=0\), admits a power majorant if there exist positive numbers \(r,c_1,c_2\) and a function \(r(t)\downarrow 0\) as \(|t|\downarrow 0\) such that \(|R(t)|<c_1r(t)\) for \(t\in\delta\) and \(r(td)\geqslant c_2d^r r(t)\) for all \(0<d\leqslant 1\) and \(t\in\delta\).

Theorem 1. Suppose the following conditions are fulfilled:

1) \(k/n\to v\) as \(n\to\infty\), \(v\) does not depend on \(n\) and belongs to the semi-interval \((0,1]\).

2) The function \(g(t)\) in a neighborhood of the point \(v\) has \(p\geqslant 1\) derivatives. In the representation

\[ g(t)=g(u)+\cdots+(t-u)^p\frac{g^{(p)}(u)}{p!}+(t-u)^p R_u^1(t-u) \]

the remainder factor \(R_u^1(t-u)\), for all \(u,\ |u-v|<\varepsilon_1\), admits one and the same power majorant.

3) \(g'(v)<g(v)/v<\alpha_-\).

4) For any \(\varepsilon_2>0\) there exists \(\varepsilon_3>0\) such that \(g(t)>\dfrac{g(v)}{v}\,t+\varepsilon_3\) for all \(t<v-\varepsilon_2\).

Then, for any fixed \(y>0\),

\[ \mathbf P\left(\eta_g=\frac{k}{n},\ \chi_g<y\right)= \]

\[ =\frac{\mathfrak m^n(\tau_\sigma)}{\sqrt{k}}\, \Psi\left(\frac{g(v)}{v},\,g'(v),\,y\right) e^{nT(\Delta,\sigma,v)} \left\{1+O\left(\frac1n,\Delta,\varkappa,r_2\left(\frac1n\right),R_v^3(\Delta)\right)\right\}. \tag{2} \]

Here \(\tau_\sigma\) is the solution of the equation \(a_\tau(v)=\sigma g(v)\), \(\sigma=\varkappa/n=1+\varkappa\), \(\Delta=k/n-v\).

\(r_2(s)\) is a majorant of the function

\[ R_u^2(s)=\frac{g(u+s)-g(u)-s g'(u)}{s}, \qquad R_v^3(s)=g'(v+s)-g'(v), \]

\[ T(t-v,\sigma,v)=T_0(t)+\sum_{k=1}^{\infty}\varkappa^k\,[T_k(t)-T_k(v)], \]

\[ T_0(t)=t\int_{a_{\tau_1}(t)/t}^{g(t)/t}\lambda(\alpha)\,d\alpha, \qquad T_k(t)=\frac{g(t)}{k!}\, \lambda^{(k-1)}\left(\frac{g(t)}{t}\right) \left(\frac{g(t)}{t}\right)^{k-1}, \quad k=1,2,\ldots \]

The function \(\Psi\left(g(v)/v,g'(v),y\right)\) depends only on its arguments and on the distribution of the \(\xi_k\) and can be written in the notation of theorem 5 from [3]. The estimate in (2) is uniform in \(k\) for \(|k/n-v|<\varepsilon_1\).

Let the following quantity be defined and finite:
\[ \tau_g=\sup\left\{\tau:\inf_{0<t<1}\bigl(g(t)-\mathfrak a_\tau(t)\bigr)>0\right\} \]
and let \(V_\varepsilon\) be the set of points of the interval \([0,1]\) for which
\[ g(t)\leqslant \mathfrak a_{\tau_g}(t)+\varepsilon, \]
and let \(V^g\) be the interval
\[ [0,\ln \mathfrak m(\tau_g)/\ln \mathfrak m(\alpha_-)]. \]
Suppose that for sufficiently small \(\varepsilon\) the intersection \(V_\varepsilon\cap V^g\) is empty. Finally, let \(\Omega\) be the union of all intervals belonging to \(V_0\), and let \(\bar A\) denote the closure of the set \(A\). For \(x=n\) the following holds.

Theorem 2. If, in addition to the assumptions made, the Lebesgue measure
\[ \operatorname{mes}(\bar V_\varepsilon-\Omega)\to 0 \quad\text{as }\varepsilon\to 0, \]
then
\[ \mathbf P(\eta_g\leqslant 1,\ \chi_g<y)\sim \sqrt{n\mathfrak m^n(\tau_g)} \int_\Omega t^{-1/2}\Psi\left(\frac{g(t)}{t},\,g'(t),\,y\right)\,dt. \]

Now let \(\operatorname{mes}V_0=0\), and let \(V_\varepsilon\), for all \(\varepsilon\) smaller than some number, be a finite number of half-intervals or intervals on which \(g\) is piecewise differentiable. In this case the asymptotic behavior of the probability
\[ \mathbf P(\eta_g\leqslant 1,\ \chi_g<y) \]
is clarified by the following theorem.

Theorem 3. Let the set \(V_\varepsilon\), for all \(\varepsilon>0\) smaller than some number, consist of one half-interval or interval, and let the set
\[ \bigcap_{k=1}^{\infty}\bar V_{1/k} \]
consist of a single point \(v\), which is a boundary point of \(V_\varepsilon\). Suppose, further, that in the half-neighborhood \(V_\varepsilon\) of the point \(v\) condition 2 of Theorem 1 is satisfied. Then, if the number \(q\) of common first derivatives of the functions \(g\) and \(\mathfrak a_{\tau_g}\) at the point \(v\) is positive and \(p\geqslant q+1\), then
\[ \mathbf P(\eta_g\leqslant 1,\ \chi_g<y)= \]
\[ = n^{q-1/2q+2}\mathfrak m^n(\tau_g)\frac{1}{q+1} \Gamma\left(\frac{1}{q+1}\right) \left[ \frac{\lambda(g(v)/v)\bigl(\mathfrak a_{\tau_g}^{(q+1)}(v)-g^{(q+1)}(v)\bigr)} {(q+1)!} \right]^{-1/(q+1)} \times \]
\[ \times \Psi\left(\frac{g(v)}{v},\,g'(v),\,y\right) \left(1+O\left(n^{-1/(q+1)},\ r_T(n^{-1/(q+1)})\right)\right). \]

Here, by derivatives of the functions \(g\) at the point \(v\), one means respectively right- or left-hand derivatives; \(r_T(t-v)\) is the power majorant of the function
\[ \frac{1}{(t-v)^{q+1}} \left\{ t\int_{\mathfrak a_{\tau_g}(t)/t}^{g(t)/t}\lambda(\alpha)\,d\alpha -(t-v)^{q+1} \frac{\lambda(g(v)/v)\bigl(g^{(q+1)}(v)-\mathfrak a_{\tau_g}^{(q+1)}(v)\bigr)} {(q+1)!} \right\}. \]

It is also possible to analyze less regular cases, when \(p=q\) or \(q=0\), etc. As is clear from Theorems 2 and 3, the asymptotics of probability (1) in the case of large deviations is determined only by the properties of the function \(g\) on the sets \(V_\varepsilon\) and by the properties of the sets \(V_\varepsilon\) themselves.

Level lines in problems of type II are defined differently. Denote
\[ \mathfrak m(\tau,\rho,t) = t\ln\mathfrak m\left(\frac{\tau}{t}\right) + (1-t)\ln\mathfrak m\left(\frac{\rho-\tau}{1-t}\right), \]
\[ \mathfrak m(\tau,\rho)=\sup_{0<t<1}\mathfrak m(\tau,\rho,t), \]
and suppose that \(\tau\) and \(\rho\), \(\tau>\rho\), are such that
\[ \mathfrak M(\tau,\rho)>-\infty. \]
We first define the level lines \(\mathfrak b_{\tau,\rho}(t)\) of the so-called local problem as the solution of the equation
\[ \mathfrak M(\mathfrak b,\rho,t)=\mathfrak m(\tau,\rho), \qquad 0<t<1. \]
In those cases where \(\mathfrak m(\alpha)\) has a discontinuity at the points \(\alpha_+\) or \(\alpha_-\), the solution may fail to exist for some \(t\). In these regions we complete the level lines (as also the level lines of type I) by line segments
\[ \mathfrak b=\alpha_-t \]
or
\[ \mathfrak b=\rho-\alpha_+(1-t). \]
This definition of the functions \(\mathfrak b_{\tau,\rho}(t)\) will be unique. The level lines \(\mathfrak b_{\tau,\rho}(t)\) themselves turn out to be convex, continuously differentiable, and glued together from pieces of, generally speaking, three regular func-

tions. Let \(D_1—D_4\) be the four sectors into which the domain \(\mathfrak b \ge \rho t,\ 0 \le t \le 1\), is divided by the straight lines \(\mathfrak b=\alpha_-t\) and \(\mathfrak b=\rho-\alpha_+(1-t)\). Then \(\mathfrak b_{\tau,\rho}(t)\) is regular in each of these sectors, on the boundaries of which the gluing also takes place. If \(t_0\) is the point at which \(\sup_{0<t<1}\mathfrak M(\tau,\rho,t)\) is attained, then \(\mathfrak b_{\tau,\rho}(t_0)=\tau\), \(\mathfrak b_{\tau,\rho}(t)\le \tau\) for any \(t\in(0,1)\). In the domain of regularity \(\mathfrak b_{\tau,\rho}(t)\) increases with increasing \(\tau\).

The required level lines of type II are obtained from the level lines of the local problem by replacing the part of the curve \(\mathfrak b_{\tau,\rho}(t)\) lying in the domain \(0<\mathfrak b<\rho\) (empty when \(\rho \le 0\)) by the part of the curve \(\mathfrak c_{\tau,\rho}(t)\) determined by the solution of the equation

\[ t\ln \mathfrak m\left(\frac{c}{t}\right)=\mathfrak m(\tau,\rho), \]

i.e., by a part of a level line of type I. We denote the newly obtained curves by \(\mathfrak d_{\tau,\rho}(t)\).

Now we can formulate some assertions concerning problems of type II. Put again \(x=n\),

\[ \tau_g=\sup\left\{\tau:\inf_{0<t<1}\bigl(g(t)-\mathfrak d_{\tau,\rho}(t)>0\bigr)\right\} \]

and denote by \(V_\varepsilon\) the set of points of the interval \([0,1]\) for which \(g(t)\le \mathfrak d_{\tau_g,\rho}(t)+\varepsilon\), and by \(V^g\) the closure of the set of those \(t\) for which the point \((t,\mathfrak d_{\tau_g,\rho}(t))\) does not belong to the lower of the sectors \(D_1—D_4\). We shall again assume that, for sufficiently small \(\varepsilon\), the intersection \(V_\varepsilon\cap V^g\) is empty.

Theorem 4. Let \(\Omega\) be the union of all intervals of the set \(V_0\). If \(\operatorname{mes}(V_\varepsilon-\Omega)\to 0\) as \(\varepsilon\to 0\), then

\[ \mathbf P(\eta_g\le 1,\ \chi_g<y,\ s_n<\rho n)\sim e^{n\mathfrak m(\tau_g,\rho)}\int_\Omega \Phi(t,y)\,dt, \]

where

\[ \Phi(t,y)=\frac{W(\gamma(t))}{\sqrt{t(1-t)}}\int_0^y e^{-y\lambda(\gamma(t))}\,d_y\Psi\left(\frac{g(t)}{t},\,g'(t),\,y\right), \]

\[ \gamma(t)=\frac{\rho-g(t)}{1-t},\qquad W(\gamma)=\frac{1}{\lambda(\gamma)\sqrt{2\pi\varphi_{\lambda}''(\lambda(\gamma),\gamma)/\varphi(\lambda(\gamma),\gamma)}},\qquad \varphi(\lambda,\gamma)=e^{\gamma\lambda}\varphi(\lambda). \]

If \(g\) is tangent to \(\mathfrak d_{\tau_g,\rho}\) at a finite number of points, then the desired value can be computed by using the following analogue of Theorem 3.

Theorem 5. If, in the notation introduced anew, the conditions of Theorem 3 are satisfied, then

\[ \mathbf P(\eta_g\le 1,\ \chi_g<y,\ s_n<\rho n)\sim \]

\[ \sim n^{-1/(q+1)}e^{n\mathfrak M(\tau_g,\rho)} \Phi(\nu,y)\frac{1}{q+1}\Gamma\left(\frac{1}{q+1}\right)U^{-1/(q+1)}, \]

where

\[ U=\frac{[\mathfrak d_{\tau_g,\rho}^{(q+1)}(\nu)-g^{(q+1)}(\nu)] [\lambda(g(\nu)/\nu)-\lambda((\rho-g(\nu))/(1-\nu))]}{(q+1)!}. \]

Institute of Mathematics with Computing Center
Siberian Branch of the Academy of Sciences of the USSR

Received
23 I 1963

REFERENCES

1. A. N. Kolmogorov, Izv. AN SSSR, Otd. matem. i estestv. nauk, 959 (1931).
2. A. V. Skorokhod, Studies in the Theory of Random Processes, Kiev, 1961.
3. A. A. Borovkov, Siberian Mathematical Journal, 3, 5, 645 (1962).