A. A. BOROVKOV, B. A. ROGOZIN

ASYMPTOTIC REPRESENTATIONS IN SOME PROBLEMS FOR TWO-DIMENSIONAL RANDOM WALKS

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

In \((^1,{}^2)\), in order to obtain limit theorems in boundary-value problems for sums of independent summands, the asymptotic behavior of the solution of the equation

\[ P_{x,t}^{y}-\int_{-\infty}^{x} P_{x-u,t-1}^{y}\,dF(u)=f(x,y,t), \tag{1} \]

was studied as \(x,t\) (and sometimes also \(y\)) increase without bound. In (1), \(F(u)\) is a distribution function; the solution is sought in the class of bounded, nonnegative functions; \(f(x,y,t)\) is determined by the probabilistic meaning of the function \(P_{x,t}^{y}\). In the present note the results of \((^1,{}^2)\) are generalized to the equation

\[ P_{x,t}^{y,s}-\int_{-\infty}^{x} dF(u)\int_{0}^{t} dG(v)\,P_{x-u,t-v}^{y,s} =f(x,y,t,s), \tag{2} \]

where \(G\) is also a distribution function, and the asymptotics of the solution is established as \(x,t\to\infty\) (and possibly also \(y,s\)). At the same time, in contrast to \((^1,{}^2)\), we shall consider the case in which \(F(u)\) is lattice-valued.

Solutions of equation (2), for suitable right-hand sides, will describe, for example, limit theorems in the following boundary-value problems for a two-dimensional random walk. Let \(\xi_1^{(1)}, \xi_2^{(1)},\ldots\) be a sequence of independent random variables with distribution function \(F(u)\), and let \(\xi_1^{(2)}, \xi_2^{(2)},\ldots\) be a sequence of nonnegative random variables, independent of one another and of the first sequence, with distribution function \(G(u)\). Consider in the plane \((t,x)\) the random walk determined by the sequence of points \((0,0)\), \((s_1^{(2)},s_1^{(1)})\), \((s_2^{(2)},s_2^{(1)})\), \(\ldots\), where \(s_k^{(i)}=\xi_1^{(i)}+\cdots+\xi_k^{(i)}\), \(i=1,2\). Denote \(s_n^{(1)}=\max_{0\le k\le n}s_k^{(1)}\); \(\eta_T\) is the time of first passage of the level \(T\) along the \(t\)-axis: \(\eta_T=\max\{k:s_k^{(2)}<T\}\). In insurance problems, problems of statistics, and others, the distributions

\[ P_{x,t}=\mathbf P(\bar s_{\eta_t}^{(1)}\ge x), \]

\[ {}_1P_{x,t}^{y,s} =\mathbf P\bigl(\bar s_{\eta_t}^{(1)}<x,\quad s_{\eta_t+1}^{(1)}=x+y,\quad s_{\eta_t+1}^{(2)}\ge t+s\bigr), \]

\[ {}_2P_{x,t}^{y,s} =\mathbf P\bigl(\bar s_{\eta_t}^{(1)}\ge x,\quad s_{\eta_t+1}^{(1)}=x-y,\quad s_{\eta_t+1}^{(2)}\ge t+s\bigr), \tag{3} \]

\[ {}_3P_{x,t}^{y,s} =\mathbf P\bigl(s_{\eta_t+1}^{(1)}=x-y,\quad s_{\eta_t+1}^{(2)}\ge t+s\bigr), \]

are of interest, where \(x,t>0\), \(s\ge0\), and \(y\) may take values of different signs. In many cases it is then possible to determine also the corresponding distributions that are integral with respect to \(s_{\eta_t+1}^{(1)}\). New distributions obtained if, in the probabilities (3), the event \(s_{\eta_t+1}^{(2)}\ge t+s\) is replaced by the event \(s_{\eta_t}^{(2)}<t-s\) are not difficult to find by using relations of the type

\[ \mathbf P\bigl(\bar s_{\eta_t}^{(1)}<x,\quad s_{\eta_t+1}^{(1)}=x-y,\quad s_{\eta_t}^{(2)}<t-u,\quad s_{\eta_t+1}^{(2)}\ge t+v\bigr) ={}_1P_{x,t-u}^{y,u+v}. \]

Denote by \({}_j Q_{x,t}^{y,s}\), \(j=1,2,3\), the probabilities obtained by replacing in (3) the event \(s_{\eta_t+1}^{(1)}=x-y\) by the event \(s_{\eta_t}^{(1)}=x-y\). It turns out to be possible to find asymptotic representations for the transforms with respect to \(t\) of the probabilities (3) and \({}_j Q_{x,t}^{y,s}\) \((j=1,2,3)\) (which are solutions of (2)), allowing one to carry out a rather complete asymptotic analysis of the distributions themselves (cf. (2)).

If \(G(+0)=0\), then our random walk can also be regarded as a jump process \(\{X_t,t>0\}\), in which changes occur at the time instants \(s_1^{(2)},s_2^{(2)},\ldots\), so that, putting \(G(t)=1-e^{-\lambda t}\), we arrive at a generalized Poisson process.

Let us introduce the necessary notation. Let \(f(\lambda)=M\lambda^{\xi_k^{(1)}}=\sum_{-\infty}^{\infty} f_k\lambda^k\), and let \((\lambda_-,\lambda_+)\) be the largest interval on which \(f(\lambda)<\infty\). If \(a\) is some notation connected with the distribution of \(\xi_k^{(2)}\), then, when necessary, we shall write \(\dot a\) if \(\xi_k^{(2)}\) is lattice-valued, and \(\widetilde a\) if \(\xi_k^{(2)}\) is nonlattice. Accordingly, put \(g(\mu)=M\mu^{\xi_k^{(2)}}=\sum_0^\infty g_k\mu^k\), \(\widetilde g(\mu)=Me^{\mu \widetilde{\xi}_k^{(2)}}\), \(\mu_+=\sup\{\mu:g(\mu)<\infty\}\).

We shall assume that:

I. \(\lambda_+-\lambda_->0\).

II. \(\dot\mu_+>1;\ \widetilde\mu_+>0,\ \limsup_{|\mu|\to\infty}|\widetilde g(i\mu)|<1\).

The greatest common divisors (g.c.d.) of possible values of \(\xi_k^{(1)}\) and \(\xi_k^{(2)}\) may, without loss of generality, be regarded as equal to 1; denote the g.c.d.’s of the differences of possible values of \(\xi_k^{(1)}\) and \(\xi_k^{(2)}\) by \(d_1\) and \(d_2\), respectively, and let \((d_1,d_2)=d\).

We have \(f''(\lambda)\ge 0\) throughout the interval \((\lambda_-,\lambda_+)\). Therefore there exist at most two real zeros \(\lambda_\pm(z)\) \((\lambda_+(z)\ge \lambda_-(z))\) of the function \(1-zf(\lambda)\), defined respectively for \(z\in(z_\pm,z_0)\), where \(z_\pm=f^{-1}(\lambda_\pm)\), \(z_0=\sup_{(\lambda_-,\lambda_+)} f^{-1}(\lambda)\). Further, the segment \([\lambda_-,\lambda_+]\) always contains a point \(\lambda_0\) at which \(f(\lambda_0)=\inf_{(\lambda_-,\lambda_+)} f(\lambda)\), so that \(z_0=f^{-1}(\lambda_0)\). For \(\lambda_0\in(\lambda_-,\lambda_+)\), the functions \(\lambda_\pm(z)\) can be analytically continued to a neighborhood of the point \(z_0\), which is for them a common branch point, at which they form one circular system. Denote by \(K_{\delta\pm}\) the domains \(\{z_\pm+\delta_1\le |z|\le z_0+\delta,\ |\arg z|<\delta\}\), where the numbers \(\delta\) and \(\delta_1=\delta_1(\delta)\) are chosen so that \(K_{\delta\pm}\) contain no other singularities of the functions \(\lambda_\pm(z)\) except the point \(z_0\), and let \(\widehat\lambda_\pm(z)\) be the functions coinciding with \(\lambda_\pm(z)\) at all points of the intervals \([z_\pm,z_0]\) and equal to \(\widehat\lambda_0\) for \(z>z_0\). Introduce the functions \(W_{z\pm}(\lambda)\) by means of the following analogue of Lemma 1 in \((1)\).

The function

\[ W_z(\lambda)=\frac{(1-zf(\lambda))\lambda}{(\lambda-\lambda_-(z))(\lambda-\lambda_+(z))} \]

for \(z\in K_{\delta+}^{\prime}\cap K_{\delta-}^{\prime}\) and sufficiently small \(\delta\) and \(\gamma\) admits a canonical factorization

\[ W_z(\lambda)=W_{z+}(\lambda)\cdot W_{z-}(\lambda) \]

with respect to \(\lambda\) in the domain \(\widehat\lambda_-(|z|)-\gamma\le |\lambda|\le \widehat\lambda_+(|z|)+\gamma\), in which the functions \(W_{z\pm}(\lambda)\) are representable in the form of absolutely convergent series \(\sum_{k=0}^{\infty}\lambda^{\pm k}w_{z\pm}(k)\).

\(W_{z\pm}(\lambda)\) are different from zero respectively for \(z\in K_{\delta+}^{\prime}\), \(|\lambda|\le \widehat\lambda_+(|z|)+\gamma\); \(z\in K_{\delta-}^{\prime}\), \(|\lambda|\ge \widehat\lambda_-(|z|)-\gamma\). In these domains the functions \(W_{z\pm}(\lambda)\) may be chosen regular jointly in the variables \(z\) and \(\lambda\).

We now define the domains \(E_\delta\) and \(K_{\delta+}\). Let \(\mu_-=0\) and \(\widetilde\mu_-=-\infty\). Then, in a neighborhood of any point of the interval \((\mu_-,\mu_+)\), the equation \(z=g(\mu)\) is uniquely solvable with respect to \(\mu=M(z)\). To define the function \(M(z)\) on the whole half-line \([0,\infty]\), set \(M(z)=\mu_-\) for \(z\in[0,g(\mu_-)]\).

and \(M(z)=\mu_+\) for \(z\in[g(\mu_+),\infty]\). Further, let \(\mu^\pm=M(z_\pm)\), \(\mu_0=M(z_0)\), \(M_\delta(z)=\min(M(z)+\delta,\mu_+)\). By \(K_{\delta\pm}\) and \(\widetilde K_{\delta\pm}\) we shall mean, respectively, the domains
\[ \{\mu^\pm+\delta_1(\delta)\leq |\mu|\leq M_\delta(z_0),\ |\arg\mu|<\delta\}, \]
\[ \{\mu^\pm+\delta_1(\delta)\leq \operatorname{Re}\mu\leq M_\delta(z_0),\ |\operatorname{Im}\mu|<\delta\}, \]
and by \(E_\delta,\widetilde E_\delta\) the domains
\[ \{|\mu|\leq M_\delta(z_0),\ |\arg\mu|<\pi/d\},\quad \{\operatorname{Re}\mu\leq M_\delta(z_0)\}, \]
from which the points of the segment \([\mu_0,\mu_0+\delta]\) have been removed.

Now we can formulate the theorems on asymptotic representations for the transforms of the probabilities (3) and \({}_jQ_{x,t}^{y,s}\). Denote
\[ {}_j\dot P_x^{y,s}(\mu)=\sum_{t=1}^{\infty}{}_j\dot p_{x,t}^{y,s}\mu^t,\qquad {}_j\widetilde P_x^{y,s}(\mu)=\int_0^\infty {}_j\widetilde P_{x,t}^{y,s}e^{\mu t}\,dt, \tag{4} \]
\[ {}_j\dot p_x^{y,s}(\mu)={}_j\dot P_x^{y,s}(\mu)-{}_j\dot P_x^{y,s+1}(\mu) \]
(correspond to the probabilities (3), local with respect to \(s_{\eta_t}^{(2)}+1\)),
\[ T(a,b)= \frac{a^{-x}b^{y+1}} {\bigl[b^2-\lambda_+(g(\mu))\lambda_-(g(\mu))\bigr]\,W_{g(\mu)+}(a)W_{g(\mu)-}(b)} \frac{g_s(\mu)}{g(\mu)}, \]
where
\[ \dot g_s(\mu)=\sum_{k=1}^{\infty}\sum_{j=k+s}^{\infty}g_j\mu^k,\qquad \widetilde g_s(\mu)=\int_0^\infty (1-G(t+s))e^{\mu t}\,dt. \]
We define the transforms \({}_jQ_x^{y,s}(\mu)\) and \(P_x(\mu)\) analogously to (4).

Theorem 1. 1)
\[ {}_jQ_x^{y,s}(\mu)={}_jP_x^{y,s}(\mu)\,g(\mu),\qquad j=1,2,3. \]

2)
\[ {}_j\dot p_x^{y,s}(\mu\theta)=\theta^{m_2k_1-s}{}_j\dot p_x^{y,s}(\mu),\qquad j=1,2,3, \]
where \(\theta=e^{2\pi i/d}\), \(k_1\) is a solution of the congruence
\[ k_1m_1\equiv x-y\pmod{d_1} \]
for \(j=2,3\), and of the congruence
\[ k_1m_1\equiv x+y\pmod{d_1} \]
for \(i=1\); \(m_i\) are residues modulo \(d_i\) of possible values \(\xi_k^{(i)}\).

3) There exist \(\delta>0,\gamma>0\) such that, as \(x\to\infty,\ y\to\infty\),
\[ P_x(\mu)=D(\mu)\lambda_+^{-x}(g(\mu)) \frac{W_{g(\mu)+}(1)} {W_{g(\mu)+}(\lambda_+(g(\mu)))} \left(1+O(e^{-\gamma x})\right) \]
\[ \left(\mu\in E_\delta\cap K_{\delta+},\quad \dot D(\mu)=\frac{\mu}{1-\mu},\quad \widetilde D(\mu)=-\frac1\mu\right), \]
\[ {}_2P_x^{y,s}(\mu)= T(\lambda_+(g(\mu)),\lambda_-(g(\mu))) \{1+\Delta(\mu)O(e^{-\gamma x}+e^{-\gamma y})\} \]
\[ (\mu\in E_\delta\cap K_{\delta+}\cap K_{\delta-}), \]
\[ {}_3P_x^{y,s}(\mu)= \begin{cases} -\,T(\lambda_+(g(\mu)),\lambda_+(g(\mu))) \{1+\Delta(\mu)O(e^{-\gamma(x-y)})\},\\ \hfill \text{for } x>y,\ \mu\in E_\delta\cap K_{\delta+},\\[4pt] T(\lambda_-(g(\mu)),\lambda_-(g(\mu))) \{1+\Delta(\mu)O(e^{\gamma(x-y)})\},\\ \hfill \text{for } x<y,\ \mu\in E_\delta\cap K_\delta. \end{cases} \]

Here the estimates are uniform in \(\mu\) and \(s\). The absolute values of the functions \(P_x(\mu)\), \({}_jP_x^{y,s}(\mu)\) \((j=2,3)\) on the contours \(|\mu|=\mathrm{const}\), \(\delta\leq|\arg\mu|<\pi/d\) in the lattice case and \(\operatorname{Re}\mu=\mathrm{const}\), \(|\arg\mu|\geq\delta\), in the nonlattice case do not exceed their absolute values at the points of contact of these contours with the boundary, respectively, of one of the domains \(K_{\delta+},K_{\delta-}\); \(\Delta(\mu)=\lambda_+(g(\mu))-\lambda_-(g(\mu))\).

If \(y\) is fixed, then the following holds.

Theorem 2. For \(\mu\in E_\delta\cap K_{\delta+}\) and \(x\to\infty\),
\[ {}_1P_x^{y,s}(\mu)= \lambda_+^{-x}(g(\mu))\frac{g_s(\mu)}{g(\mu)} \frac{w_{g(\mu)+}(y)} {W_{g(\mu)+}(\lambda_+(g(\mu)))} \left(1+O(e^{-\gamma x})\right), \qquad \text{if } y\geq0, \]
\[ {}_1P_x^{y,s}(\mu)= \lambda_+^{-x}(g(\mu))\frac{g_s(\mu)}{g(\mu)} \frac{v_{g(\mu)+}(y)} {W_{g(\mu)+}(\lambda_+(g(\mu)))} \left(1+O(e^{-\gamma x})\right), \qquad \text{if } y<0, \]
where the estimates are uniform in \(\mu,s\) (and for \(y>0\), also in \(y\)); \(v_z(y)\) are the coefficients

for \(\lambda^y\) in the expansion of the function

\[ -\lambda\bigl[(\lambda-\lambda_{-}(z))(\lambda-\lambda_{+}(z))\,W_{z-}(\lambda)\bigr]^{-1}. \tag{5} \]

The values of \({}_1P_x^{y,s}(\mu)\) for \(\mu\in E_\delta\) and \(\mu\in K_{\delta+}\) are estimated in the same way as in Theorem 1.

The asymptotic analysis of the distributions themselves
\(P_{x,t}^{y,s}\), \({}_1p_{x,t}^{y,s}\), \({}_jp_{x,t}^{y,s}\), \({}_jQ_{x,t}^{y,s}\)
is carried out by means of the saddle-point method. The results obtained make it possible, for example, to describe completely the large deviations for Kolmogorov–Smirnov statistics over the entire conceivable range of deviations. Here we give only one theorem, clarifying the physical meaning of the functions \(W_{z\pm}(\lambda)\).

Theorem 3. Let \(x/t=\tau\), \(\lim_{t\to\infty}\tau=a\geqslant 0\), \(m(\lambda)=\lambda^{-\tau}M^{-1}(f^{-1}(\lambda))\), let \(\lambda_\tau\) be the point at which \(\min_\lambda m(\lambda)\) is attained, and let \(g_\tau=f^{-1}(\lambda_\tau)\). Then, if \(\lambda_a\) is located in the domain of regularity of the function \(m(\lambda)\), then for \(y\geqslant 0\) and \(x\to\infty\)

\[ {}_1\bar P_{x,t}^{y,s} = A\Phi(\lambda_a)\,g_s\bigl(M(g_a)\bigr)\,w_{g_a+}(y)\, t^{-1/2}m^t(\lambda_\tau)\, \Xi\!\left(\frac1t,\tau-a\right), \tag{6} \]

where \(A=\tau\) for \(a=0\), and \(A=A(\lambda_a)>0\) for \(a>0\); \(\Xi\!\left(\frac1t,\tau-a\right)\) is an asymptotic expansion in powers of \(1/t\) and \(\tau-a\). The functions \(A(\lambda_a)\), \(\Phi(\lambda_a)\) are known and do not depend on \(x,t,y,s\). The function \(m^t(\lambda_\tau)\) can also be represented in the form \(m^t(\lambda_a)e^{tH(\tau-a)}\), where \(H(\tau-a)\) is the generalized Cramér series.

The value of \({}_1P_{x,t}^{y,s}\) for \(y<0\) is obtained if in formula (6) \(w_{g_a+}(y)\) is replaced by \(v_{g_a}(y)\), and \(\Phi(\lambda_a)\) by the likewise known value \(\Phi_1(\lambda_a)\).

Thus the functions \(W_{z+}(\lambda)\) and (5) turn out to be the generating functions of the corresponding limiting conditional distributions. The functions
\(W_{z+}(\lambda)(\lambda-\lambda_+(z))\) and
\(W_{z-}(\lambda)(\lambda-\lambda_-(z))\lambda^{-1}\), which in a narrower domain effect the factorization of the function \(1-zf(\lambda)\), can also be given a certain probabilistic meaning. If we put (cf. (3))

\[ r_+(z,\lambda) = \sum_{n=1}^{\infty}\sum_{x=1}^{\infty} \mathbf P\{\bar s_{n-1}=0,\ s_n=x\}\,z^n\lambda^x, \]

\[ r_-(z,\lambda) = \sum_{n=1}^{\infty}\sum_{x=-\infty}^{0} \mathbf P\{\bar s_n=0,\ \max(s_1,\ldots,s_{n-1})<s_n,\ s_n=x\}\,(z^n\lambda^x), \]

then

\[ (1-zf(\lambda))=(1-r_+(z,\lambda))(1-r_-(z,\lambda)). \]

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

Received
23 I 1963

References Cited

\({}^{1}\) A. A. Borovkov, DAN, 143, No. 3, 510 (1962).
\({}^{2}\) A. A. Borovkov, Siberian Mathematical Journal, 3, 5, 645 (1962).
\({}^{3}\) G. Baxter, Pacific J. Math., 8, 4, 649 (1958).