Abstract
Full Text
Reports of the Academy of Sciences of the USSR
- Volume 170, No. 4
MATHEMATICS
E. S. Shtatland
ASYMPTOTICS OF THE DISTRIBUTION OF THE MAXIMUM FOR A CERTAIN CLASS OF PROCESSES WITH INDEPENDENT INCREMENTS
(Presented by Academician Yu. V. Linnik on 13 I 1966)
Let \(\xi(t)\) be a homogeneous process with independent increments, having only positive jumps and a negative drift. Then \((^{1})\)
\[ \mathbf{M}e^{-s\xi(t)}=e^{tA(s)} \qquad (s=\sigma+i\tau,\ \sigma>0), \]
where
\[ A(s)=\beta s+\int_{0}^{\infty}\left(e^{-sx}-1+\frac{sx}{1+x^{2}}\right)dN(x),\quad -\beta-\int_{0}^{\infty}\frac{x}{1+x^{2}}\,dN(x)<0. \]
Introduce the notation
\[ F(t,x)=\mathbf{P}\left\{\sup_{0<u\le t}\xi(u)<x\right\};\qquad F(x)=\mathbf{P}\left\{\sup_{0\le t<\infty}\xi(t)<x\right\}; \]
\[ \varphi(z,x)=\int_{0}^{\infty}e^{-zt}F(t,x)\,dt \qquad (\operatorname{Re} z>0); \]
\[ \bar{\varphi}(z,s)=\int_{0}^{\infty}\int_{0}^{\infty}e^{-zt-sx}F(t,x)\,dt\,dx \qquad (\operatorname{Re} z>0,\ \operatorname{Re} s>0). \]
Regarding the spectral function \(N(x)\), assume the following:
I. The absolutely continuous component of the function \(N(x)\) is different from zero.
\[ \text{II. }\quad a=\inf\left\{\sigma:\int_{\varepsilon}^{\infty}e^{-\sigma x}\,dN(x)<\infty\right\}<0 \qquad (\varepsilon>0). \]
Under these assumptions \(A(s)\) will be an analytic function in the half-plane \(\operatorname{Re}s>a\). For real \(s\), \(A(s)\) is a convex downward function. Denote by \(z_{0}\) the unique real minimum of the function \(A(s)\), \(z_{0}=A(s_{0})\). If \(\mathbf{M}\xi(1)<0\), then \(s_{0}<0,\ z_{0}<0\); if \(\mathbf{M}\xi(1)>0\), then \(s_{0}>0,\ z_{0}<0\), and when \(\mathbf{M}\xi(1)=0\) we have \(s_{0}=z_{0}=0\).
On the ray \(z>z_{0}\) the functions \(\eta_{+}(z)\) and \(\eta_{-}(z)\) are defined—the real roots of the equation \(A(s)=z\). By the implicit-function theorem, \(\eta_{\pm}(z)\) will be analytic in some neighborhood (in the \(z\)-plane) of the ray \(z>z_{0}\). The only singular point of the functions \(\eta_{\pm}(z)\) in this region is \(z_{0}\), a branch point of the second order. Using condition I, one can prove the following assertions \((^{2,3})\):
Lemma 1. For every \(\sigma>a\) and \(\tau_{0}>0\) there exists an \(\varepsilon>0\) such that for \(|\tau|>\tau_{0}\) the inequality
\[ A(\sigma)>\operatorname{Re}A(\sigma+i\tau)+\varepsilon \]
holds.
Remark. From this lemma it follows, in particular, that the function \(A(s)\) maps the line \(\operatorname{Re}s=s_{0}\) into a certain contour \(\mathcal{T}\) lying in the half-plane \(\operatorname{Re}z\le z_{0}\).
Lemma 2. The function \(\eta_{+}(z)\) is analytic in the domain \(D\), situated to the right of the contour \(\mathcal J\), and for \(z \in D\) is the unique root of the equation \(A(s)=z\) in the half-plane \(\operatorname{Re}s \geq s_0\).
Lemma 3. For any sufficiently small \(\delta>0\) there exists a \(\gamma>0\) (common for all \(z\) on the line \(\operatorname{Re}z=b,\ b\geq z_0\)) such that, when \(|\operatorname{Im}z|>\delta\), all roots of the equation \(A(s)=z\) belonging to the half-plane \(\operatorname{Re}s<s_0\) lie to the left of the vertical line \(\operatorname{Re}s=\eta_{-}(\operatorname{Re}z)-\gamma\).
For the function \(\varphi(z,s)\) the formula \((5)\) holds:
\[
\varphi(z,s)=\frac{\eta_{+}(x)-s}{s\eta_{+}(z-A(s))}.
\]
The results of Lemmas 1–3 make it possible to pass from the double Laplace transform \(\overline{\varphi}(z,s)\) to the asymptotics of the distribution function \(F(t,x)\). A consequence of Lemmas 2, 3 and the residue theorem \((6)\) is
Lemma 4. As \(x\to\infty\) and \(|\operatorname{Im}z|<\delta,\ \operatorname{Re}z>z_0-\delta\),
\[
\varphi(z,x)=\frac{1}{z}
-
e^{\eta_{-}(z)x}
\frac{\eta_{+}(z)-\eta_{-}(z)}
{\eta_{+}(z)\eta_{-}(z)A'(\eta_{-}(z))}
+
\frac{1}{\eta_{+}(z)}
O\!\left(e^{(\eta_{-}(\operatorname{Re}z)-\gamma)x}\right).
\]
If \(|\operatorname{Im}z|>\delta,\ \operatorname{Re}z\geq z_0\), then
\[
\varphi(z,x)=\frac{1}{z}
+
\frac{1}{\eta_{+}(z)}
O\!\left(e^{(\eta_{-}(\operatorname{Re}z)-\gamma)x}\right)
\]
(\(\gamma\) and \(\delta\) are the constants appearing in Lemma 3).
The further asymptotic analysis is carried out by the saddle-point method or with the aid of a certain modification of this method \(({}^{4}),\) Lemma 12). Following the arguments of A. A. Borovkov \(({}^{4})\), we obtain theorems that are analogues of the corresponding theorems for sums of independent identically distributed random variables.
Theorem 1. Suppose \(M\xi(1)\neq0\), and \(x\) and \(t\) tend to infinity, with \(x=o(t)\); then
\[
\mathbf P\left\{\sup_{0\leq u\leq t}\xi(u)<x\right\}
=
\]
\[
=
F(x)
+
\frac{\omega_1^{+}x}
{2\sqrt{\pi c s_0^{2}t^{3/2}}}
e^{s_0x-z_0t+H(\tau)t}
\left(1+\frac{1}{t}E_{1,2}\!\left(\frac{1}{t},\tau\right)\right)
+
O\!\left(e^{(s_0-\gamma)x+z_0t}\right);
\]
\[
F(x)=1-\frac{M\xi(1)}{A'(R)}e^{Rx}+O\!\left(e^{(R-\gamma)x}\right)
\quad \text{when } M\xi(1)<0.
\]
If \(M\xi(1)>0\), then \(F(x)\equiv0\).
Here \(\tau=x/t;\ R=\eta_{-}(0);\ H(\tau)\) is a function analytic at the point \(\tau=0\):
\[
H(\tau)=-\frac{(\omega_1^{-})^2}{4}\tau^2
+
\frac{(\omega_1^{-})^2\omega_2^{-}}{4}\tau^3+\cdots;
\]
\(\omega_k^{\pm}\) are the coefficients in the expansion of the functions
\(\omega_{\pm}(p)=\eta_{\pm}(z_0+p^2)\) in powers of \(p\) at the point \(p=0\)
(\(p=(z-z_0)^{1/2}\), and by \((z-z_0)^{1/2}\) we mean the principal value of the root):
\[
\omega_{+}(p)=\eta_{+}(z_0+p^2)=s_0+\omega_1^{+}p+\omega_2^{+}p^2+\cdots,
\]
\[
\omega_{-}(p)=\eta_{-}(z_0+p^2)=s_0+\omega_1^{-}p+\omega_2^{-}p^2+\cdots;
\]
\(c\) is the coefficient of \((s-s_0)^2\) in the Taylor expansion of the function \(A(s)\) in a neighborhood of \(s_0\); \(E_{1,2}(1/t,\tau)\) is an expansion in powers of \(1/t\) and \(\tau\), containing no terms of order \((1/t)^m\tau^n\) where simultaneously \(m<1,\ n<2\) (the coefficients of this expansion are known).
Theorem 2. Let \(\mathbf{M}\xi(1)=0\), \(x=X\sqrt t\), and \(X=o(t^{1/6})\). Then, as \(t\to\infty\),
\[ \mathbf{P}\left\{\sup_{0\leqslant u\leqslant t}\xi(u)<X\sqrt t\right\} = 1-\sqrt{\frac{2}{\pi}}\int_X^\infty e^{-u^2/2}\,du - e^{-\frac12 X^2}\sum_{j=1}^\infty \frac{1}{t^{j/2}}\Pi_{3j-1}(X) + O(e^{-\gamma x}), \]
where \(\Pi_{3j-1}(X)\) are polynomials of degree \(3j-1\) with known coefficients.
(For simplicity we assume that \(\mathbf{D}\xi(1)=1\).)
Theorem 3. Let
\[ \lim_{t\to\infty}\frac{x}{t}=\alpha>0 \]
and \(\alpha=A'(\eta_-(0))\); then, as \(t\to\infty\),
\[ \mathbf{P}\left\{\sup_{0\leqslant u\leqslant t}\xi(u)<x\right\} = 1-(1-F(x))\Delta(z_\alpha) - \frac{\eta_+(z_\alpha)-\eta_-(z_\alpha)} {\sqrt{2\pi\alpha\eta_-''(z_\alpha)\eta_+(z_\alpha)\eta_-(z_\alpha)A'(\eta_-(z_\alpha))}} e^{tz_\alpha+xs_\alpha+tH_\alpha(\varepsilon)} \left(1+\Xi_{1,1}\left(\frac1t,\varepsilon\right)\right). \]
Here \(z_\alpha\) is the real point at which the unique minimum of the function \(z+\alpha\eta_-(z)\) is attained \((z_0<z_\alpha<\infty)\); \(s_\alpha=\eta_-(z_\alpha)\); \(\varepsilon=x/t-\alpha\); \(\tau=x/t\); \(\tau=\alpha+\varepsilon\); \(H_\alpha(\varepsilon)\) is a function analytic at the point \(\varepsilon=0\):
\[ H_\alpha(\varepsilon)=h_1\varepsilon+h_2\varepsilon^2+\cdots; \]
the \(h_k\) depend on the derivatives of the function \(\eta_-(z)\) at the point \(z_\alpha\) and on the coefficients \(c_k\) in the expansion
\[ z_\tau=z_\alpha+\sum_{k=1}^\infty c_k\varepsilon^k \]
(the coefficients \(c_k\) are found from the identity \(1+\tau\eta_-'(z_\tau)\equiv0\)); the value of \(F(x)\) is given in Theorem 1; \(\Xi_{1,1}(1/t,\varepsilon)\) is an expansion in powers of \(1/t\) and \(\varepsilon\) with known coefficients, and this expansion contains no terms of order \((1/t)^m\varepsilon^n\), where simultaneously \(m<1\), \(n<1\);
\[ \Delta(t)= \begin{cases} 1, & t<0,\\ 0, & t>0. \end{cases} \]
In conclusion, the author expresses sincere gratitude to A. V. Skorokhod for assistance in writing this work.
Kyiv State University
named after T. G. Shevchenko
Received
27 XII 1965
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