数理解析研究所講究録
第 1620 巻 2009 年 109-118

On a$J_{1}$

109-convergence theorem for stochastic processes on $D[0, \infty)$

having monotone sample paths and its applications
琉球大学理学部 山里 眞 (Makoto Yamazato)
Dept. Mathematical Sciences, Faculty of Science,
University of the Ryukyus
Abstract
In this paper, we first show that Bingham’s assertion on weak convergence under
topology of stochastic processes with monotone sample paths is incorrect by giving a counterexample and then remark that his assertion is valid if the limit process is continuous. lburthermore, we give a simple sufficient condition for weak convergence under -topology ofstochastic processes with monotone sample paths. We apply the result to the weak convergence under -topology of hitting time processes of l-dimensional generalized diffusion
processes.
$J_{1^{-}}$$J_{1}$$J_{1}$

1

Introduction

Let $D[0, \infty)$ be the class of functions defined on
to which have left limits on $(0, \infty)$
and are right continuous on. Bingham asserted in [2] that a sequence of randomvariablesin $D[0, \infty)$ , which has monotone sample paths, converges weakly to an$D|0,$
in -topology if
converges tois continuous in probability andin finite
dimensional distributions sense.
In this paper, we remark that his proof contains an error and give an example whichshows that his statement does not hold. We also remark that his conclusion is valid if wechange the continuity in probability assumption forto continuity of all sample functions.
We, then, give an additional condition so that the conclusion of Bingham’s assertion
holds and apply the result to the weak convergence in -topology of additive processes
and sums of independent random variables. The former result is then applied directly to
weak convergences under -topology of hitting time processes of l-dimensional generalized
diffusion processes to selfsimilar additive processes.
Define$[0, \infty)$$\mathbb{R}$$[0, \infty)$$\{X_{n}\}$$\infty)$$J_{1}$$X_{\infty}\in$$X_{\infty}$$X_{n}$$X_{\infty}$$X_{\infty}$$J_{1}$$J_{1}$$\Delta(c,X,T)=\sup_{0\leq t-c<t_{1}<t<t_{2}<t+c\leq T}\min(|X(t_{1})-X(t)|, |X(t_{2})-X(t)|)$.

Stone [7] proved the following theorem.

conTheorem 1.1 (Stone) If $\{X_{n} : n=1,2, \ldots\}$ are random variables in $D[0, \infty),$verges to a random variable
in $D[0, \infty)$ as $narrow\infty$ weakly under the -topology on$D[0, \infty)$ if and only if$(a)$ The finite-dimensional distribu tions ofconverge to those ofas $narrow\infty$$(b) \lim_{carrow 0}\lim\sup_{narrow\infty}P(\Delta(c,X_{n}, T)>\epsilon)=0$ for eachand each $0<T<\infty$ .
$X_{n}$$X_{\infty}$$J_{1}$$X_{n}$$X_{\infty}$$\epsilon>0$

Bingham’s assertion in next section reduces the proof to the above Stone’s theorem.

110

2

On Bingham’s assertion

Bingham asserted the following assertion as Theorem 3 in [2].
Bingham’s assertion : Letbe a sequence of stochastic processes whose pathfunctions lie in $D[0, \infty)$ . If(i) the finite-dimensional distributions ofconverge to those of,(ii) the process
is continuous in probability and(iii) the processes
have monotone path-functions,
thenconverges to
weakly under the -topology on $D[0, \infty)$
He defines the quantity$\{X_{n}\}_{n\geq 1}$$X_{n}$$X_{\infty}$$X_{\infty}$$X_{n}$$X_{n}$$X_{\infty}$$J_{1}$

r=1,2, \ldots, k\}$ .

$D(k, X, T)= \max\{|X(rT/k)-X((r-1)T/k)| :He asserts that, by stochastic continuity of$X_{\infty}$,$P(\Delta(Tk^{-1}, X_{\infty},T)>\epsilon)arrow 0$

as

(1)

$karrow\infty$

for each
and $0<T<\infty$ (for this convergence, the stochastic continuity assumption
necessaryis notif
is in $D([0, \infty))$ , because $\Delta(Tk^{-1}, X,T)arrow 0$ as $karrow\infty$ for $X\in$$\epsilon>0$$X_{\infty}$$D([0, \infty))).$

Then he asserts that the inequalities(2)$\frac{1}{2}\Delta(2Tk^{-1}, X,T)\leq D(k, X,T)\leq\Delta(Tk^{-1}, X,T)$

holds and using these inequalities, he shows the validity of the condition (b) in Stone’s
theorem. But, the second inequality in (2) does not hold even for the following simple

function:$x(t)=\{\begin{array}{l}0, 0\leq t<a,1, a\leq t\leq T,\end{array}$where $0<a<T$ . For this function, $D(k,x,T)=1$ while $\Delta(Tk^{-1},x,T)=0$ for large
may consider, instead of (1),$P(D(k, X_{\infty}, T)>\epsilon)arrow 0$

as

$k$

. One
(3)

$karrow\infty$

and $0<T<\infty$ . But this does not hold even for a simple stochastic process.
. Then,
Let $X(t)$ be a Poisson process with parameter and choose
for each$\epsilon>0$$\epsilon\in(0,1)$$\lambda$$P(D(k, X_{\infty}, T) \leq\epsilon)=\prod_{r=1}^{k}P(X(rT/k)-X((r-1)T/k)\leq\epsilon)=e^{-\lambda T}$.

Moreover, the above assertion does not hold under the assumtions in the threorem. The
followin is a counter example : Let $T=1$ . Let$g$$X_{\infty}(t,\omega)$$=$$2II_{[\xi(\omega),1]}(t)$$X_{n}(t,\omega)$

$=$

II

,

$1^{(t)}\xi(\omega)_{n}-A,1]+II_{[\xi(\omega)+_{n},1]}A(t)$,Then
and $X_{n}(t)$ arewhereis a random variable uniformly distributed on
is stochastically continuous.
non-decreasing and right continuous with left limits. Hence,$X_{n}(t)$ converges in finite dimensional distributions, but does not converge in-topology.
. This example
The last assertion is immediate from the fact that $\Delta(c, X_{n},T)=1$ foris stated as Problem 12.5 in [1].
$\xi(\omega)$$[ \frac{1}{3},$$\frac{2}{3}1\cdot$$X_{\infty}(t)$$X_{\infty}$$J_{1}$$\frac{1}{n}<c$111
Remark 2.1 If we substitu $te$ the condition ”continuous in probability“ $by$ “continuo us withprobability 1 “, then th above theorem is valid $sin$ ce in this case, $D(k, X_{\infty}(\omega), T)arrow 0$ as$narrow\infty$ wi th probability 1 and hence (3) holds as $karrow\infty$ for each
and $0<T<\infty$ .
By the assumption (i), $\lim\sup_{narrow\infty}P(D(k, X_{n}, T)>2\epsilon)\leq P(D(k,X_{\infty}, T)>\epsilon)$ . Using thisineq uality, the convergence in (3) and th left inequ ality in (2), we have the weak $con$ vergence
under the -topology.
$e$$\epsilon>0$$e$$J_{1}$

If the above Bingham’s assertion were tme, then the statement would be very useful. In fact,many papers use Bingham’s “ Theorem 3”. For example, in Corollary 3.5 and Corollary 3.11
in [5] and Corollary 3.4 in [4]. Fortunately, sample paths of the limit processes in Corollary
3.11 in [5] and Corollary 3.4 in [4] are contiouous, the conclusion is valid by the above remark.
In Corollary 3.5 in [5], limit processes are purely discontinuous. Hence the result in Remark
2.1 can not be applied.
the following section, we investigate what additional condition is needed so that weak
convergence under -topology holds. The result is described as Theorem 3.1, in the following
section. The conclusion of Corollary 3.5 in [5] is valid by Theorem 3.3, which is an applicationof Theorem 3.1.
$\ln$

$J_{1}$

3

Main results

Lemma 3.1 Let $x\in D[0,1]$ with non-decreasing path. Let$x_{1}(t)= \sum_{0\leq\epsilon\leq t,x(s)-x(s-)>\epsilon}(x(s)-x(s-))$and let $x_{2}(t)=x(t)-x_{1}(s)$ . Assume thatin $[0,1]$ . Then, for $r<s<t$ ,$x_{1}(t)$has at most one jump in a sub interval $(r,t]$$(x(s)-x(r))\wedge(x(t)-x(s))\leq x_{2}(t)-x_{2}(r)$Proof First assume that there is no jump greater than$\epsilon$.

in $(r,t]$ . Then, for $r\leq s\leq t$ ,$(x(s)-x(r))\wedge(x(t)-x(s))\leq(x(t)-x(r))/2=(x_{2}(t)-x_{2}(r))/2$ .

Next, assume that $r<s_{0}\leq t$ be the jumping point of $x_{1}(t)$ . We devide into three cases.
(1) $x(s_{0}-)< \frac{1}{2}(x(t)-x(r))<x(s_{0})$ ,(2) $x(s_{0})\leq\tilde{2}1(x(t)-x(r))$ and(3) $\frac{1}{2}(x(t)-x(r))\leq x(s_{0}-)$ .
If (1) holds, then$(x(s)-x(r))$ A $(x(t)-x(s))\leq(x(s_{0}-)-x(r))\vee(x(t)-x(so))\leq x_{2}(t)-x_{2}(r)$ .

Now, assume that the case (2) or (3). Then$(x(s)-x(r))$ A $(x(t)-x(s))\leq(x(t)-x(r))/2\leq x_{2}(t)-x_{2}(r)$ .
$\square$

Let $X$ be a random variable in $D[0, \infty)$ . For each$\epsilon>0$,$X^{1,\epsilon}(t)= \sum_{0<\epsilon\leq t,X(s)-X(s-)>\epsilon}(X(s)-X(s-))$be a set of such that $X(t,\omega)$ has at most one
and $X^{2,\epsilon}(t)=X(t)-X^{1}$ ,‘ $(t)$ . Letjump greater than in any semi closed interval ( $r,r+\delta|$ contained in $(0,T]$ , where $\delta>0$ .
$\Omega_{\delta}^{\epsilon}(X,T)$$\epsilon$$\omega$

112
is a seq uence of stochastic processes in $D[0, \infty)$ with
Theorem 3.1 Suppose thaton-decreasin sample pa $ths$ which converges in finite dimensional distribu tions sense to astochastic process $X_{\infty}\in D[0, \infty)$ as $narrow\infty$ . Ifconverges to(i) for each
as $narrow\infty$ in finite dimensional distributions senseand,(ii) for each $T>0,$lim
converges to
weakly in -topology on $D[0, \infty)$ .
then$\{X_{n}\}$

$n$

$g$

$\epsilon>0,$$X_{n}^{2,\epsilon}$$X_{\infty}^{2_{2}\epsilon}$$supnarrow\infty^{P(\Omega_{\delta}^{\epsilon}(X_{n},T))}=1$$\lim_{\delta\downarrow 0}$$\epsilon,$$X_{n}$$J_{1}$$X_{\infty}$. If
Proof Let $X$ be a random variable in $D[0, \infty)$ and let$k-1$ , then, by Lemma 3.1,
and $s- \frac{T}{2k}<r<s<t<s+\pi T$ for $\ell=1,$$\frac{2\ell-1}{2k}T\leq s\leq$$\omega\in\Omega_{\tau/k}^{\epsilon}(X, T)$$\frac{2\ell+1}{2k}T$$\ldots,$$(X(t)-X(s)) \wedge(X(s)-X(r)\leq X^{2}(\frac{\ell+1}{k}T)-X^{2}(\frac{\ell-1}{k}T)$

.

Hence
$\Delta(\frac{1}{2k}T, X,T)\leq\max_{1\leq l\leq k-1}\{X^{2}(\frac{\ell+1}{k}T)-X^{2}(\frac{\ell-1}{k}T)\}$.

Denote the right hand side of the above inequality $E(k,X^{2})$ . We have$\lim_{narrow}\sup_{\infty}P(E(k, X_{n}^{2,e})>2\epsilon)\leq P(E(k, X_{\infty}^{2\epsilon}\rangle)\geq 2\epsilon)$by (i). If

is sufficiently large, then$k$$E(k,X_{\infty}^{2,\epsilon}( \omega))<\frac{3}{2}\epsilon$. Hence$\lim_{karrow\infty}E(k, X_{\infty}^{2,\epsilon}(\omega))\vee\frac{3}{2}\epsilon-\frac{3}{2}\epsilon=0$

.

Then,
$P(E(k, X_{\infty}^{2,\epsilon})>2\epsilon)$$\leq$$=$$P(E(k, X_{\infty}^{2,\epsilon}) \vee\frac{3}{2}\epsilon>2\epsilon)$$P(E(k, X_{\infty}^{2_{1}\epsilon}) \vee\frac{3}{2}\epsilon-\frac{3}{2}\epsilon>\frac{1}{2}\epsilon)arrow 0$

as$karrow\infty$.

(4)This shows that
$P( \Delta(\frac{1}{2k}, X_{n}, T)>2\epsilon)$$\leq$$P( \Delta(\frac{1}{2k}, X_{n},T)>2\epsilon:\Omega_{T/k}^{\epsilon}(X_{n}, T))+P((\Omega_{T/k}^{\epsilon}(X_{n},T)^{c})$$\leq$$P(E(k, X_{n}^{2,\epsilon})>2\epsilon)+P((\Omega_{T/k}^{\epsilon}(X_{n}, T)^{c})$By assumption (ii) and (4), we have, for eqch $T>0$ and$\epsilon>0$,$\lim_{karrow\infty}\lim_{narrow}\sup_{\infty}P(\Delta(\frac{1}{2k}, X_{n}, T)>2\epsilon)=0$

Appealing to Stone’s theorem, we have the conclusion.

.

$\square$We say that a stochastic process $X$ is an additive process in law if it has independent
increments, $X(0)=0$ a.s. and it is stochastically continuous. Moreover, if the sample paths
are contained in $D[0, \infty)$ with probability one, then we call $X$ simply additive process. See[3] for detail.
be a seq uence of additive processes having non-decreasing sample
Theorem 3.2 Let
in finite dimensional
converges to a stochastically continuous process
paths. If$X$$D[0,$
with non-decreasing
processin
anisadditivesense,
then there
distributions$X$
weakly under -topology
converges toandsample paths which is equivalent to$\{X_{n}\}$$X_{\infty}$$X_{n}$$\infty)$$X_{\infty}$as$narrow\infty$.

$X_{n}$$J_{1}$

113
Proof If the finite d\’imensional distributions ofconverge to the finite dimensional dis, thentributions ofis an additive process in law. It has a modification $X$ whose
sample patlis are contained inand non-decreasing. For the convergence oftoin finite dimensional distributions sense, it is necessary and sufficient that for every $t\geq 0$
the distribution of $X_{n}(t)$ converges to the distribution of. Let$X_{n}$$X_{\infty}$$X_{\infty}$$[0, \infty)$$X_{n}$$X_{\infty}$$X_{\infty}(t)$$Ee^{-zX_{n}(t)}=\exp(-\gamma_{n}(t)+/o^{\infty}(e^{-zx}-1)\nu_{n}(t, dx))$

and$Ee^{-zX_{\infty}(t)}=\exp(-\gamma_{\infty}(t)+/o^{\infty}(e^{-zx}-1)\nu\infty(t, dx))$.

For the convergence of the distribution of $X_{n}(t)$ to the distribution of, it is necessary
and sufficient that the L\’evy measure $\nu_{n}(t, A)$ converges to the L\’evy measureforevery continuity Borel set ofwhich is outside of a neighbourhood of and $t\geq 0$ and([3]). This shows that for any
converges toas $narrow\infty$in finite dimensional distributions sense since
. Let
for
$\ell=0,1,$$k-1$ . If the number of jumps ofandfor $\ell=0,1,$$\ell=0,1,$$\ell=0,1,$
contained in each interval$k-1$ and$k-2$
are atmost one, then
. Probability that the number of jumps ofcontained in each intervals $(ss],$ $\ell=0,1,$$k-1$ are atmost one is$X_{\infty}(t)$$\nu_{\infty}(t, A)$

$A$

$0$

$\nu_{\infty}$$\gamma_{n}(t)arrow\gamma(t)_{\infty}$$\epsilon>0,$$X_{n}^{2,\epsilon}$$X_{\infty}^{2_{\}\epsilon}$$\nu_{\infty}^{2,\epsilon}(t, \cdot)=\nu_{n}(t,$$k$$t_{\ell}= \frac{2\ell+1}{2k}T$$\ldots,$$s \ell=\frac{l}{k}T$$\cdot\cap(0, \epsilon|)$$X_{n}^{1,\epsilon}(t,\omega)$$\ldots,$$(s\ell, s_{\ell+1}],$$(t_{\ell},$$\ldots,$$t_{\ell+1}|,$$\ldots,$$X_{n^{Z}}^{1\epsilon}(t,\omega)$$\omega\in\Omega_{T/k}^{\epsilon}(X_{n},T)$$\ldots,$$\prod_{\ell=0}^{k-1}e^{-\nu_{n}((\epsilon_{\ell},s_{\ell+1}],(\epsilon,\infty))}(1+\nu_{n}((s\ell, s_{\ell+1}], (\epsilon, \infty)))$$=$By the convergence of$e^{-\nu_{n}((0,T],(\epsilon,\infty))} \prod_{\ell=0}^{k-1}(1+\nu_{n}((S\ell, S\ell+1], (\epsilon, \infty)))$$X_{n}$

to$X_{\infty}$.

in finite dimensional distributions sense,$\lim_{narrow}\inf_{\infty}e^{-\nu_{n}((0,T],(e_{J}\infty))}\prod_{\ell=0}^{k-1}(1+\nu_{n}((s_{\ell}, s_{\ell+1}], (\epsilon, \infty)))$$\geq$$e^{-\nu_{\infty}((0,T](\epsilon_{1}\infty))}) \prod_{\ell=0}^{k-1}(1+\nu_{\infty}((s\ell, s_{\ell+1}], (\epsilon, \infty)))$as$narrow\infty$.

We have$e^{-\nu_{\infty}((0,T],(\epsilon,\infty))} \prod_{\ell=0}^{k-1}(1+\nu_{\infty}((s\ell, s_{\ell+1}], (\epsilon, \infty)))$$\geq$$\exp[-\nu_{\infty}((0, T],$ $(\epsilon, \infty))+\nu_{\infty}((0, T],$ $( \epsilon, \infty))(1-0\max\frac{1}{2}\nu_{\infty}((s_{\ell}, s_{\ell+1}|, (\epsilon, \infty)))]$is stochastically continuous,by the inequality $\log(1+x)\geq x-\frac{1}{2}x^{2}$ for $x\geq 0$ . Since
as $karrow\infty$ .
is continuous in and hence$karrow\infty$ .
Similarly, we have that
The right hand side of the above inequality tends to 1 as,contained in each intervals
the probability that the number of jumps of$karrow\infty$$narrow\infty$.
Hence$P=0,1,$$k-2$ are atmost one tends to 1 asand then$X_{\infty}$$t$$\nu_{\infty}(t, \cdot)$$\max_{0\leq\ell\leq k-1}\nu_{\infty}((\frac{\ell}{k}T, \frac{\ell+1}{k}T],$ $(\epsilon, \infty))arrow 0$$(t_{\ell}, t_{\ell+1}]$$X_{n}^{1,\epsilon}(t,\omega)$$\ldots,$.

$\lim_{karrow\infty}\lim_{narrow}\sup_{\infty}P(\Omega_{T/k}^{\epsilon}(X_{n},T))=1$

By appealing to Theorem 3.1 we have the weak converge under$J_{1}$-topology of $X_{n}arrow X$ .

$\square$114
Skorohod [6] obtains the same result for general L\’evy processes.
Theorem 3.3 Let for each $n\geq 1,$
be a stric $tly$ increasing seq uence of positive$th
at\lim_{karrow\infty}t_{n_{1}k}=\infty$$\lim_{narrow\infty}\sup_{k\geq1}(t_{n,k+1}-t_{nk,\rangle})=0$ .
umbers suchandLet for each$n\geq 1,$
be a sequence of independent on negative ran $dom$ variables such that for,
each$\{t_{n,k}\}_{k=1}^{\infty}$$t1$$\{Y_{n,k}\}_{k=1}^{\infty}$$n$$\epsilon>0$$\lim_{narrow\infty}\max_{1}P(Y_{n,k}k\geq>\epsilon)=0$

If there is a measure$\nu(t)$on$(0, \infty)$.

continuous in $t>0$ such that for each $t>0$ ,(5)$\lim_{narrow\infty}\sum_{t_{n,k}\leq t}P(Y_{n,k}>x)=\nu(t, (x, \infty))$at every $con$ tinuity point $x>0$ of$\nu(t)$and$\lim_{\epsilonarrow 0}\lim\sup_{t}\sum_{\leq t}narrow\infty n,k(\int_{0<x<\epsilon}x^{2}P(Y_{n,k}\in dx)-(/0<x<\epsilon xP(Y_{n_{J}k}\in dx))^{2})$$=$$\lim_{\epsilonarrow 0}\lim_{narrow}\inf_{\infty}\sum_{t_{n,k}\leq t}(1_{0<x<\epsilon}^{x^{2}P(Y_{n_{t}k}}\in dx)-(/0<x<\epsilon xP(Y_{n,k}\in dx))^{2})=0$$\lim_{\epsilonarrow 0}\lim_{narrow\infty}\sum_{t_{n.k}\leq t}\int_{0<x<\epsilon}xP(Y_{n,k}\in dx)=\gamma(t)$converges weakly underthenwith non decreasing sample paths such th at$X_{n}(t)= \sum_{t_{n.k}\leq t}Y_{n_{t}k}$$J_{1}$,,-topology to an additive process $\{X(t)\}$$E( e^{-\theta X(t)})=\exp(-\gamma(t)+\int_{0}^{\infty}(e^{-\theta x}-1)\nu(t, dx))$

Proof Convergence of finite dimensional distributions of $X_{n}(t)$ to those of $X(t)$ is straightforward by the convergence of the distribution of triangular array of independent randomvariables to a infinitely divisible distribution. We show weak convergence under -topology
applying Theorem 3.1. Let $T>0$ and let
for $\ell=0,1,$
for
and$\ell=0,1,$$k-1$ . If the number of jumps of,contained in each interval$\ell=0,1,$$\ell=0,1,$$k-2$ are atmost one, then$k-1$ and.
The probability that there is no jump of $X_{n}(t)$ greater than in the interval $(ss|$ isisand the probability that there is one jump of $X_{n}(t)$ in$J_{1}$$k$$s_{\ell}= \frac{\ell}{k}T$$t_{\ell}= \frac{2\ell+1}{2k}T$$\ldots,$$(s\ell, s_{\ell+1}]$$X_{n}^{1,\epsilon}(t,\omega)$$\ldots,$$(t_{1},t_{\ell+1}],$$\ldots,$$\omega\in\Omega_{T/k}^{\epsilon}(X_{n},T)$$\ldots,$$\epsilon$$(s\ell,$ $s_{\ell+1}|$$\prod_{s\ell<t_{n,j}}{}_{\leq s_{\ell+\iota)}}P_{nj}([0, \epsilon])$$\sum_{s\ell<t_{\mathfrak{n},j}\leq s\ell+1}P_{n,j}((\epsilon, \infty))x\prod_{1k\neq j,t_{n.k}\in(\epsilon_{\ell},\epsilon p+1}P_{n,k}([0, \epsilon|)$The probability that there is atmost one jump of $X_{n}(t)$ in each.

$(8\ell, S\ell+1]$

is$\prod_{\ell=0}^{k-1}(\prod_{\epsilon\ell<t_{n,j}\leq\epsilon\ell+1}(1-P_{n_{2}j}((\epsilon, \infty))))(1+\sum_{\partial\ell<t_{\mathfrak{n},j}\leq s\ell+1}P_{n,j}((\epsilon, \infty))/P_{n,j}([0, \epsilon|))$.

(6)

Now, we use the inequalities $\log(1+x)\geq x-\frac{1}{2}x^{2}$ for $x>0$ and $\log(1-x)\geq-x-x^{2}$ for$0 \leq x\leq\frac{1}{2}$. We have115(6)$\geq$$\exp(\sum_{\ell=0}^{k-1}[\sum_{e\ell<t_{n,j}\leq sp+1}\log(1-P_{n,j}((\epsilon, \infty)))$$+ \log(1+\sum_{s\ell<t_{n,j}\leq s_{\ell+1}}P_{n,j}((\epsilon, \infty))/P_{n,j}([0, \epsilon|))])$$\geq$$\exp[\sum_{\ell=0}^{k-1}\{\sum_{s\ell<t_{\mathfrak{n},j}\leq s_{\ell+1}}(-P_{n,j}((\epsilon, \infty))-\{P_{n,j}((\epsilon, \infty))\}^{2})$$+ \sum_{s\ell<t_{n,j}\leq s_{\ell+\iota}}P_{n,j}((\epsilon, \infty))/P_{n,j}([0_{7}\epsilon])-\frac{1}{2}()$$\geq$$\exp[\sum_{\ell=0}^{k-1}\{-\sum_{s\ell<t_{n,j}\leq s\ell+\iota}P_{n,j}((\epsilon, \infty))(1+\max_{i}P_{n,i}((\epsilon, \infty)))$.

$+$$\frac{1}{\max_{i}P_{n,j}([0,\epsilon])}\sum_{s_{\ell}<t_{n,j}\leq s_{\ell+1}}P_{n,j}((\epsilon, \infty))(1-\frac{1}{2\min_{i}P_{n,i}([0,\epsilon])}\sum_{sp<t_{n,l}\leq s_{\ell+1}}P_{n,i}((\epsilon, \infty)))\}]$

Since $\lim_{narrow\infty}\max_{i}P_{n,i}((\epsilon, \infty))=0$ andby the assumption,
the assumption (5) yields that the inferior limit as $narrow\infty$ of the last term of the above
inequalities is greater than or equal to$\lim_{narrow\infty}\min_{i}P_{n_{2}i}([0, \epsilon])=1$$\exp[\sum_{\ell=0}^{k-1}\{\ell,\ell+11-\frac{1}{2}\nu((\ell+1\cdot$

Here $\nu([(s, t], \cdot)=\nu(t, \cdot)-\nu(s, \cdot)$ . By the continuity ofin , the above quantity
tends to 1 as $karrow\infty$ . In the same way we have that the probability that $X_{n}(t)$ has
atmost one jump greater than in each semiclosed interval$\ell=0,$$k-2$ ,
tends to 1 as $narrow\infty$ and then $karrow\infty$ . Hence. Since$\nu(t, \cdot)$$t$$(t_{\ell},$$\epsilon$$t_{\ell+1}|,$$\ldots,$$\lim_{\deltaarrow 0}\lim\sup_{narrow\infty}P(\Omega_{\delta}^{\epsilon}(X_{n},T))$

and $P(Y_{n_{i}j}1_{[0,\epsilon]}(Y_{n,j})\in B)=P_{n_{2}j}((\epsilon, \infty))\delta_{0}(B)+P_{n_{2}j}(B\cap$, convergence oftofor eqachin finite dimensional distributions sense
is automatic from the convergence in finite distributions sense of
to $X$ .
$X_{n}^{2,\epsilon}(t)= \sum_{t_{j}\leq t}Y_{n_{l}j}1_{[0,\epsilon]}(Y_{n,j})$$[0, \epsilon])$$X_{n}^{2,\epsilon}$$X^{2,\epsilon}$$\epsilon>0$$X_{n}$

The above theorem guarantees the validity of Corollary 3.5 in [5].
Theorem 3.4 Let for each $n\geq 1,$
random variables and let$Y_{n}(t)=k$$\{Y_{n,i}\}_{i\geq 1}$

is a sequence of independent nonnegative

if $\sum_{j<k}Y_{n,j}<t\leq\sum_{j\leq k}Y_{n_{J}j}$ .

If $\{X_{n}(t)=c(n)Y_{n}(nt)\}$ converges to X $(t)\}$ in finite dimensional distributions sense with$c(n)$ satisfying $\lim_{narrow\infty}c(n)=0$ , thenconverges to $X$ weakly under -topology as$\{$$X_{n}$$narrow\infty$$J_{1}$.

Proof Since
for $\epsilon>c(n)$ , the processes
satisfies the condition (i) of
Theorem 3.1 and hence there is no jump greater than . This means that the condition (ii)is also satisfied. Hence
converges weakly under -topology.
$X_{n}^{2,\epsilon}(t)=X_{n}(t)$$\{X_{n}\}$$\epsilon$$X_{n}$$J_{1}$

116
Limit theorems for hitting times of generalized diffusions
4, whichto
Let $m$ be a non-decreasing, right continuous function defined on$m(\infty)=\infty,$$(0,\infty)$$m(O-)=0$ . We set, satis $\Phi ingm(-\infty)=-$ oo
has left limit on$[-\infty, \infty]$$[-\infty, \infty]$$\ell_{1}$$=$$\sup\{x<0:m(x)=-oo\}$$\ell_{2}$$=$$\inf\{x>0:m(x)=\infty\}$. Therethe support of the measure induced by $m$ restricted to
We denote by, called l-dimensional generalized difnaturally corresponds a strong Markov process, to $m$ by changing the, whose formal infinitesimal generator isfusion process ontime of the Brownian motion (see [8]). The measure $m(dx)$ is called the speed measure of
can be regarded as aby . The hitting time
. Denote the hitting time of for$x\in
E_{m}$
independent
increments. Moreover, thiswithstochastic process with time parameter. This extended process is called generalized hitting
process can be extended toof when the process starts at the origin anddistribution
limit
time process in [8]. In [8],dimensionaltends to . Some of results in [8] are concemed with convergence ofdistributions. Since generalized hitting time process has non-decreasing sample paths, byTheorem 3.2, finite dimensional distributions leads to -convergence.
Following [8], we introduce assumptions for asymptotics of $m$ .
$(C_{\gamma})(0\leq\gamma<1):\ell_{2}=\infty$ and $m(x)\sim x^{\gamma/(1-\gamma)}L(x)/(1-\gamma)$ as $xarrow\infty$ with a function$(\ell_{1},p_{2})$$E_{m}$$\{X_{t}\}$$E_{m}$$\frac{d}{dm}\frac{d}{dx}$$\{X_{t}\}$$x$$\{X_{t}\}$$\tau_{x}$$\tau_{x}$$x\in(\ell_{1}, \ell_{2})$$\tau_{x}$$i\overline{m}ite$$x$$\ell_{2}$$J_{1}$$L$

slowly varying at .
$(C_{1})\ell_{2}=\infty$ and there is a function $s(x)$ regularly varying at
such that the derivative $s’(s)$ is positive and monotone and$\infty$and differentiable for large$\infty$$x$$m(s(x))\sim e^{x}L(e^{x})$with a function$L$slowly varying at$\infty$as$xarrow\infty$

.

and
$K$
slowly varying at .
hmctionand $m(x)\sim(\ell_{2}-x)^{-1}K(\ell_{2}-x)^{-1}$ as:varying at .
Define $a(x)= \int_{0}^{x}m(y)dy$ .

$m(x)\sim(\ell_{2}-x)^{\gamma/(1-\gamma)}K(\ell_{2}-x)/(\gamma-1)$$(C_{\gamma})(1<\gamma<\infty):\ell_{2}<\infty$as$xarrow\ell_{2}$with a$0$$(C_{\infty})$$\ell_{2}<\infty$$xarrow\ell_{2}$with a function $K$ slowly$0$

Theorem 4.1 Assume that $\ell_{1}=-\infty,$$(C_{\gamma})$holds for some $0\leq\gamma<1$ and$m(-x)/m(x)arrow c\in[0, \infty)$as$xarrow\infty$.

Then, the normalized generalized hitting time process $\{\tau_{tx}/a(x)\}$ with any starting point
withconverges weakly under -topology on $D[0, \infty)$ to a selfsimilar additive process
is the hitting time process of the generalized
exponent $(1-\gamma)^{-1}$ as $xarrow\infty$ . The process
diffusion process wi th speed measure$\{\tau_{t}^{0}\}$$J_{1}$$\{\tau_{t}^{0}\}$$m^{0}(x)=\{\begin{array}{ll}x^{\gamma/(1-\gamma)}/(1-\gamma), x\geq 0,-c|x|^{\gamma/(1-\gamma)}/(1-\gamma), x<0.\end{array}$starting at the origin

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Theorem 4.2 Assume that $p_{1}=-\infty$ .
$\int_{0}^{x}|m(-y)|dy$ . Assume thatsatisfies$\tilde{m}$Let $\tilde{m}(x)=-m((-x)+)+m(0)$ andfor some $0\leq\gamma<1$ and$\overline{a}(x)=$$(C_{\gamma})$$a(x)/\tilde{a}(x)arrow 0$

as$xarrow\infty$.

Then, the normalized generalized hitting time processwith any starting point
converges weakly under -topology on $D[0, \infty)$ to a selfsimilar additi processwithexponent $(1-\gamma)^{-1}$ as $narrow\infty$ . The process$time$
is the hitting
pro cess of the generalized
diffusion process with speed measure$\{\tau_{tx}/\tilde{a}(x)\}$$J_{1}$$r\prime e$$\{\tau_{t}^{0}\}$$\{\tau_{t}^{0}\}$$x\geq 0$$m^{0}(x)=\{$$0-c|x|^{\gamma/(1-\gamma)}/(1-\gamma)$,,$x<0$ ,

starting at the origin
Theorem 4.3 Assume that $\ell_{1}=-$ oo,

holds and$(C_{1})$$x|m(-tx)|/a(x)arrow 0$ as $xarrow\infty$.

Then, the normalized generaJized hitting time process $\{\tau_{s(t+x)}/a(s(x))\}$ with any starting
point converges weakly under -topology on $D[0, \infty)$ to a process
as $xarrow\infty$ . The pro$time$process of th generalized diffusion process with speed measurecessis the hitting$J_{1}$$\{\tau_{t}^{0}\}$$\{\tau_{t}^{0}\}$$m^{0}(t)=e^{t},$$e$$t\in \mathbb{R}$and starting $point-\infty$ .

$\{\tau_{\log t}^{0}\}_{t\geq 0}$

exponent 1.

is a selfsimilar additive process with

Theorem 4.4 Assume that $\ell_{1}=-$ oo,
holds for some $1<\gamma<\infty$ . Then, the normalized generalized hit ting time process
wi th any starting point converges$D[0,\infty)$is theweakly under -topology on. The process
to a processashitting time process of the generalized diffusion process with speed measure$(C_{\gamma})$$\{\tau_{\ell_{2}-y|t|}/a(\ell_{2}-y)\}_{t<0}$$J_{1}$$\{\tau_{t}^{0}\}$$m^{0}(x)=\{$and starting poin $t-\infty$ .

$\{\tau_{t}^{0_{-1}}\}_{t\geq 0}$$\infty|x|^{\gamma/(1-\gamma)}/(\gamma-1)$,$y\downarrow 0$$x>0x<0$$\{\tau_{t}^{0}\}$,is a selfSimilar additive process with exponent $(\gamma-1)^{-1}$ .

Theorem 4.1 corresponds to Theorem 2, Theorem 4.2 corresponds to Theorem 3, Theorem
4.3 corresponds to Theorem 4 and Theorem 4.4 corresponds to Theorem 6 in [8], respectively.

References[1] P. Billingsley. Convergence of Probability Measures. Wiley series in probability andstatistics. John Wiley&Sons, Inc, New York, second edition, 1999.
[2] N. H. Bingham. Limit theorems for occupation times of markov processes. Z. Wahrscheinlichkeitstheorie verw. Geb., 17:1-22, 1971.
[3] K. Sato. L\’evy Processes and Infinitely Divisible Distributions, volume 68 of Cambridge
Studies in Advanced Mathematics. Cambridge University Press, 1999.
[4] M. M. Meerschaert; Hans-Peter SchefHer. Limit theorems for continuous-time random
walks with infinite mean waiting times. J. Appl. Probab., $41(3):623-638$ , 2004.
[5 M. M. Meerschaert; Hans-Peter Scheffler. Stochastic model for ultraslow dithmsion.
Stochastic Process. A ppl., $116(9):1215-1235$ , 2006.
$|$

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[6] A. V. Skorohod. Limit theorems for stochastic processes with independent increments.
Theory of probability and its applications, $2(2):138-171\rangle$ 1957.
[7] C. Stone. Limit theorems for random walks, birth and death processes, and diffusion
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diffusions. Nagoya Math. J., 152:1-37, December 1998.
Department of Mathematical Sciences

Faculty of Science
University of the Ryukyus
Nishihara-cho, Okinawa 903-0213
JAPAN