﻿ On the Increments of Stable Subordinators

Applied Mathematics
Vol.08 No.05(2017), Article ID:76331,8 pages
10.4236/am.2017.85053

On the Increments of Stable Subordinators

1Department of Mathematics, Djillali Liabes University, Sidi-Bel-Abbes, Algeria

2Department of Mathematics, Faculty of Science, Qassim University, Saudi Arabia    Received: April 16, 2017; Accepted: May 20, 2017; Published: May 23, 2017

ABSTRACT

Let $\left\{X\left(t\right),t\ge 0\right\}$ be a stable subordinator defined on a probability space $\left(\Omega ,\mathcal{F},\mathcal{A}\right)$ and let ${a}_{t}$ for $t>0$ be a non-negative valued function. In this paper, it is shown that under varying conditions on ${a}_{t}$ , there exists a function ${\lambda }_{\beta }\left(t\right)$ such that

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{\left(X\left(t+{a}_{T}\right)-X\left(t\right)\right)}{{\lambda }_{\beta }\left(t\right)}=1\text{ }a.s.,$

where ${\lambda }_{\beta }\left(t\right)={\theta }_{\alpha }{a}_{t}^{\frac{1}{\alpha }}{\left(\mathrm{log}\frac{t}{{a}_{t}}+\beta \mathrm{log}\mathrm{log}t+\left(1-\beta \right)\mathrm{log}\mathrm{log}{a}_{t}\right)}^{\frac{\alpha -1}{\alpha }}$ , $0\le \beta \le 1$ , ${\theta }_{\alpha }={\left(B\left(\alpha \right)\right)}^{\frac{1-\alpha }{\alpha }}$ and $B\left(\alpha \right)=\left(1-\alpha \right){\alpha }^{\frac{\alpha }{1-\alpha }}{\left(\mathrm{cos}\left(\frac{\text{π}\alpha }{2}\right)\right)}^{\frac{1}{\alpha -1}}.$

Keywords:

Increments, Stable Subordinators, Iterated Logarithm Laws 1. Introduction

Let $\left\{X\left(t\right),t\ge 0\right\}$ be a stable ordinator with exponent $\alpha$ with $0<\alpha <1$ , defined on a probability space $\left(\Omega ,\mathcal{F},\mathcal{A}\right)$ . Let ${a}_{t}$ for $t>0$ be a non-negative valued function and $Y\left(t\right)=X\left(t+{a}_{t}\right)-X\left(t\right)$ , $t>0$ . Define

${\lambda }_{\beta }\left(t\right)={\theta }_{\alpha }{a}_{t}^{\frac{1}{\alpha }}{\left(\mathrm{log}\frac{t}{{a}_{t}}+\beta \mathrm{log}\mathrm{log}t+\left(1-\beta \right)\mathrm{log}\mathrm{log}{a}_{t}\right)}^{\frac{\alpha -1}{\alpha }}$ ,

where $0\le \beta \le 1$ ,

${\theta }_{\alpha }={\left(B\left(\alpha \right)\right)}^{\frac{1-\alpha }{\alpha }}$ and $B\left(\alpha \right)=\left(1-\alpha \right){\alpha }^{\frac{\alpha }{1-\alpha }}{\left(\mathrm{cos}\left(\frac{\text{π}\alpha }{2}\right)\right)}^{\frac{1}{\alpha -1}}.$

For any value of t, the characteristic function of $X\left(t\right)$ is of the form

$E\left({\text{e}}^{iuX\left(t\right)}\right)=\mathrm{exp}\left(-t{|u|}^{\alpha }\left(1-\frac{ui}{|u|}\mathrm{tan}\left(\frac{\text{π}\alpha }{2}\right)\right)\right),\text{ }0<\alpha <1.$

Limit theorems on the increments of stable subordinators have been investigated in various directions by many authors  -  . Among the above many results, we are interested in Fristedt  and Vasudeva and Divanji  whose results are the following limit theorems on the increments of stable subordinators.

Theorem 1 (  )

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}{\theta }_{\alpha }{t}^{-\frac{1}{\alpha }}{\left(\mathrm{log}\mathrm{log}t\right)}^{\frac{1-\alpha }{\alpha }}X\left(t\right)=1\text{ }\text{ }\text{almost}\text{\hspace{0.17em}}\text{surely}\text{ }\text{ }\left(a.s\right).$

Theorem 2 (  ) Let $0<{a}_{t}$ for $t>0$ , be a non-decreasing function of $t$ such that

(i) $0<{a}_{t}\le t$ for $t>0$ ,

(ii) ${a}_{t}\to \infty$ as $t\to \infty$ , and

(iii) ${a}_{t}/t$ is non-increasing. Then

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{\left(X\left(t+{a}_{t}\right)-X\left(t\right)\right)}{\xi \left(t\right)}=1\text{ }a.s.,$ (1)

where $\xi \left(t\right)={\theta }_{\alpha }{a}_{t}^{\frac{1}{\alpha }}{\left(\mathrm{log}\frac{t}{{a}_{t}}+\mathrm{log}\mathrm{log}t\right)}^{\frac{\alpha -1}{\alpha }}.$

In this paper, our aim is to investigate Liminf behaviors of the increments of Y. We establish that, under certain conditions on ${a}_{t}$ ,

$\begin{array}{l}\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{Y\left(t\right)}{{\lambda }_{\beta }\left(t\right)}=1\text{ }a.s.,\text{ }\text{ }\\ \text{where}\text{ }\text{\hspace{0.17em}}Y\left(t\right)=X\left(t+{a}_{t}\right)-X\left(t\right).\end{array}$ (2)

Throughout the paper c and k (integer), with or without suffix, stand for positive constants. i.o. means infinitely often. We shall define for each $u\ge 0$ the functions $\mathrm{log}u=\mathrm{log}\left(\mathrm{max}\left(u,1\right)\right)$ and $\mathrm{log}\mathrm{log}u=\mathrm{log}\mathrm{log}\left(\mathrm{max}\left(u,3\right)\right)$ .

2. Main Result

In this section, we reformulate the result obtained in Theorem 2 and establish our main result using ${\lambda }_{\beta }\left(t\right)$ with $0\le \beta \le 1$ instead of $\xi \left(t\right)$ .

Theorem 3 Let ${a}_{t}$ , $t>0$ , be a non-decreasing function of $t$ such that

(i) $0<{a}_{t}\le t$ for $t>0$ ,

(ii) ${a}_{t}\to \infty$ as $t\to \infty$ , and

(iii) ${a}_{t}/t$ is non-increasing. Then

$\mathrm{lim}{\mathrm{inf}}_{t\to \infty }\frac{Y\left(t\right)}{{\lambda }_{\beta }\left(t\right)}=1\text{ }a.s.$

Remark 1 Let us mention some particular cases

1. For ${a}_{t}=t$ we obtain Fristedt’s iterated logarithm laws (see Thorem 1).

2. If $\beta =1$ we have Vasudeva and Divanji theorem (see Theorem 2).

3. If $\beta =0$ under assumptions (i), (ii) and (iii) of Theorem 3 we also have

$\underset{t\to \infty }{\mathrm{lim}\mathrm{inf}}\frac{Y\left(t\right)}{{\lambda }_{0}\left(t\right)}=1\text{ }a.s.$

In order to prove Theorem 3, we need the following Lemma

Lemma 1 (see  or  ) Let ${X}_{1}$ be a positive stable random variable with characteristic function

$E\left(\mathrm{exp}\left\{iu{X}_{1}\right\}\right)=\mathrm{exp}\left\{-{|u|}^{\alpha }\left(1-\frac{iu}{|u|}\mathrm{tan}\left(\frac{\text{π}\alpha }{2}\right)\right)\right\},\text{\hspace{0.17em}}0<\alpha <1.$

Then, as $x\to 0,$

$P\left({X}_{1}\le x\right)\simeq \frac{{x}^{\frac{\alpha }{2\left(1-\alpha \right)}}}{\sqrt{2\text{π}\alpha B\left(\alpha \right)}}\mathrm{exp}\left\{-B\left(\alpha \right){x}^{\frac{\alpha }{\alpha -1}}\right\}$

where

$B\left(\alpha \right)=\left(1-\alpha \right){\alpha }^{\frac{\alpha -1}{\alpha }}{\left(\mathrm{cos}\left(\frac{\text{π}\alpha }{2}\right)\right)}^{\frac{1}{\alpha -1}}.$

Proof of Theorem 3. Firstly, we show that for any given $\epsilon >0$ , as $t\to \infty ,$

$P\left(Y\left(t\right)\le \left(1+\epsilon \right){\lambda }_{\beta }\left(t\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}i.o\right)=1.$ (3)

Let ${u}_{1}$ be a number such that ${a}_{{u}_{1}}>1$ . Define a sequence $\left({u}_{k}\right)$ through ${u}_{k+1}={u}_{k}+{a}_{{u}_{k}}$ , for $k=1,2,\cdots .$ Now we show that

$P\left(Y\left({u}_{k}\right)\le \left(1+\epsilon \right){\lambda }_{\beta }\left({u}_{k}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}i.o\right)=1.$

From Mijhneer  , we have

$P\left(Y\left({u}_{k}\right)\le \left(1+\epsilon \right){\lambda }_{\beta }\left({u}_{k}\right)\right)=P\left(X\left(1\right)\le \frac{\left(1+\epsilon \right){\lambda }_{\beta }\left({u}_{k}\right)}{{a}_{{u}_{k}}^{\frac{1}{\alpha }}}\right).$ (4)

But

$\frac{{\lambda }_{\beta }\left({u}_{k}\right)}{{a}_{{u}_{k}}^{\frac{1}{\alpha }}}={\theta }_{\alpha }{\left(\mathrm{log}\frac{{u}_{k}}{{a}_{{u}_{k}}}+\beta \mathrm{log}\mathrm{log}{u}_{k}+\left(1-\beta \right)\mathrm{log}\mathrm{log}{a}_{{u}_{k}}\right)}^{\frac{\alpha -1}{\alpha }}.$

Applying Lemma 1 with

$x=\left(1+\epsilon \right){\theta }_{\alpha }{\left(\mathrm{log}\frac{{u}_{k}}{{a}_{{u}_{k}}}+\beta \mathrm{log}\mathrm{log}{u}_{k}+\left(1-\beta \right)\mathrm{log}\mathrm{log}{a}_{{u}_{k}}\right)}^{\frac{\alpha -1}{\alpha }},$

one can find a ${k}_{0}$ such that, for all $k\ge {k}_{0}$ ,

$\begin{array}{l}P\left(X\left(1\right)\le \frac{\left(1+\epsilon \right){\lambda }_{\beta }\left({u}_{k}\right)}{{a}_{{u}_{k}}^{\frac{1}{\alpha }}}\right)\\ \ge \frac{{c}_{0}}{2{\left(\mathrm{log}\left(\frac{{u}_{k}{\left(\mathrm{log}{u}_{k}\right)}^{\beta }{\left(\mathrm{log}{a}_{{u}_{k}}\right)}^{1-\beta }}{{a}_{{u}_{k}}}\right)\right)}^{1/2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\mathrm{exp}\left\{-{\left(1+\epsilon \right)}^{\alpha /\left(\alpha -1\right)}\mathrm{log}\left(\frac{{u}_{k}{\left(\mathrm{log}{u}_{k}\right)}^{\beta }{\left(\mathrm{log}{a}_{{u}_{k}}\right)}^{1-\beta }}{{a}_{{u}_{k}}}\right)\right\},\end{array}$

where ${c}_{0}$ is some positive constant. Notice that

${\left(1+\epsilon \right)}^{\frac{\alpha }{\alpha -1}}=\left(1-{\epsilon }_{1}\right)<1\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{for}\text{\hspace{0.17em}}\text{some}\text{ }\text{\hspace{0.17em}}{\epsilon }_{1}>0.$

Hence

$\begin{array}{l}P\left(X\left(1\right)\le \frac{\left(1+\epsilon \right){\lambda }_{\beta }\left({u}_{k}\right)}{{a}_{{u}_{k}}^{\frac{1}{\alpha }}}\right)\\ \ge \frac{{c}_{0}}{2{\left(\mathrm{log}\left(\frac{{u}_{k}{\left(\mathrm{log}{u}_{k}\right)}^{\beta }{\left(\mathrm{log}{a}_{{u}_{k}}\right)}^{1-\beta }}{{a}_{{u}_{k}}}\right)\right)}^{1/2}}\left(\frac{{a}_{{u}_{k}}}{{u}_{k}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }×{\left(\frac{{u}_{k}}{{a}_{{u}_{k}}}\right)}^{{\epsilon }_{1}}\frac{1}{{\left({\left(\mathrm{log}{u}_{k}\right)}^{\beta }{\left(\mathrm{log}{a}_{{u}_{k}}\right)}^{1-\beta }\right)}^{\left(1-{\epsilon }_{1}\right)}}\\ =\frac{{c}_{0}}{2{\left(\mathrm{log}\left(\frac{{u}_{k}{\left(\mathrm{log}{u}_{k}\right)}^{\beta }{\left(\mathrm{log}{a}_{{u}_{k}}\right)}^{1-\beta }}{{a}_{{u}_{k}}}\right)\right)}^{1/2}}\left(\frac{{u}_{k+1}-{u}_{k}}{{u}_{k}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{ }×{\left(\frac{{u}_{k}}{{a}_{{u}_{k}}}\right)}^{{\epsilon }_{1}}\frac{1}{{\left({\left(\mathrm{log}{u}_{k}\right)}^{\beta }{\left(\mathrm{log}{a}_{{u}_{k}}\right)}^{1-\beta }\right)}^{\left(1-{\epsilon }_{1}\right)}}.\end{array}$

Let ${1}_{k}={u}_{k}/{a}_{{u}_{k}}$ and ${m}_{k}={\left(\mathrm{log}{u}_{k}\right)}^{\beta }{\left(\mathrm{log}{a}_{{u}_{k}}\right)}^{1-\beta }$ . Note that 1k is non-decreasing and ${m}_{k}\to \infty$ as $k\to \infty$ . In turn one finds a ${k}_{1}\ge {k}_{0},$ such that

$\frac{{1}_{k}^{{\epsilon }_{1}}{m}_{k}^{{\epsilon }_{1}}}{{\left(log{1}_{k}{m}_{k}\right)}^{1/2}}\ge 1,\text{ }\text{ }\text{whenever}\text{ }\text{\hspace{0.17em}}k\ge {k}_{1}.$

Therefore, for all $k\ge {k}_{1}$ , we have

$\begin{array}{l}P\left(X\left(1\right)\le \frac{\left(1+\epsilon \right){\lambda }_{\beta }\left({u}_{k}\right)}{{a}_{{u}_{k}}^{\frac{1}{\alpha }}}\right)\\ \ge {c}_{0}\frac{\left({u}_{k+1}-{u}_{k}\right)}{2{u}_{k}{\left(\mathrm{log}{u}_{k}\right)}^{\beta }{\left(\mathrm{log}{a}_{{u}_{k}}\right)}^{1-\beta }}={c}_{0}\frac{\left({u}_{k+1}-{u}_{k}\right)}{2{u}_{k}}{\left(\frac{\mathrm{log}{a}_{{u}_{k}}}{\mathrm{log}{u}_{k}}\right)}^{\beta }\frac{1}{\mathrm{log}{a}_{{u}_{k}}}\\ \ge {c}_{0}\frac{\left({u}_{k+1}-{u}_{k}\right)}{2{u}_{k}}\left(\frac{\mathrm{log}{a}_{{u}_{k}}}{\mathrm{log}{u}_{k}}\right)\frac{1}{\mathrm{log}{a}_{{u}_{k}}}={c}_{0}\frac{\left({u}_{k+1}-{u}_{k}\right)}{2{u}_{k}\mathrm{log}{u}_{k}}.\end{array}$ (5)

Observe that

${\int }_{{k}_{1}}^{\infty }\frac{\text{d}t}{t\mathrm{log}t}\le \underset{k={k}_{1}}{\overset{\infty }{\sum }}\frac{\left({u}_{k+1}-{u}_{k}\right)}{{u}_{k}\mathrm{log}{u}_{k}}.$ (6)

From the fact that ${\int }_{{k}_{1}}^{\infty }\frac{\text{d}t}{t\mathrm{log}t}=\infty$ and from (4), (5), and (6) one gets

$\underset{k=1}{\overset{\infty }{\sum }}\text{ }\text{ }P\left(Y\left({u}_{k}\right)\le \left(1+\epsilon \right){\lambda }_{\beta }\left({u}_{k}\right)\right)=\infty .$

Observe that $\left(Y\left({u}_{k}\right)\right)$ is a sequence of mutually independent random variables (for, ${u}_{k+1}={u}_{k}+{a}_{{u}_{k}}$ ) and by applying Borel-Cantelli lemma, we get

$P\left(Y\left({u}_{k}\right)\le \left(1+\epsilon \right){\lambda }_{\beta }\left({u}_{k}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}i.o\right)=1$

which establishes (3).

Now we complete the proof by showing that, for any $\epsilon \in \left(0,1\right)$ ,

$P\left(Y\left(t\right)\le \left(1-\epsilon \right){\lambda }_{\beta }\left({t}_{k}\right)\text{ }i.o\right)=0.$ (7)

Define a subsequence $\left({t}_{k}\right)$ , such that

${a}_{{t}_{k}}=\left({t}_{k+1}-{t}_{k}\right)/{\left(\mathrm{log}{t}_{k}\right)}^{\left(1-\beta \right)\left(1+\epsilon \right)},\text{\hspace{0.17em}}k=1,2,\cdots$ (8)

and the events ${A}_{t}$ and ${B}_{k}$ as

${A}_{t}=\left\{Y\left(t\right)\le \left(1-\epsilon \right){\lambda }_{\beta }\left(t\right)\right\}$

and

${B}_{k}=\left\{\underset{{t}_{k}\le t\le {t}_{k+1}}{\mathrm{inf}}Y\left(t\right)\le \left(1-\epsilon \right){\lambda }_{\beta }\left({t}_{k+1}\right)\right\},\text{ }k=1,2,\cdots .$

Note that

$\left({A}_{t}\text{ }i.o.,t\to \infty \right)\subset \left({B}_{k}\text{ }i.o.,k\to \infty \right).$

Further, define

${C}_{k}=\left\{X\left({t}_{k}+{a}_{{t}_{k}}\right)-X\left({t}_{k+1}\right)\le \left(1-\epsilon \right){\lambda }_{\beta }\left({t}_{k+1}\right)\right\}$

and observe that

$\left({B}_{k}\text{ }i.o.,k\to \infty \right)\subset \left({C}_{k}\text{ }i.o.,k\to \infty \right).$

Hence in order to prove (7) it is enough to show that

$P\left({C}_{k}\text{ }i.o.\right)=0.$ (9)

We have

$P\left(X\left({t}_{k}+{a}_{{t}_{k}}\right)-X\left({t}_{k+1}\right)\le \left(1-\epsilon \right){\lambda }_{\beta }\left({t}_{k+1}\right)\right)=P\left(X\left(1\right)\le \frac{\left(1-\epsilon \right){\lambda }_{\beta }\left({t}_{k+1}\right)}{{\left({a}_{{t}_{k}}+{t}_{k}-{t}_{k+1}\right)}^{1/\alpha }}\right)$

and

$\begin{array}{l}\frac{\left(1-\epsilon \right){\lambda }_{\beta }\left({t}_{k+1}\right)}{{\left({a}_{{t}_{k}}+{t}_{k}-{t}_{k+1}\right)}^{1/\alpha }}\\ \simeq \left(1-\epsilon \right){\theta }_{\alpha }{\left(\frac{{a}_{{t}_{k+1}}}{{a}_{{t}_{k}}}\right)}^{1/\alpha }{\left(\mathrm{log}\left(\frac{{t}_{k+1}{\left(\mathrm{log}{t}_{k+1}\right)}^{\beta }{\left(\mathrm{log}{a}_{{t}_{k}}\right)}^{1-\beta }}{{a}_{{t}_{k}}}\right)\right)}^{\left(\alpha -1\right)/\alpha }.\end{array}$

The fact that ${a}_{t}/t$ is non-increasing as $t\to \infty$ implies that

$\frac{{a}_{{t}_{k+1}}}{{t}_{k+1}}\le \frac{{a}_{{t}_{k}}}{{t}_{k}}\text{ }\text{ }\text{or}\text{ }\text{ }\frac{{a}_{{t}_{k+1}}}{{a}_{{t}_{k}}}\le \frac{{t}_{k+1}}{{t}_{k}}.$

Hence for a given ${\epsilon }_{1}>0$ satisfying $\left(1-\epsilon \right){\left(1+{\epsilon }_{1}\right)}^{1/\alpha }<1,$ there exists a ${k}_{2}$ such that

${a}_{{t}_{k+1}}/{a}_{{t}_{k}}\le \left(1+{\epsilon }_{1}\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\text{for}\text{\hspace{0.17em}}\text{all}\text{ }\text{\hspace{0.17em}}k\ge {k}_{2}.$

Let $\left(1-\epsilon \right)\right){\left(1+{\epsilon }_{1}\right)}^{1/\alpha }=\left(1-{\epsilon }_{2}\right)$ . Then, for $k\ge {k}_{2}$ ,

$P\left({C}_{k}\right)\le P\left(X\left(1\right)\le \left(1-{\epsilon }_{2}\right){\theta }_{\alpha }{\left(\mathrm{log}\frac{{t}_{k+1}}{{a}_{{t}_{k+1}}}{\left(\mathrm{log}{t}_{k+1}\right)}^{\beta }{\left(\mathrm{log}{a}_{{t}_{k+1}}\right)}^{1-\beta }\right)}^{\left(\alpha -1\right)/\alpha }\right).$

From lemma 1, we can find a ${k}_{3}\left(\ge {k}_{2}\right)$ such that for all $k\ge {k}_{3}$ ,

$\begin{array}{c}P\left({C}_{k}\right)\le {c}_{1}{\left(\mathrm{log}\frac{{t}_{k+1}}{{a}_{{t}_{k+1}}}{\left(\mathrm{log}{t}_{k+1}\right)}^{\beta }{\left(\mathrm{log}{a}_{{t}_{k+1}}\right)}^{1-\beta }\right)}^{-\frac{1}{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\mathrm{exp}\left\{{\left(1-{\epsilon }_{2}\right)}^{\alpha /\left(\alpha -1\right)}\left(\mathrm{log}\frac{{t}_{k+1}}{{a}_{{t}_{k}}}{\left(\mathrm{log}{t}_{k+1}\right)}^{\beta }{\left(\mathrm{log}{a}_{{t}_{k+1}}\right)}^{1-\beta }\right)\right\},\end{array}$

where ${c}_{1}$ is a positive constant.

Let ${\left(1-{\epsilon }_{2}\right)}^{\alpha /\left(\alpha -1\right)}=\left(1+{\epsilon }_{3}\right)$ , ${\epsilon }_{3}>0.$ Then, for all $k\ge {k}_{3}$ ,

$\begin{array}{l}P\left({C}_{k}\right)\le {c}_{1}{\left(\mathrm{log}\frac{{t}_{k+1}}{{a}_{{t}_{k}}}{\left(\mathrm{log}{t}_{k+1}\right)}^{\beta }{\left(\mathrm{log}{a}_{{t}_{k+1}}\right)}^{1-\beta }\right)}^{-1/2}{\left(\frac{{a}_{{t}_{k+1}}}{{t}_{k}}\right)}^{\left(1+{\epsilon }_{3}\right)}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left({\left(\mathrm{log}{t}_{k+1}\right)}^{\beta }{\left(\mathrm{log}{a}_{{t}_{k+1}}\right)}^{1-\beta }\right)}^{-\left(1+{\epsilon }_{3}\right)}.\end{array}$

Since

${\left({a}_{{t}_{k+1}}/{t}_{k+1}\right)}^{\left(1+{\epsilon }_{3}\right)}\le {\left({a}_{{t}_{k}}/{t}_{k}\right)}^{\left(1+{\epsilon }_{3}\right)}\le {a}_{{t}_{k}}/{t}_{k},$

then from (8) and for all $k\ge {k}_{3}$ , we have

$P\left({C}_{k}\right)\le {c}_{1}{\left(log\frac{{t}_{k}}{{a}_{{t}_{k}}}{\left(log{t}_{k}\right)}^{\beta }{\left(log{a}_{{t}_{k}}\right)}^{1-\beta }\right)}^{-1/2}\left(\frac{{a}_{{t}_{k}}}{{t}_{k}}\right){\left({\left(log{t}_{k}\right)}^{\beta }{\left(log{a}_{{t}_{k}}\right)}^{1-\beta }\right)}^{-\left(1+{\epsilon }_{3}\right)}.$

$\begin{array}{c}P\left({C}_{k}\right)\le {c}_{1}{\left(\mathrm{log}\frac{{t}_{k}}{{a}_{{t}_{k}}}{\left(\mathrm{log}{t}_{k}\right)}^{\beta }{\left(\mathrm{log}{a}_{{t}_{k}}\right)}^{1-\beta }\right)}^{-1/2}\left(\frac{{t}_{k+1}-{t}_{k}}{{t}_{k}}\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}×\frac{1}{{\left(\mathrm{log}{t}_{k}\right)}^{1+{\epsilon }_{3}}}\frac{1}{{\left(\mathrm{log}{a}_{{t}_{k+1}}\right)}^{\left(1-\beta \right)\left(1+{\epsilon }_{3}\right)}}\\ \le {c}_{1}\left(\frac{{t}_{k+1}-{t}_{k}}{{t}_{k}}\right)\frac{1}{{\left(\mathrm{log}{t}_{k}\right)}^{\left(1+{\epsilon }_{3}\right)}}.\end{array}$

Observe that

${\int }_{{k}_{4}}^{\infty }\frac{\text{d}t}{t{\left(\mathrm{log}t\right)}^{\left(1+{\epsilon }_{3}\right)}}\ge \underset{k={k}_{4}}{\overset{\infty }{\sum }}\frac{\left({t}_{k+1}-{t}_{k}\right)}{{t}_{k+1}{\left(\mathrm{log}{t}_{k+1}\right)}^{\left(1+{\epsilon }_{3}\right)}},$

and

$\frac{\left({t}_{k+1}-{t}_{k}\right)}{{t}_{k+1}{\left(\mathrm{log}{t}_{k+1}\right)}^{\left(1+{\epsilon }_{3}\right)}}\simeq \frac{\left({t}_{k+1}-{t}_{k}\right)}{{t}_{k}{\left(\mathrm{log}{t}_{k}\right)}^{\left(1+{\epsilon }_{3}\right)}}.$

Hence

$\underset{k={k}_{4}}{\overset{\infty }{\sum }}\frac{\left({t}_{k+1}-{t}_{k}\right)}{{t}_{k}{\left(\mathrm{log}{t}_{k}\right)}^{\left(1+{\epsilon }_{3}\right)}}<\infty .$

Now we get ${\sum }_{k={k}_{4}}^{\infty }P\left({C}_{k}\right)<\infty$ , which in turn establishes (9) by applying to the Borel-Cantelli lemma. The proof of Theorem 3 is complete.

3. Conclusion

In this paper, we developed some limit theorems on increments of stable subordinators. We reformulated the result obtained by Vasudeva and Divanji  , and established our result by using ${\lambda }_{\beta }\left(t\right)$ .

Acknowledgments

Our thanks to the experts who have contributed towards development of our paper.

Cite this paper

Bahram, A. and Almohaimeed, B. (2017) On the Increments of Stable Subordinators. Applied Mathematics, 8, 663-670. https://doi.org/10.4236/am.2017.85053

References

1. 1. Bahram, A. and Almohaimeed, B. (2016) Some Liminf Results for the Increments of Stable Subordinators. Theoretical Mathematics and Applications, 28, 117-124.

2. 2. Hawkes, J.A. (1971) Lower Lipschitz Condition for Stable Subordinator. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 17, 23-32. https://doi.org/10.1007/BF00538471

3. 3. Vasudeva, R and Divanji, G. (1988) Law of Iterated Logarithm for Increments of Stable Subordinators. Stochastic Processes and Their Applications, 28, 293-300.

4. 4. Fristedt, B. (1964) The Behaviour of Increasing Stable Process for Both Small and Large Times. Journal of Applied Mathematics and Mechanics, 13, 849-856.

5. 5. Fristedt, B and Pruit, W.E. (1971) Lower Functions of Increasing Random Walks and Subordinators. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 18, 167-182. https://doi.org/10.1007/BF00563135

6. 6. Fristedt, B and Pruit, W.E. (1972) Uniform Lower Functions for Subordinators. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, 24, 63-70. https://doi.org/10.1007/BF00532463

7. 7. Mijhneer, J. L. (1975) Sample Path Properties of Stable Process. Mathematisch Centrum, Amsterdam.

8. 8. Mijhneer, J.L. (1995) On the Law of Iterated Logarithm for Subsequences for a Stable Subordinator. Journal of Mathematical Sciences, 76, 2283-2286.https://doi.org/10.1007/BF02362699