A note on a deterministic property to obtain the long run behavior of the range of a stochastic process

The range of a real valued process $X=(X_{t})_{t\in T}$ is defined to be the "measure" (in a certain sense) of the set of values generated by this process up to time $t$. We shall denote the range of the process $X$ by $R_{t}(X)$. When $X$ is discrete, i.e $X=(X_{n})_{n\in\mathbb{N}}$, $R_{n}(X)$ simplifies to the number of explored sites up to time $n$ by the process $X$, i.e $R_{n}(X)=|\{X_{0},...,X_{n}\}|$. The range of discrete processes (random walks, Markov chains etc…) has been investigated in several seminal works [4, 5, 2, 15, 16]. In particular, we cite Kesten-Spitzer-Whitman theorem [12] which says that the average of the range of a nearest neighbor simple random walk (referred to as its speed in [1]) converges to the probability of never returning to the origin (the starting point more generally). That is, let $(\xi_{n})_{n}$ be a sequence of i.i.d Redmacher random variables of parameter $p$ and set $X_{n}=\xi_{1}+...+\xi_{n}$. Then

$$\frac{R_{n}(X)}{n}\overset{a.e}{\underset{n\rightarrow\infty}{\longrightarrow}}\mathbb{P}(X\,\,\text{never returns to the origin}).$$

(2)

Note that $\mathbb{P}(X\,\,\text{never returns to the origin})$ is simply $2p-1=|\mathbb{E}(X_{1})|$. In [1], the authors showed that (2) is a deterministic property, and no requirement to involve randomness to get it. In fact they showed the following result.

Theorem (1) shows that the asymptotic behavior of the range of any nearest neighbor random walk follows immediately from the asymptotic behavior of the random walk itself. This deterministic result covers also the case of random walks in random environments [11]. An immediate consequence of theorem (1) is that if $X$ is a random walk for which the Birkoff ergodic theorem [10] applies then

$$\frac{R_{n}(X)}{n}\underset{{\scriptscriptstyle n\rightarrow+\infty}}{\longrightarrow}|\mathbb{E}(X_{1})|.$$

(3)

In continuous time, the range of a process $X=(X_{t})_{t\geq 0}$ is defined similarly. It is the Lebesgue measure of the interval $[\sup_{0\leq s\leq t}X_{s}-\inf_{0\leq s\leq t}X_{s}]$. In other words

$$R_{t}(X)=\sup_{0\leq s\leq t}X_{s}-\inf_{0\leq s\leq t}X_{s}.$$

(4)

One should notice that the path of $R_{t}(X)$ is non-decreasing. Thus, we can define its right continuous inverse

$$\theta_{X}(a):=\inf\{t\geq 0\mid R_{t}(X)>a\}.$$

Note that $R_{t}(X)$ and $\theta_{X}(a)$ share the same monotonicity. In addition, the following two relations encodes the duality in between them:

• •

  $R_{t}(X)<a\Longleftrightarrow\theta_{X}(a)<t.$

• •

  $\theta_{X}(R_{t}(X))\geq t.$

A typical process to study its range in continuous time is obviously the Brownian motion as well as its derivatives [13, 6, 14]. The density of the range of a standard Brownian motion $B=B_{t}$ is computed in [4] for a fixed time $t$ while the Laplace transform of $\theta_{B}(a)$ is given in [6, 14]. In [13], the authors considered the case of a positive drifted Brownian motion $V_{t}^{\eta}:=B_{t}+\eta t$ where $\eta>0$. In section $4,$ they provided the distribution functions of $R_{t}(V^{\eta})$ and $\theta_{V^{\eta}}(a)$. In addition, they proved two limit theorems for $\theta_{V^{\eta}}(a)$ that can be seen as a law of large numbers and a central limit theorem. What matters for us in this work is their result about the long run behavior of the range. They showed that the range of $V_{t}^{\eta}$ is asymptotically equivalent to $\eta t$ (a.e), or equivalently

$$\frac{R_{t}(V^{\eta})}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}\eta.$$

However, their proof is quite technical and it is based on the study of the right continuous inverse process $\theta_{V^{\eta}}(a):=\inf\{t\geq 0\mid R_{t}(V^{\eta})>a\}$, combined with the use of the subadditive ergodic theorem. Their approach will be discussed in the sequel. What we have discovered is that the comportment of the range of $V_{t}^{\eta}$ at infinity does not require that whole machinery to obtain. In fact, it is a deterministic property that follows directly from the behavior of the underlying process. More generally, we show that the long run behavior of the range of a function $f:[0,\infty)\rightarrow\mathbb{R}$ is deducible from the long run behavior of the function itself. To this end, recall that the range of $f$ is defined exactly as in (4), i.e

$$R_{t}(f)=\sup_{0\leq s\leq t}f_{s}-\inf_{0\leq s\leq t}f_{s}.$$

When $f$ is continuous then its range path is also continuous. Moreover, when $f$ s monotonic then its range increases over the time. Finally note that

$$R_{t}(f)=R_{t}(-f).$$

(5)

Now, we are ready to state the main result of the paper which is inspired from the work in [1] but has more consequences.

Note that theorem (2) covers also the case when $\ell$ is infinite. This is done by a minor change in the previous proof. Using the same assumptions and notations of theorem (2), the case when $\ell$ is zero is similar and it is the subject of the next theorem.

The long run behavior of the the range of the drifted Brownian motion $V^{\eta}_{t}$ follows easily by looking at the behavior of $V^{\eta}_{t}$ itself followed by applying theorems (2) and (3).

In particular we obtain the behavior of the range of $B_{t}$

$$\frac{R_{t}(B)}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}0$$

which is not covered in [13]. The reason we think is because the authors derived the limit (12) by proving fist the following limit of $\theta_{V^{\eta}}(a)$

$$\frac{\theta_{V^{\eta}}(a)}{a}\overset{a.e}{\underset{a\rightarrow\infty}{\longrightarrow}}\frac{1}{|\eta|}$$

which itself requires computations in terms of $\frac{1}{\eta}$ (valid only when $\eta\neq 0$). Then they deduced the asymptotic behavior of the range $R_{t}(V^{\eta})$ leaning on the fact that $\theta_{V^{\eta}}(a)$ is the right continuous process of $R_{t}(V^{\eta})$. Now by theorems (2) and (3), the asymptotic comportment of the range of $V^{\eta}$ are obtained independently of $\theta_{V^{\eta}}(x)$ and follows exclusively from the behavior of the underlying process $V^{\eta}_{t}$.

Now we show that the behavior of $\theta_{V^{\eta}}(a)$ is obtainable from that of $R_{t}(V^{\eta})$ and vice versa. The main tool to do so is the following result relating the asymptotic behaviors of a non decreasing function and its generalized inverse. For a survey about generalized inverses, we refer the reader to [3].

A straightforward consequence of proposition (5) is the equivalence between the comportment of $\theta_{V^{\eta}}(x)$ and $R_{t}(V^{\eta})$. This appeared only as implication in [13].

The following diagram illustrates the difference between our approach and that in [13].

Another application of proposition (5) is the recovery of the limit theorem for renewal processes (We refer the reader to [8] for more details). That is, let $N_{t}$ be a renewal counting process associated to some renewal sequence $T_{n}$ defined by

$$N_{t}=\sup\{n\mid T_{n}\leq t\}.$$

(15)

Note that the RHS of (15) is simply $\sum_{n=1}^{\infty}\mathbf{1}_{\{T_{n}\leq t\}}$. The process $N_{t}$ is the generalized inverse of the renewal sequence $T_{n}$ (the right continuous one). More precisely, $T_{n}$ is obtainable from $N_{t}$ by

$$T_{n}=\inf\{t\mid N_{t}\geq n\}.$$

That is, we recover the following equivalence

$$\frac{T_{n}}{n}\overset{a.e}{\underset{n\rightarrow\infty}{\longrightarrow}}\gamma\Longleftrightarrow\frac{N_{t}}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}\frac{1}{\gamma}$$

valid even in the case when the sequence of inter-arrivals $T_{n}-T_{n-1}$ is ergodic. Both theorems (2) and (3) can be seen as the continuous version of the main result stated in [1]. However, the discrete version can follow easily from the continuous case. That is, let $\left(x_{n}\right)$ be a sequence of integers mimicking a simple walk on $\mathbb{Z}$ and let $r_{n}$ be the range of $\left(x_{n}\right)$, i.e the number of explored sites up to time $n$ by $\left(x_{n}\right)$. Let $x_{t}^{\star}$ be the function obtained from $\left(x_{n}\right)$ by connecting the dots. It is clear that

$$|R_{t}(x^{\star})-r_{\lfloor t\rfloor}|\leq 2.$$

Hence, in case of existence of limits, we obtain

$$\lim_{n\rightarrow\infty}\frac{r_{n}}{n}=\lim_{\lfloor t\rfloor\rightarrow\infty}\frac{r_{\lfloor t\rfloor}}{\lfloor t\rfloor}=\lim_{t\rightarrow\infty}\frac{R_{t}(x^{\star})}{t}.$$

A typical process derived from Brownian motion is the so called Bessel process. Bessel processes are important as they interfere in financial modeling of stock market prices etc. A $n$-dimensional Bessel process, denoted by $BES(n)$, is defined by

$$BES(n)=\|\boldsymbol{B}_{t}\|$$

where $\|\cdot\|$ is the Euclidean norm and $\mathbf{B}_{t}$ is a $n$-dimensional Brownian motion ⁶⁶6People often add the starting point as a subscript.. Another alternative definition of Bessel processes uses SDE theory. More precisely, a Bessel process is the solution of the following SDE

$$dX_{t}=dB_{t}+\frac{n-1}{2}\frac{dt}{X_{t}}.$$

We refer the reader to [7] for more details about the subject. The determination of the statistic of the range of a Bessel process is not an easy task, not to mention its long run behavior. Now, theorems (2) and (3) allows us to obtain the asymptotic behavior of such a range by knowing only the behavior of the process itself. However, we provide a result that covers more than the case of Bessel processes.

The following result illustrates the relation between a deterministic function $f$ and its underlying supremum.

In other words, the four functions $f,\sup_{0\leq s\leq t}f_{s},R_{t}(f),R_{t}(\sup f)$ have the same long run behavior under the assumptions of proposition (10). Note that proposition (10) is still valid when $\ell=0$ provided that $f$ becomes eventually positive after certain time. The following result is a consequence of proposition (10). It concerns the underlying supremum process of a drifted Brownian motion.

$$\frac{\widetilde{V}_{t}^{\eta}}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}\eta$$

and

$$\frac{R_{t}(\widetilde{V}^{\eta})}{t}\overset{a.e}{\underset{t\rightarrow\infty}{\longrightarrow}}\eta.$$

A corollary of proposition (12) is that the supremum of a Brownian motion, as well as its range are negligible against the time evolution. However, it would be interesting to derive estimates for the range like the law of iterated logarithm for Brownian motion, and on top of that investigate whether such potential properties are deterministic or not.