Convergence to a Lévy Process in the Skorohod ${{\mathcal {M}}}_1$ and ${{\mathcal {M}}}_2$ Topologies for Nonuniformly Hyperbolic Systems, Including Billiards with Cusps

Abstract

We prove convergence to a Lévy process for a class of dispersing billiards with cusps. For such examples, convergence to a stable law was proved by Jung and Zhang. For the corresponding functional limit law, convergence is not possible in the usual Skorohod ${{\mathcal {J}}}_1$ topology. Our main results yield elementary geometric conditions for convergence (i) in ${{\mathcal {M}}}_1$, (ii) in ${{\mathcal {M}}}_2$ but not ${{\mathcal {M}}}_1$. In general, we show for a large class of nonuniformly hyperbolic systems how to deduce functional limit laws once convergence to the corresponding stable law is known.

Appendices

Inducing Stable Laws in Both Directions

We assume the set up from Sect. 2.1 with measure-preserving transformations f, $F=f^\tau$ and ${{\hat{f}}}$ on probability spaces $(X,\mu _X)$, $(Y,\mu _Y)$ and $(\Delta ,\mu _\Delta )$ respectively. Let $\pi :\Delta \rightarrow X$ be the measure-preserving semiconjugacy $\pi (y,\ell )=f^\ell y$ and set ${\bar{\tau }}=\int _Y\tau \,d\mu _Y$. We assume in addition that the probability measures $\mu _X$, $\mu _Y$, $\mu _\Delta$ are ergodic.

In the following result, based on [25, 36], we relate limit theorems on X and Y.

Theorem A.1

Let $V\in L^1(X)$ with $\int _X V\,d\mu _X=0$. Define the induced observable

$$\begin{aligned} \textstyle V^Y:Y\rightarrow {{\mathbb {R}}},\qquad V^Y=\sum _{\ell =0}^{\tau -1}V\circ f^\ell , \end{aligned}$$

and the Birkhoff sums

$$\begin{aligned} \textstyle V_n=\sum _{j=0}^{n-1}V\circ f^j, \qquad V^Y_n=\sum _{j=0}^{n-1}V^Y\circ F^j, \qquad \tau _n=\sum _{j=0}^{n-1}\tau \circ F^j, \quad n\ge 1. \end{aligned}$$

Let G be a random variable. Let $b_n>0$ be a sequence with $b_n\rightarrow \infty$, such that $\inf _{n\ge 1} b_n/b_{[{\bar{\tau }}^{-1}n+cb_n]}>0$ for each $c>0$. Assume that $b_n^{-1}(\tau _n-n{\bar{\tau }})\rightarrow _p 0$ as $n\rightarrow \infty$. Then the following are equivalent:

• (a) $b_n^{-1}V_n\rightarrow _d G$ on $(X,\mu _X)$ as $n\rightarrow \infty$.

• (b) $b_n^{-1}V^Y_{[n/{\bar{\tau }}]}\rightarrow _d G$ on $(Y,\mu _Y)$ as $n\rightarrow \infty$.

Remark A.2

It is a special case of [25, Theorem A.1] that (b) implies (a). Moreover, instead of condition $b_n^{-1}(\tau _n-n{\bar{\tau }})\rightarrow _p0$ it suffices that $b_n^{-1}(\tau _n-n{\bar{\tau }})$ is tight in [25, Theorem A.1].

Proof

Note that

$$\begin{aligned} \int _Y|V^Y|\,d\mu _Y\le \int _Y\sum _{\ell =0}^{\tau -1}|V\circ f^\ell |\,d\mu _Y&=\int _Y\sum _{\ell =0}^{\tau (y)-1}|V\circ \pi (y,\ell )|\,d\mu _Y(y) \\&= {\bar{\tau }}\int _\Delta |V|\circ \pi \,d\mu _\Delta = {\bar{\tau }}\int _X |V|\,d\mu _X<\infty . \end{aligned}$$

So $V^Y\in L^1(Y)$ and similarly $\int _Y V^Y\,d\mu _Y=0$.

Define ${{\widehat{V}}}=V\circ \pi :\Delta \rightarrow {{\mathbb {R}}}$ and ${{\widehat{V}}}_n=\sum _{j=0}^{n-1}{{\widehat{V}}}\circ {{\hat{f}}}^j$. Since $\pi$ is a measure-preserving semiconjugacy, condition (a) is equivalent to

• ($\hbox {a}'$) $b_n^{-1}{{\widehat{V}}}_n\rightarrow _d G$ on $(\Delta ,\mu _\Delta )$ as $n\rightarrow \infty$.

Note that $\mu _Y$ can be viewed as a probability measure on $\Delta$ supported on Y. As such, $\mu _Y\ll \mu _\Delta$. By Remark 3.3, we obtain that condition ($\hbox {a}'$) is equivalent to

• ($\hbox {a}''$) $b_n^{-1}{{\widehat{V}}}_n\rightarrow _d G$ on $(Y,\mu _Y)$ as $n\rightarrow \infty$.

The lap number $N_n:Y\rightarrow {{\mathbb {Z}}}^+$ is defined by the relation

$$\begin{aligned} \tau _{N_n(y)}(y)\le n < \tau _{N_n(y)+1}(y). \end{aligned}$$

For initial conditions $y\in Y$, we write

$$\begin{aligned} {{\widehat{V}}}_n(y)=V^Y_{N_n(y)}(y)+H({{\hat{f}}}^ny), \end{aligned}$$

where $H:\Delta \rightarrow {{\mathbb {R}}}$ is given by $H(y,\ell )=\sum _{\ell '=0}^{\ell -1}{{\widehat{V}}}(y,\ell ')$. Now

$$\begin{aligned} \mu _Y(y\in Y:b_n^{-1}|H({{\hat{f}}}^ny)|\ge a)&={\bar{\tau }} \mu _\Delta (y\in Y:b_n^{-1}|H({{\hat{f}}}^ny)|\ge a) \\&\le {\bar{\tau }} \mu _\Delta (x\in \Delta :b_n^{-1}|H({{\hat{f}}}^nx)|\ge a) \\&= {\bar{\tau }} \mu _\Delta (x\in \Delta :b_n^{-1}|H(x)|\ge a)\rightarrow 0 \end{aligned}$$

as $n\rightarrow \infty$ since H is measurable. Hence condition (a$''$) is equivalent to

• ($\hbox {a}'''$) $b_n^{-1}V^Y_{N_n}\rightarrow _d G$ on $(Y,\mu _Y)$ as $n\rightarrow \infty$.

It remains to prove that conditions (a$'''$) and (b) are equivalent. In other words, we must show that $b_n^{-1}(V^Y_{N_n}-V^Y_{[n/{\bar{\tau }}]})\rightarrow _p0$ on $(Y,\mu _Y)$.

We recall some properties of the lap number. By the ergodic theorem, $\lim _{n\rightarrow \infty }n^{-1}N_n= {\bar{\tau }}^{-1}$ a.e. Also, $\tau _k\le n$ if and only if $N_n\ge k$. Let $c>0$ and set $k=k(n)=[n/{\bar{\tau }}+cb_n]$. A calculation shows that if $b_n^{-1}|N_n-n/{\bar{\tau }}|>c$ then $b_n^{-1}|\tau _k-k{\bar{\tau }}|\ge c{\bar{\tau }}+O(b_n^{-1})$, so $b_k^{-1}|\tau _k-k{\bar{\tau }}|\ge cb_k^{-1}b_n{\bar{\tau }}+O(b_k^{-1})$. It follows from the assumptions on $\tau _n$ and $b_n$ that

$$\begin{aligned} \mu _Y\big (b_k^{-1}|\tau _k-k{\bar{\tau }}|\ge cb_k^{-1}b_n{\bar{\tau }}+O(b_k^{-1})\big )\rightarrow 0 \quad \hbox { as}\ n\rightarrow \infty , \end{aligned}$$

and hence that

$$\begin{aligned} b_n^{-1}(N_n-[n/{\bar{\tau }}])\rightarrow _p 0\quad \text {as }n\rightarrow \infty . \end{aligned}$$

(A.1)

Passing to the natural extension, we can suppose without loss that F is invertible. For $n\le -1$, we write $V^Y_n=\sum _{j=n}^{-1}V^Y\circ F^j$. Then

$$\begin{aligned} V^Y_{N_n(y)}(y)-V^Y_{[n/{\bar{\tau }}]}(y)= V^Y_{{{\widetilde{N}}}_n(y)}(F^{[n/{\bar{\tau }}]}y) \quad \text {where}\quad {{\widetilde{N}}}_n(y)=N_n(y)-[n/{\bar{\tau }}]. \end{aligned}$$

Since F is measure-preserving, it suffices to show that $b_n^{-1}V^Y_{{{\widetilde{N}}}_n}\rightarrow _p0$.

By the ergodic theorem, $n^{-1}V^Y_n\rightarrow 0$ a.e. and hence in probability as $n\rightarrow \pm \infty$. Let $\epsilon >0$. We can choose ${{\widetilde{Y}}}\subset Y$ with $\mu _Y({{\widetilde{Y}}})>1-\epsilon$ and $N_0\ge 1$ such that $|n^{-1}V^Y_n|<\epsilon$ on ${{\widetilde{Y}}}$ for all $|n|\ge N_0$.

For each $n\ge 1$, define

$$\begin{aligned} Y_n'=\{y\in Y:|{{\widetilde{N}}}_n(y)|\le N_0\}, \qquad Y_n''=\{y\in Y:|{{\widetilde{N}}}_n(y)|> N_0\}. \end{aligned}$$

For $y\in Y_n'$, we have $|V^Y_{{{\widetilde{N}}}_n(y)}|\le \Psi$, where $\Psi (y)=\sum _{j=-N_0}^{N_0-1}|V^Y(F^jy)|$. Note that $|\Psi |_1\le 2N_0|V^Y|_1<\infty$, so

$$\begin{aligned} \mu _Y(y\in Y_n':|b_n^{-1}V^Y_{{{\widetilde{N}}}_n(y)}(y)|>\epsilon )&\le \mu _Y(b_n^{-1}\Psi >\epsilon )<\epsilon \end{aligned}$$

for n sufficiently large.

For $y\in Y_n''\cap {{\widetilde{Y}}}$, we have $\Big |\frac{1}{|{{\widetilde{N}}}_n|}V^Y_{{{\widetilde{N}}}_n}\Big |<\epsilon$, and hence $|b_n^{-1}V^Y_{{{\widetilde{N}}}_n(y)}|< \epsilon b_n^{-1}|{{\widetilde{N}}}_n|$, so that

$$\begin{aligned} \mu _Y(y\in Y_n'':|b_n^{-1}V^Y_{{{\widetilde{N}}}_n(y)}(y)|\ge \epsilon )&\le \mu _Y(b_n^{-1}|{{\widetilde{N}}}_n|\ge 1) + \epsilon . \end{aligned}$$

By (A.1), $b_n^{-1}|{{\widetilde{N}}}_n|\rightarrow _p0$. Hence $b_n^{-1}V^Y_{{{\widetilde{N}}}_n}\rightarrow _p0$, completing the proof. $\quad \square$

Remark A.3

Suppose that $b_n$ is regularly varying of index $1/\alpha$ with $\alpha >1$. Then the assumptions on $b_n$ in Theorem A.1 are satisfied, and condition (b) can be restated as $b_n^{-1}V^Y_n\rightarrow _d {\bar{\tau }}^{1/\alpha }G$.

The Skorohod Topologies on $D\hbox {[0,\,1]}$

Let D[0, 1] denote the càdlàg space of right-continuous functions $g:[0,1]\rightarrow {{\mathbb {R}}}$ with left limits. The uniform topology on D[0, 1] is not suitable for many purposes; on the theoretical side it is not separable, and for applications it is too strong since functions must have jumps in exactly the same place in order to be close to each other.

To circumvent these issues, Skorohod [43] introduced four topologies on D[0, 1] that are separable and sufficiently strong for theoretical purposes, whilst being sufficiently weak to allow the flexibility for functions to be close to each other in reasonable situations. The four topologies are ordered by

$$\begin{aligned} {{\mathcal {J}}}_1> {{\mathcal {J}}}_2> {{\mathcal {M}}}_2 \quad \text {and}\quad {{\mathcal {J}}}_1> {{\mathcal {M}}}_1 > {{\mathcal {M}}}_2 \end{aligned}$$

where > means stronger than. The ${{\mathcal {M}}}_1$ and ${{\mathcal {J}}}_2$ topologies are not comparable. All these topologies are weaker than the uniform topology. The ${{\mathcal {J}}}_2$ topology plays no role in this paper; we define the remaining topologies below. For simplicity, we restrict to the interval [0, 1]. (The differences between D[0, 1] and $D[0,\infty )$ are of a purely technical nature.) We refer the reader to [43, 47] for more details and proofs.

The Skorohod${{\mathcal {J}}}_1$topology The first Skorohod topology, the ${{\mathcal {J}}}_1$ topology, is metrizable and is defined through the metric $d_{J_1}$ given by

$$\begin{aligned} d_{J_1} (g_1,g_2) =\inf _{\lambda \in \Lambda } \big \{ \Vert g_2\circ \lambda -g_1\Vert \,\vee \, \Vert \lambda -id\Vert \big \} \end{aligned}$$

for $g_1, g_2\in D[0,1]$, where $\Lambda$ denotes the space of strictly increasing reparametrizations mapping [0, 1] onto itself and $\Vert \cdot \Vert$ denotes the uniform norm. This strong topology, which coincides with the uniform topology on the subspace of continuous functions $C[0,1] \subset D[0,1]$, is suitable to define convergence of discontinuous functions when discontinuities and magnitudes of the jumps are close. For instance, if $a_n \rightarrow 1$ then the function $g_n=a_n 1_{[\frac{1}{2}-\frac{1}{n},1]}$ converges to the function $g=1_{[\frac{1}{2},1]}$ in the ${{\mathcal {J}}}_1$ topology as $n\rightarrow \infty$. (Note that $\Vert g_n-g\Vert =|a_n|$ for all n, so there is no convergence in the uniform topology.)

The Skorohod${{\mathcal {M}}}_1$topology In many situations, a single jump in the limit function g corresponds to multiple smaller jumps in the functions $g_n$. In this paper, as in [37], the jumps of $g_n$ are o(1) and the limit function g has jumps, so a more flexible topology on D[0, 1] is required.

The ${{\mathcal {M}}}_1$ topology on D[0, 1] is again metrizable and is defined in terms of the Hausdorff distance between completed graphs of elements of D[0, 1]. Given $g\in D[0,1]$, the completed graph of g is the set

$$\begin{aligned} \Gamma (g)=\{(t,s) \in [0,1]\times {{\mathbb {R}}}: s=\alpha g(t^-) +(1-\alpha ) g(t), \; \alpha \in [0,1] \}. \end{aligned}$$

Let $\Lambda ^*(g)$ denotes the space of parameterizations $G=(\lambda , \gamma ):[0,1] \rightarrow \Gamma (g)$ such that $t'<t$ implies either $\lambda (t')<\lambda (t)$, or $\lambda (t')=\lambda (t)$ and $|\gamma (t)-g(\lambda (t))| \le |\gamma (t')-g(\lambda (t'))|$. Then the ${{\mathcal {M}}}_1$-metric is defined by

$$\begin{aligned} d_{{{\mathcal {M}}}_1} (g_1,g_2) = \inf _{G_i=(\lambda _i, \gamma _i) \in \Lambda ^*(g_i)} \big \{ \Vert \gamma _1 -\gamma _2\Vert \,\vee \, \Vert \lambda _1-\lambda _2\Vert \big \}. \end{aligned}$$

An example in the spirit of Fig. 2a is obtained by defining $g_n=\frac{3}{4} 1_{[\frac{1}{2}-\frac{1}{n},\frac{1}{2})}+a_n 1_{[\frac{1}{2},1]}$. If $a_n \rightarrow 1$ then $g_n$ converges to $g=1_{[\frac{1}{2},1]}$ in the ${{\mathcal {M}}}_1$ topology as $n\rightarrow \infty$, but not in the ${{\mathcal {J}}}_1$ topology.

The Skorohod${{\mathcal {M}}}_2$topology The ${{\mathcal {M}}}_2$ topology on D[0, 1] is also defined in terms of the Hausdorff distance between completed graphs of elements of D[0, 1], namely $d_{{{\mathcal {M}}}_2}(g_1,g_2) = \rho (\Gamma (g_1),\Gamma (g_2))\vee \rho (\Gamma (g_2),\Gamma (g_1))$ where

$$\begin{aligned} \rho (\Gamma (g_1),\Gamma (g_2))&=\sup _{(t_1,s_1) \in \Gamma (g_1)} \inf _{(t_2,s_2) \in \Gamma (g_2)} \Vert (t_1,s_1) - (t_2,s_2) \Vert . \end{aligned}$$

(Here, $\Vert (t_1,s_1) - (t_2,s_2) \Vert =|t_1-t_2|+|s_1-s_2|$.) An example in the spirit of Fig. 2b is obtained by defining $g_n=\frac{3}{4} 1_{[\frac{1}{2}-\frac{1}{n},\frac{1}{2})}+\frac{1}{3} 1_{[\frac{1}{2},\frac{1}{2}+\frac{1}{n})} +a_n 1_{[\frac{1}{2}+\frac{1}{n},1]}$. If $a_n \rightarrow 1$ then $g_n$ converges to $g=1_{[\frac{1}{2},1]}$ in the ${{\mathcal {M}}}_2$ topology as $n\rightarrow \infty$, but not in the ${{\mathcal {M}}}_1$ topology.

An example in the spirit of Fig. 2c is obtained by defining $g_n=\frac{5}{4} 1_{[\frac{1}{2}-\frac{1}{n},\frac{1}{2})}+a_n 1_{[\frac{1}{2},1]}$, where $a_n \rightarrow 1$. Then $g_n$ fails to converge in any of the Skorokhod topologies.

We end this appendix with the following instrumental lemma.

Lemma B.1

Given $g\in D[a,b]$ take ${\bar{g}}\in D[a,b]$ given by ${\bar{g}}=1_{[a,b)} g(a) + 1_{\{b\}} g(b)$. Then

$$\begin{aligned} d_{{{\mathcal {M}}}_2, [a,b]}(g,{\bar{g}}) \le b-a + A\wedge B, \end{aligned}$$

where

$$\begin{aligned} A&= \sup _{t\in [a,b]} (g(a)-g(t)) + \sup _{t\in [a,b]} (g(t)-g(b)), \\ B&= \sup _{t\in [a,b]} (g(t)-g(a)) + \sup _{t\in [a,b]} (g(b)-g(t)). \end{aligned}$$

Proof

We assume that $g(b)\ge g(a)$ (the case $g(b)<g(a)$ is entirely analogous). Then $\Gamma ({\bar{g}})=\{(t,g(a)):a\le t\le b\}\cup \{(b,s):g(a)\le s\le g(b)\}$. Also $\Gamma (g)\subset [a,b]\times {{\mathbb {R}}}$ and intersects every horizontal line between $s=g(a)$ and $s=g(b)$.

For every $(t,s)\in \Gamma ({\bar{g}})$, there exists $t'\in [a,b]$ such that $(t',s) \in \Gamma (g)$. Then $\Vert (t,s)-(t',s)\Vert \le b-a$ and hence $\rho (\Gamma ({\bar{g}}),\Gamma (g))\le b-a$.

It remains to estimate $\rho (\Gamma (g),\Gamma ({\bar{g}}))$. Let $(t,s)\in \Gamma (g)$.

• If $s\in [g(a),g(b)]$, then $(b,s)\in \Gamma ({\bar{g}})$ and $\Vert (t,s)-(b,s)\Vert \le b-a$.

• If $s<g(a)$, then $(t,g(a))\in \Gamma ({\bar{g}})$ and $g(t)\le s<g(a)$, so $\Vert (t,s)-(t,g(a))\Vert = g(a)-s\le g(a)-g(t)=(g(a)-g(t))\wedge (g(b)-g(t)) \le A\wedge B$.

• If $s>g(b)$, then $(b,g(b))\in \Gamma ({\bar{g}})$ and there exists $t'\in [a,b]$ such that $g(t')\ge s>g(b)$. Hence $\Vert (t,s)-(b,g(b))\Vert \le b-a+s-g(b)\le b-a+g(t')-g(b)=b-a+(g(t')-g(b))\wedge (g(t')-g(a))\le b-a+A\wedge B$.

In all cases, $\inf _{({\bar{t}},{\bar{s}})\in \Gamma ({\bar{g}})}\Vert (t,s)-({\bar{t}},{\bar{s}})\Vert \le b-a+A\wedge B$ so $\rho (\Gamma (g),\Gamma ({\bar{g}}))\le b-a+A\wedge B$ completing the proof. $\quad \square$