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The shark fin function: asymptotic behavior of the filtered derivative for point processes in case of change points

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Abstract

A multiple filter test for the analysis and detection of rate change points in point processes on the line has been proposed recently. The underlying statistical test investigates the null hypothesis of constant rate. For that purpose, multiple filtered derivative processes are observed simultaneously. Under the null hypothesis, each process G asymptotically takes the form

$$\begin{aligned} G \sim L, \end{aligned}$$

while L is a zero-mean Gaussian process with unit variance. This result is used to derive a rejection threshold for statistical hypothesis testing. The purpose of this paper is to describe the behavior of G under the alternative hypothesis of rate changes and potential simultaneous variance changes. We derive the approximation

$$\begin{aligned} G \sim \Delta \cdot \left( \Lambda + L\right) \!, \end{aligned}$$

with deterministic functions \(\Delta \) and \(\Lambda \). The function \(\Lambda \) accounts for the systematic deviation of G in the neighborhood of a change point. When only the rate changes, \(\Lambda \) is hat shaped. When also the variance changes, \(\Lambda \) takes the form of a shark’s fin. In addition, the parameter estimates required in practical application are not consistent in the neighborhood of a change point. Therefore, we derive the factor \(\Delta \) termed here the distortion function. It accounts for the lack in consistency and describes the local parameter estimating process relative to the true scaling of the filtered derivative process.

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Acknowledgments

This work was supported by the German Federal Ministry of Education and Research (BMBF) within the framework of the e:Med research and funding concept (Grant number: 01ZX1404B) and by the Priority Program 1665 of the DFG. We thank Brooks Ferebee for helpful comments on the manuscript.

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Correspondence to Michael Messer.

Appendix

Appendix

Unless otherwise specified, we use the following notation (compare Construction 2.3): Let \(T>0\), \(h\in (0,T/2]\), \(t\in \tau _h\) and \(c\in (0,T)\). Further, let \(\{\xi _{1,j}\}_{j \ge 1}\), \(\{\xi _{2,j}\}_{j \ge 1}\) and \(\{\xi _j^{(n)}\}_{j\ge 1}\) denote the sequences of life times that correspond to \(\Phi _1\), \(\Phi _2\) and to the compound process \(\Phi ^{(n)}\), respectively. Analogously, let \((N_{1,t})_{t\ge 0}\), \((N_{2,t})_{t\ge 0}\) and \((N_t^{(n)})_{t\ge 0}\) denote the associated counting processes [see Eq. (1)]. Further, let \((W_{1,t})_{t\ge 0}\) and \((W_{2,t})_{t\ge 0}\) be independent standard Brownian motions.

1.1 Proof of Proposition 2.5

Outline We show the joint convergence in distribution of the rescaled counting processes \((N_{1,t})_{t}\) and \((N_{2,t})_{t}\) to a function of \((W_{1,t})_{t}\) and \((W_{2,t})_{t}\) (compare 24). Then, at time t both processes refer to the information of the entire time interval (0, t]. In a second step, the processes are continuously mapped to the scenario of the two windows \((t-h,t]\) and \((t,t+h]\) which refers to the filtered derivative process \((\Gamma _t)_t\).

Proof of Proposition 2.5

For \(i=1,2\) let the rescaled random walk \((X_{i,t}^{(n)})_{t\ge 0}\) and the rescaled counting process \((Z_{i,t}^{(n)})_{t\ge 0}\) concerning \(\Phi _i\) be given as

$$\begin{aligned} X_{i,t}^{(n)} := \frac{1}{\sigma _i\sqrt{n}}\sum _{j=1}^{[nt]}(\xi _{i,j}-\mu _i)\quad \text {and}\quad Z_{i,t}^{(n)} := \frac{N_{i,n t} - n t/\mu _i}{\sqrt{n\sigma _i^2/\mu _i^3}}, \end{aligned}$$
(23)

for \(t\ge 0\). According to Donsker’s theorem [in the case of RPVVs apply Messer et al. (2014, Proposition A.8.)], we find in \((D[0,\infty ),d_{SK})\) as \(n\rightarrow \infty \) that

$$\begin{aligned} (X_{i,t}^{(n)})_{t\ge 0}\mathop {\longrightarrow }\limits ^{d} (W_{i,t})_{t\ge 0} \quad \text {for}\quad i=1,2, \end{aligned}$$

implying weak convergence of \((Z_{i,t}^{(n)})_{t\ge 0}\), i.e., it holds in \((D[0,\infty ),d_{SK})\) as \(n\rightarrow \infty \) that \((Z_{i,t}^{(n)})_{t\ge 0}\mathop {\longrightarrow }\limits ^{d} (W_{i,t})_{t\ge 0}\) for \(i=1,2\), as stated in Billingsley (1999, Theorem 14.6.).

We use a different scaling and set

$$\begin{aligned} \widetilde{Z}_{i,t}^{(n)} := \frac{N_{i,n t} - n t/\mu _i}{s_{t}^{(n)}}, \quad t\ge 0, \end{aligned}$$

where \(s_{t}^{(n)}, t\in [0,\infty )\) is given in Definition 2.4. Then for \(i=1,2\), we find in \((D[0,\infty ),d_{SK})\) for \(n\rightarrow \infty \)

$$\begin{aligned} \left( \widetilde{Z}_{i,t}^{(n)} \right) _{t\ge 0}\mathop {\longrightarrow }\limits ^{d} \left( \frac{\sqrt{\sigma _i^2/\mu _i^3}}{s_{t}^{(1)}} W_{i,t} \right) _{t\ge 0} \end{aligned}$$

because \(\left( {\sqrt{n}\sqrt{\sigma _i^2/\mu _i^3}}/{s_{t}^{(n)}}\right) _{t} = \left( {\sqrt{\sigma _i^2/\mu _i^3}}/{s_{t}^{(1)}}\right) _{t} \) is continuous in t and does not depend on n.

Let now \((\widetilde{Z}_{1,t}^{(n)})_{t\ge 0}\) and \((\widetilde{Z}_{2,t}^{(n)})_{t\ge 0}\) denote the processes derived from \(\Phi _1\) and \(\Phi _2\), respectively. Due to independence of \(\Phi _1\) and \(\Phi _2\), we obtain joint convergence in \((D[0,\infty )\times D[0,\infty ), d_{SK} \otimes d_{SK})\) for \(n\rightarrow \infty \)

$$\begin{aligned} \left( \left( \widetilde{Z}_{1,t}^{(n)} \right) _{t\ge 0}, \left( \widetilde{Z}_{2,t}^{(n)} \right) _{t\ge 0}\right) \mathop {\longrightarrow }\limits ^{d} \left( \left( \frac{\sqrt{\sigma _1^2/\mu _1^3}}{s_{t}^{(1)}} W_{1,t} \right) _{t\ge 0},\left( \frac{\sqrt{\sigma _2^2/\mu _2^3}}{s_{t}^{(1)}} W_{2,t} \right) _{t\ge 0} \right) . \end{aligned}$$
(24)

We consider the continuous map \(\varphi : (D[0,\infty )\times D[0,\infty ), d_{SK} \otimes d_{SK}) \rightarrow (D[h,T-h],d_{SK})\) given by

$$\begin{aligned}&((f(t))_{t\ge 0},(g(t))_{t\ge 0})\\&\quad \mathop {\longmapsto }\limits ^{\varphi } \left( \begin{array}[c]{l} (f(t+h)-f(t)) - (f(t)-f(t-h)){\mathbbm {1}}_{[h,c-h)}(t) \\ \quad +(g(t+h)-g(c))+(f(c)-f(t)) - (f(t)-f(t-h)){\mathbbm {1}}_{[c-h,c)}(t) \\ \quad +(g(t+h)-g(t))-(g(t)-g(c)) - (f(c)-f(t-h)){\mathbbm {1}}_{[c,c+h)}(t)\\ \quad +(g(t+h)-g(t)) - (g(t)-g(t-h)){\mathbbm {1}}_{[c+h,T-h]}(t)\\ \end{array}\right) _{t\in \tau _h}. \end{aligned}$$

The continuous mapping theorem applied to (24) with map \(\varphi \) yields in \((D[h,T-h],d_{SK})\) for \(n\rightarrow \infty \)

$$\begin{aligned} \varphi \left( \left( \widetilde{Z}_{1,t}^{(n)} \right) _{t\ge 0}, \left( \widetilde{Z}_{2,t}^{(n)} \right) _{t\ge 0}\right) \mathop {\longrightarrow }\limits ^{d} \varphi \left( \left( \frac{\sqrt{\sigma _1^2/\mu _1^3}}{s_{t}^{(1)}} W_{1,t} \right) _{t\ge 0},\left( \frac{\sqrt{\sigma _2^2/\mu _2^3}}{s_{t}^{(1)}} W_{2,t} \right) _{t\ge 0} \right) . \end{aligned}$$

Thus, it remains to be shown that

$$\begin{aligned}&\displaystyle \left( \Gamma _t^{(n)}\right) _{t\in \tau _h}= \varphi \left( \left( \widetilde{Z}_{1,t}^{(n)} \right) _{t\ge 0}, \left( \widetilde{Z}_{2,t}^{(n)} \right) _{t\ge 0}\right) ,&\end{aligned}$$
(25)
$$\begin{aligned}&\displaystyle \left( L_t\right) _{t\in \tau _h}\sim \varphi \left( \left( \frac{\sqrt{\sigma _1^2/\mu _1^3}}{s_{t}^{(1)}} W_{1,t} \right) _{t\ge 0},\left( \frac{\sqrt{\sigma _2^2/\mu _2^3}}{s_{t}^{(1)}} W_{2,t} \right) _{t\ge 0} \right) ,&\end{aligned}$$
(26)

where \(\sim \) denotes equality in distribution. In order to show (25) and (26) we differentiate the four cases \(t\in [h,c-h)\), \( t\in [c-h, c)\), \(t \in [c, c+h)\) and \(t\in [c+h,T-h]\).

Derivation of (25)

Case \(t<c-h:\)

$$\begin{aligned} \left. \varphi \left( \left( \widetilde{Z}_{1,t}^{(n)} \right) _{t\ge 0}, \left( \widetilde{Z}_{2,t}^{(n)} \right) _{t\ge 0}\right) \right| _t&= \frac{\left( N_{1,n(t+h)}-N_{1,nt}\right) -\left( N_{1,nt}-N_{1,n(t-h)}\right) }{s_{t}^{(n)}}\\&= \frac{\left[ \left( N_{n(t+h)}^{(n)}-N_{nt}^{(n)}\right) -\left( N_{nt}^{(n)}-N_{n(t-h)}^{(n)}\right) \right] -m_{t}^{(n)}}{s_{t}^{(n)}} = \Gamma _{t}^{(n)}. \end{aligned}$$

For \(t\ge c+h\) we obtain analogous results by exchanging subscripts. For \(t\in [c-h,c)\) we obtain

$$\begin{aligned}&\left. \varphi \left( \left( \widetilde{Z}_{1,t}^{(n)} \right) _{t\ge 0}, \left( \widetilde{Z}_{2,t}^{(n)} \right) _{t\ge 0}\right) \right| _t\\&\quad = \frac{\left( N_{2,n(t+h)}-N_{2,nc}\right) \!+\! \left( N_{1,nc}-N_{1,nt}\right) -\left( N_{1,nt}-N_{1,n(t-h)}\right) \!- n \left( \frac{(t+h)-c}{\mu _2}-\frac{(t+h)-c}{\mu _1}\right) }{s_{t}^{(n)}}\\&\quad = \frac{\left[ \left( N_{n(t+h)}^{(n)}-N_{nt}^{(n)}\right) -\left( N_{nt}^{(n)}-N_{n(t-h)}^{(n)}\right) \right] -m_{t}^{(n)}}{s_{t}^{(n)}} = \Gamma _{t}^{(n)}. \end{aligned}$$

Analogously, we obtain \(c\le t < c+h\), which proves (25).

Derivation of (26)

For \(t< c-h\) we obtain

$$\begin{aligned} \left. \varphi \left( \left( \frac{\sqrt{\sigma _1^2/\mu _1^3}}{s_{t}^{(1)}} W_{1,t} \right) _{t\ge 0},\left( \frac{\sqrt{\sigma _2^2/\mu _2^3}}{s_{t}^{(1)}} W_{2,t} \right) _{t\ge 0} \right) \right| _t = \frac{\left( W_{1,t+h}-W_{1,t}\right) -\left( W_{1,t}-W_{1,t-h}\right) }{\sqrt{2h}} = L_{t}. \end{aligned}$$
(27)

The same holds for \(t\ge c+h\) with the subscript exchanged. In the case \(c-h\le t < c\) we obtain

$$\begin{aligned}&\left. \varphi \left( \left( \frac{\sqrt{\sigma _1^2/\mu _1^3}}{s_{t}^{(1)}} W_{1,t} \right) _{t\ge 0},\left( \frac{\sqrt{\sigma _2^2/\mu _2^3}}{s_{t}^{(1)}} W_{2,t} \right) _{t\ge 0} \right) \right| _t\nonumber \\&= \frac{\sqrt{\sigma _2^2/\mu _2^3} \left( W_{2,t+h} - W_{2,c}\right) + \sqrt{\sigma _1^2/\mu _1^3} \left[ \left( W_{1,c}- W_{1,t}\right) - \left( W_{1,t}- W_{1,t-h}\right) \right] }{s_{t}^{(1)}}= L_{t}. \end{aligned}$$
(28)

Analogously, we obtain for \(c\le t < c+h\)

$$\begin{aligned}&\left. \varphi \left( \left( \frac{\sqrt{\sigma _1^2/\mu _1^3}}{s_{t}^{(1)}} W_{1,t} \right) _{t\ge 0},\left( \frac{\sqrt{\sigma _2^2/\mu _2^3}}{s_{t}^{(1)}} W_{2,t} \right) _{t\ge 0} \right) \right| _t\nonumber \\&\quad = \frac{\sqrt{\sigma _2^2/\mu _2^3} \left[ \left( W_{2,t+h} - W_{2,t}\right) -\left( W_{2,t} - W_{2,c}\right) \right] + \sqrt{\sigma _1^2/\mu _1^3} \left( W_{1,c}- W_{1,t-h}\right) }{s_{t}^{(1)}}= L_{t}. \end{aligned}$$
(29)

Now let \((W_t)_{t\ge 0}\) be a standard Brownian motion, i.e., \((W_t)_{t\ge 0} \sim (W_{1,t})_{t\ge 0} \sim (W_{2,t})_{t\ge 0}\). The process defined in (27), (28) and (29) has continuous sample paths and is given as a function of increments of disjoint intervals of the processes \((W_{1,t})_{t\ge 0}\) and \((W_{2,t})_{t\ge 0}\). Therefore, we can omit the subscripts one and two in (27), (28) and (29) and obtain a process that has continuous sample paths and the same distribution as the former one. By omitting the subscripts, we obtain the limit process L as defined in Eq. (8), which completes the proof of Proposition 2.5. \(\square \)

1.2 Proof of Lemma 3.2

The Proof of Lemma 3.2 works as follows: The uniform convergence \((s_t^{(n)}/\hat{s}_t^{(n)})_{t\in \tau _h}\rightarrow (\Delta )_{t\in \tau _h}\) a.s. is equivalent to the uniform convergence \((\tilde{s}_t^{(n)}/\hat{s}_t^{(n)})_{t\in \tau _h}\rightarrow (1)_{t\in \tau _h}\) a.s. as \(n\rightarrow \infty \). The terms \(\tilde{s}_t^{(n)}\) and \(\hat{s}_t^{(n)}\) are functions of the estimators \(\hat{\mu }_{le},\hat{\mu }_{ri},\hat{\sigma }_{le}^2\) and \(\hat{\sigma }_{ri}^2\) as given in (15). We show the uniform a.s. convergence to their counterparts \(\mu _{le},\mu _{ri},\sigma _{le}^2\) and \(\sigma _{ri}^2\) defined in (19) and (20). More precisely, we show the uniform a.s. convergence of \((\hat{\mu }_{le})_{t\in \tau _h}\) to \((\mu _{le})_{t\in \tau _h}\) and \((\hat{\mu }_{ri})_{t\in \tau _h}\) to \((\mu _{ri})_{t\in \tau _h}\) in Lemma 4.2, and the uniform a.s. convergence of \((\hat{\sigma }^2_{le})_{t\in \tau _h}\) to \((\sigma _{le}^2)_{t\in \tau _h}\) and \((\hat{\sigma }^2_{ri})_{t\in \tau _h}\) to \((\sigma _{ri}^2)_{t\in \tau _h}\) in Lemma 4.3. Thus, the assertion of the Proposition holds true by the structure of the estimator \(\hat{s}^2\) in (15) and the function \(\tilde{s}^2\) in (18) and because convergence of sums and products of càdlàg-valued functions in supremum norm is preserved when the limits are constant. \(\square \)

For completeness of the proof, we show the consistency of the estimators \(\hat{\mu }_{le}\) and \(\hat{\mu }_{ri}\) in Lemma 4.2 and the consistency of \(\hat{\sigma }_{le}^2\) and \(\hat{\sigma }_{ri}^2\) in Lemma 4.3. For that we first show a functional version of the SLLN in the following Lemma.

Lemma 4.1

For the counting process \(N_t^{(n)}\) that corresponds to the process \(\Phi ^{(n)}\), it holds in \((D[h,T-h],d_{\Vert \cdot \Vert })\) as \(n\rightarrow \infty \) almost surely

$$\begin{aligned} \left( \frac{N_{n(t+h)}^{(n)}-N_{nt}^{(n)}}{nh}\right) _{t\in \tau _h}&\longrightarrow \left( \frac{1}{\mu _{ri}(h,t)}\right) _{t\in \tau _h}, \end{aligned}$$
(30)
$$\begin{aligned} \left( \frac{N_{nt}^{(n)}-N_{n(t-h)}^{(n)}}{nh}\right) _{t\in \tau _h}&\longrightarrow \left( \frac{1}{\mu _{le}(h,t)}\right) _{t\in \tau _h}. \end{aligned}$$
(31)

Proof

Outline: We show the convergence of the right window half as stated in (30). The statement for the left window half follows analogously.

First, we show that for all \(t\ge 0\) and all \(h>0\) it holds almost surely as \(n\rightarrow \infty \)

$$\begin{aligned} \frac{N_{n(t+h)}^{(n)}-N_{nt}^{(n)}}{nh} \longrightarrow \frac{1}{\mu _{ri}(h,t)}. \end{aligned}$$
(32)

Then, by a discretization argument this result is extended to hold true in \((D[h,T-h],d_{\Vert \,\cdot \,\Vert })\), as stated in (30).

Derivation of (32)

In order to show the convergence in (32), we distinguish between three cases. First assume \(t\le c-h\). Here, for all \(n=1,2,\ldots \), the corresponding window \((nt,n(t+h)]\) lies left of the change point nc. Thus, the counting process \(N_t^{(n)}\) completely refers to the first RP \(\Phi _1(\mu _1,\sigma _1^2)\), i.e., \(N_{nt}^{(n)}=N_{1,nt}\), while \((N_{1,t})_{t\ge 0}\) denotes the counting process associated with \(\Phi _1\). Then, it can be shown that it holds almost surely for \(n\rightarrow \infty \)

$$\begin{aligned} \frac{N_{n(t+h)}^{(n)}-N_{nt}^{(n)}}{nh} = \frac{N_{1,n(t+h)}-N_{1,nt}}{nh} \longrightarrow \frac{1}{\mu _1}=\frac{1}{\mu _{ri}(h,t)}, \end{aligned}$$
(33)

compare e.g., Messer et al. (2014). An analogous statement holds for \(t>c\).

For \(t\in (c-h,c]\), the right window half refers partially to \(\Phi _1\) and \(\Phi _2\). The section (ntnc] refers to \(\Phi _1\) and the section \((nc,n(t+h)]\) corresponds to \(\Phi _2\). Thus, we decompose \(N_{n(t+h)}^{(n)}-N_{nt}^{(n)} = (N_{2,n(t+h)}-N_{2,nc}) + (N_{1,nc}-N_{1,nt})\). We obtain almost surely for \(n\rightarrow \infty \)

$$\begin{aligned} \frac{N_{n(t+h)}^{(n)}-N_{nt}^{(n)}}{nh}&= \frac{(N_{2,n(t+h)}-N_{2,nc}) + (N_{1,nc}-N_{1,nt})}{nh} \nonumber \\&= \frac{t+h-c}{h}\;\; \frac{N_{2,n(t+h)}-N_{2,nc}}{n(t+h-c)} + \frac{c-t}{h}\;\;\frac{N_{1,nc}-N_{1,nt}}{n(c-t)}\nonumber \\&\longrightarrow \frac{t+h-c}{h}\;\; \frac{1}{\mu _2} + \frac{c-t}{h}\;\; \frac{1}{\mu _1} = \frac{1}{\mu _{ri}(h,t)}. \end{aligned}$$
(34)

In total, the convergences (33) and (34) yield (32).

Derivation of (30)

In order to show that also convergence in \((D[h,T-h],d_{\Vert \cdot \Vert })\) holds, we even show the convergence in (30) on \([0,T-h]\). It is sufficient to show that almost surely

$$\begin{aligned} \lim _{n\rightarrow \infty } \sup _{t\in [0,T-h]} \frac{N_{n(t+h)}^{(n)}-N_{nt}^{(n)}}{nh/\mu _{ri}(h,t)} \le 1 \qquad \text {and} \qquad \lim _{n\rightarrow \infty } \inf _{t\in [0,T-h]} \frac{N_{n(t+h)}^{(n)}-N_{nt}^{(n)}}{nh/\mu _{ri}(h,t)} \ge 1. \end{aligned}$$
(35)

We show the left inequality of (35). The right one follows analogously. We use a discretization argument. For \(x\in \mathbb {R}\) let \(|\lceil x \rceil |:= \lceil x \rceil + 1\). For \(\varepsilon >0\) with \(T/\varepsilon \in \mathbb {N}\) we decompose the time interval (0, nT] into equidistant sections of length \(n\varepsilon \) (Fig. 5). Then we observe a set \(S_\varepsilon :=\{(kn\varepsilon ,kn\varepsilon + n|\lceil h/\varepsilon \rceil | \varepsilon ] :k=0,1,\ldots ,T/\varepsilon - |\lceil h/\varepsilon \rceil |\}\) of finitely many windows of size \(n|\lceil h/\varepsilon \rceil |\varepsilon \). The windows of \(S_\varepsilon \) are slightly larger than nh and for every \(t\in (0,T-h]\) we find an element of \(S_\varepsilon \) that overlaps the window \((nh,n(t+h)]\) (blue window in Fig. 5). We bound

$$\begin{aligned} \sup _{t\in [0,T-h]} \frac{N_{n(t+h)}^{(n)}-N_{nt}^{(n)}}{nh/\mu _{ri}(h,t)} \le&\max _{k=0,1,\ldots ,T/\varepsilon - |\lceil h/\varepsilon \rceil |} \frac{N_{kn\varepsilon + n|\lceil h/\varepsilon \rceil | \varepsilon }^{(n)} - N_{kn\varepsilon }^{(n)}}{nh / \mu _{ri}(h,k\varepsilon )}. \end{aligned}$$
(36)

Now we make use of the fact that the convergence in (30) holds true for a finite number of windows. By letting \(n\rightarrow \infty \) the right hand side of (36) converges to \(1+\delta _\varepsilon \), with

$$\begin{aligned} \delta _\varepsilon&\le \frac{\max (\mu _1,\mu _2)}{\min (\mu _1,\mu _2)} \,\,\frac{|\lceil h/\varepsilon \rceil |\varepsilon -h}{h}. \end{aligned}$$

The expression \(\delta _\varepsilon >0\) accounts for the additional portion that results from the enlarged windows. Then, by letting \(\varepsilon \downarrow 0\) the summand \(\delta _\varepsilon \) vanishes, which yields the first inequality in (35). Analogously, for the lower bound of (35) we find finitely many smaller windows of length \(n(\lfloor h/\varepsilon \rfloor - 1)\varepsilon \), such that every window \((nh,n(t+h)])\) contains such a smaller window (red window in Fig. 5). Then, the limit of the infimum can be bounded from below by \(1-\delta _\varepsilon '\) with \(\delta _\varepsilon '>0\) and such that \(\delta _\varepsilon '\rightarrow 0\) as \(\varepsilon \downarrow 0\). Here, \(\delta _\varepsilon '\) refers to the portion that is not covered by choosing the finitely many windows to be slightly smaller than the true window size nh. \(\square \)

Fig. 5
figure 5

Schematic representation of the discretization of the time horizon (0, nT] into equidistant sections of length \(n\varepsilon \). All windows of length nh (black) are contained in one of finitely many windows of length \(n(\lceil h/\varepsilon \rceil + 1)\varepsilon \) (blue) and contain one of finitely many windows of length \(n(\lfloor h/\varepsilon \rfloor - 1)\varepsilon \) (red). By letting \(\varepsilon \downarrow 0\), the size of the finitely many blue and red windows gets arbitrarily close to the true window size nh. (Color figure online)

Next, we show the uniform a.s. convergences \((\hat{\mu }_{ri}(nh,nt))_{t\in \tau _h}\rightarrow (\mu _{ri}(h,t))_{t\in \tau _h}\) and

\((\hat{\mu }_{le}(nh,nt))_{t\in \tau _h}\rightarrow (\mu _{le}(h,t))_{t\in \tau _h}\) as \(n\rightarrow \infty \). The estimators are given as

$$\begin{aligned} \hat{\mu }_{ri} =\hat{\mu }_{ri}(nh,nt)=\frac{1}{N_{n(t+h)}^{(n)}-N_{nt}^{(n)}-1}\sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\xi _i^{(n)}, \quad \text {if}\quad N_{n(t+h)}^{(n)}-N_{nt}^{(n)}>1, \end{aligned}$$

and \(\hat{\mu }_{ri}=0\) otherwise and \(\hat{\mu }_{le}\) is given analogously.

Lemma 4.2

For the estimators \(\hat{\mu }_{ri}(nh,nt)\) and \(\hat{\mu }_{le}(nh,nt)\) as given in (15), it holds in \((D[h,T-h],d_{\Vert \cdot \Vert })\) as \(n\rightarrow \infty \) almost surely

$$\begin{aligned} (\hat{\mu }_{ri}(nh,nt))_{t\in \tau _h} \longrightarrow (\mu _{ri}(h,t))_{t\in \tau _h}, \end{aligned}$$
(37)
$$\begin{aligned} (\hat{\mu }_{le}(nh,nt))_{t\in \tau _h} \longrightarrow (\mu _{le}(h,t))_{t\in \tau _h}. \end{aligned}$$
(38)

Proof

We show the convergence of the right window half as stated in (37). The assertion for the left window half follows analogously. We proceed as in the proof Lemma 4.1. First, we show that for all \(t\ge 0\) and \(h>0\), it holds almost surely as \(n\rightarrow \infty \)

$$\begin{aligned} \frac{1}{nh}\sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\xi _i^{(n)}\longrightarrow 1, \end{aligned}$$
(39)

i.e., the sum of the life times in the window half asymptotically equals the window length. Then, this result is extended to \((D[h,T-h],d_{\Vert \,\cdot \,\Vert })\) to conclude the convergence in (37).

Derivation of (39)

Assertion (37) has been shown in Messer et al. (2014) to hold for the individual processes \(\Phi _j(\mu _j,\sigma _j^2)\). Therefore, as we show (37) for the right window half, convergence (39) holds true for \(t\in (0,c-h]\) and \(t\ge c\). For \(t\in (c-h,c]\), the right window half contains parts of \(\Phi _1\) and of \(\Phi _2\). We therefore decompose \((nt,n(t+h)] = (nt,nc] \cup (nc,n(t+h)]\). The section (ntnc] refers to \(\Phi _1\) and the section \((nc,n(t+h)]\) corresponds to \(\Phi _2\). The life time at the change point c results from \(\Phi _1\) and \(\Phi _2\), and we therefore bound

$$\begin{aligned} \sum _{i=N_{1,nt}+2}^{N_{1,nc}}\xi _{1,i} + \sum _{i=N_{2,nc}+2}^{N_{2,n(t+h)}}\xi _{2,i}\le \sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\xi _i^{(n)} \le \sum _{i=N_{1,nt}+2}^{N_{1,nc}+1}\xi _{1,i} + \sum _{i=N_{2,nc}+1}^{N_{2,n(t+h)}}\xi _{2,i}, \end{aligned}$$
(40)

which allows to use the properties of the individual processes. For the right hand side in (40) it holds as \(n\rightarrow \infty \)

$$\begin{aligned} \frac{1}{nh}\sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\xi _i^{(n)}&\le \left( \frac{c-t}{h} \,\,\,\frac{1}{n(c-t)} \sum _{i=N_{1,nt}+2}^{N_{1,nc}+1}\xi _{1,i} \right) \nonumber \\&\quad +\left( \frac{t+h-c}{h} \,\,\,\frac{1}{n(t+h-c)} \sum _{i=N_{2,nc}+1}^{N_{2,n(t+h)}}\xi _{2,i}\right) \nonumber \\&\longrightarrow \frac{(c-t)}{h} +\frac{t+h-c}{h}\,\, = 1. \end{aligned}$$

Analogously we obtain the lower bound, such that assertion (39) holds true.

Derivation of (37)

In order to show convergence (39) in \((D[h,T-h],d_{\Vert \,\cdot \,\Vert })\) we even prove it on the interval \([0,T-h]\). For that, we show that almost surely

$$\begin{aligned} \lim _{n\rightarrow \infty } \sup _{t\in [0,T-h]} \frac{1}{nh} \sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\xi _i^{(n)} \,\le \, 1 \qquad \text {and} \qquad \lim _{n\rightarrow \infty } \inf _{t\in [0,T-h]} \frac{1}{nh} \sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\xi _i^{(n)} \ge 1. \end{aligned}$$
(41)

We show here the left inequality of (41). We use the same discretization argument as in the proof of Lemma 4.1 and decompose the interval (0, nT] into equidistant sections of length \(n \varepsilon \) (Fig. 5). Then we bound

$$\begin{aligned} \sup _{t\in [0,T-h]} \frac{1}{nh} \sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\xi _i^{(n)}&\le \max _{k=0,1,\ldots ,T/\varepsilon - |\lceil h/\varepsilon \rceil |} \frac{1}{nh} \sum _{i=N_{kn\varepsilon }^{(n)}}^{N_{kn\varepsilon +n|\lceil h/\varepsilon \rceil |\varepsilon }^{(n)}}\xi _i^{(n)}\\&\le \frac{|\lceil h/\varepsilon \rceil | \varepsilon -h}{h} + \max _{k=0,1,\ldots ,T/\varepsilon - |\lceil h/\varepsilon \rceil |} \frac{1}{nh} \sum _{i=N_{kn\varepsilon }^{(n)}}^{N_{kn\varepsilon +nh}^{(n)}}\xi _i^{(n)}. \end{aligned}$$

The first summand tends to zero as \(\varepsilon \downarrow 0\) and is independent of n. Further, for every \(\varepsilon >0\), the second summand converges to unity almost surely as \(n \rightarrow \infty \), according to equation (39). Thus, the first inequality in (41) holds. The second inequality in (41) can be shown similarly. Thus, convergence (39) holds in \((D[h,T-h],d_{\Vert \,\cdot \,\Vert })\), and together with Lemma 4.1 the assertion (37) holds true by Slutsky’s Theorem. \(\square \)

To finish the proof of Lemma 3.2, we need to show the uniform a.s. convergences \((\hat{\sigma }_{ri}^2(nh,nt))_{t\in \tau _h}\rightarrow (\sigma _{ri}^2(h,t))_{t\in \tau _h}\) and \((\hat{\sigma }_{ri}^2(nh,nt))_{t\in \tau _h}\rightarrow (\sigma _{ri}^2(h,t))_{t\in \tau _h}\) as \(n\rightarrow \infty \). The estimator \(\hat{\sigma }_{ri}^2\) is given as

$$\begin{aligned} \hat{\sigma }^2_{ri}(nh,nt) =\frac{1}{N_{n(t+h)}^{(n)}-N_{nt}^{(n)}-2}\sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\left( \xi _i^{(n)}-\hat{\mu }(nh,nt)\right) ^2 , \quad \text {if}\quad N_{n(t+h)}^{(n)}-N_{nt}^{(n)}>2, \end{aligned}$$

and \(\hat{\sigma }^2_{ri}=0\) otherwise. Similarly \(\hat{\sigma }^2_{le}\) is given.

Lemma 4.3

For the estimators \(\hat{\sigma }_{ri}^2(nh,nt)\) and \(\hat{\sigma }_{le}^2(nh,nt)\) as given in (15) it holds in \((D[h,T-h],d_{\Vert \cdot \Vert })\) as \(n\rightarrow \infty \) almost surely

$$\begin{aligned} (\hat{\sigma }_{ri}^2(nh,nt))_{t\in \tau _h} \longrightarrow (\sigma _{ri}^2(h,t))_{t\in \tau _h},\\ (\hat{\sigma }_{le}^2(nh,nt))_{t\in \tau _h} \longrightarrow (\sigma _{le}^2(h,t))_{t\in \tau _h}. \end{aligned}$$
(42)

Proof

Again we show the convergence of the right window half as given in (42), while the statement for the left window half follows analogously.

First, we show that for all \(t\ge 0\) and \(h>0\), it holds almost surely as \(n\rightarrow \infty \)

$$\begin{aligned} \frac{\mu _{ri}(h,t)}{nh}\sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\left( \xi _i^{(n)}-\hat{\mu }_{ri}(nh,nt)\right) ^2\longrightarrow \sigma _{ri}^2(h,t). \end{aligned}$$
(43)

Then, this result is extended to \((D[h,T-h],d_{\Vert \,\cdot \,\Vert })\) which yields (42).

Derivation of (44)

In the following, let \(\hat{\mu }_{j,ri}(nh,nt)\) denote the estimator that corresponds to \(\Phi _j(\mu _j,\sigma _j^2)\). As before, \(\hat{\mu }_{ri}(nh,nt)\) denotes the estimator that refers to the compound process \(\Phi ^{(n)}\).

Note that (44) was shown in Messer et al. (2014) to hold for the individual processes \(\Phi _j(\mu _j,\sigma _j^2)\). Therefore, as we show (42) for the right window, (44) holds for \(t\in (0,c-h]\) and \(t\ge c\). For the remaining case \(t\in (c-h,c]\), we recall that the right window half partially corresponds to \(\Phi _1\) and \(\Phi _2\). Again, we decompose \((nt,n(t+h)] = (nt,nc] \cup (nc,n(t+h)]\), where the sections (ntnc] and \((nc,n(t+h)]\) refer to \(\Phi _1\) and \(\Phi _2\), respectively. We decompose

$$\begin{aligned}&\frac{\mu _{ri}(h,t)}{nh}\sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\left( \xi _i^{(n)}-\hat{\mu }_{ri}(nh,nt)\right) ^2\nonumber \\&\qquad = \left( \frac{(c-t)\mu _{ri}(h,t)}{h\mu _1}\,\,\frac{\mu _1}{n(c-t)} \sum _{i=N_{1,nt}^{(n)}+2}^{N_{1,nc}^{(n)}}\left( \xi _{1,i}-\hat{\mu }_{ri}(nh,nt)\right) ^2\right) \nonumber \\&\qquad \qquad +\left( \frac{(t+h-c)\mu _{ri}(h,t)}{h\mu _2}\,\,\frac{\mu _2}{n(t+h-c)} \sum _{i=N_{2,nc}^{(n)}+2}^{N_{2,n(t+h)}^{(n)}}\left( \xi _{2,i}-\hat{\mu }_{ri}(nh,nt)\right) ^2\right) + o_{a.s.}(1) \end{aligned}$$
(44)

The term \(o_{a.s.}(1)\) accounts for the summand that corresponds to the single life time \(\xi _{N_{nc}^{(n)}+1}^{(n)}\) that overlaps the change point and that is not respected in the first two terms of (45). By Borel-Cantelli Lemma, the sequence \(\{(\xi _{N_{nc}^{(n)}+1}^{(n)}-\hat{\mu }_{ri}(nh,nt))^2/nh\}_{n=1,2,\ldots }\) can be shown to vanish almost surely for \(n\rightarrow \infty \) and is therefore abbreviated with \(o_{a.s.}(1)\). For the first summand, we find almost surely as \(n\rightarrow \infty \)

$$\begin{aligned}&\frac{\mu _1}{n(c-t)} \sum _{i=N_{1,nt}^{(n)}+2}^{N_{1,nc}^{(n)}}\left( \xi _{1,i}-\hat{\mu }_{ri}(nh,nt)\right) ^2\nonumber \\&\qquad = \frac{\mu _1}{n(c-t)}\sum _{i=N_{1,nt}^{(n)}+2}^{N_{1,nc}^{(n)}}\Big (\left[ \xi _{1,i}-\hat{\mu }_{1,ri}(n(c-t),nt)\right] \, + \, \left[ \hat{\mu }_{1,ri}(n(c-t),nt)-\hat{\mu }_{ri}(nh,nt)\right] \Big )^2\nonumber \\&\qquad =\frac{\mu _1}{n(c-t)}\sum _{i=N_{1,nt}^{(n)}+2}^{N_{1,nc}^{(n)}} \left[ \xi _{1,i}-\hat{\mu }_{1,ri}(n(c-t),nt)\right] ^2\nonumber \\&\qquad \,\,\quad + 2\left[ \hat{\mu }_{1,ri}(n(c-t),nt)-\hat{\mu }_{ri}(nh,nt)\right] \left( \frac{\mu _1}{n(c-t)}\sum _{i=N_{1,nt}^{(n)}+2}^{N_{1,nc}^{(n)}} \left[ \xi _{1,i}-\hat{\mu }_{1,ri}(n(c-t),nt)\right] \right) \nonumber \\&\qquad \qquad \!\!+ \frac{\mu _1}{n(c-t)}\left( N_{1,nc}^{(n)}-N_{1,nt}^{(n)}-1\right) \left[ \hat{\mu }_{1,ri}(n(c-t),nt)-\hat{\mu }_{ri}(nh,nt)\right] ^2\nonumber \\&\qquad \longrightarrow \sigma _1^2 + \left( \mu _1-\mu _{ri}(h,t)\right) ^2. \end{aligned}$$
(45)

The first summand in (46) shows the a.s. convergence to \(\sigma _1^2\) because it refers only to \(\Phi _1\). The second summand in (46) vanishes a.s. since the left term converges a.s. according to Lemma 4.2 and the right term tends to zero a.s. according to Lemmas 4.1 and 4.2. The third summand in (46) tends to \((\mu _1-\mu _{ri}(h,t))^2\) a.s., since the term in the squared brackets converges to \((\mu _1-\mu _{ri}(h,t))^2\) a.s. according to Lemma 4.2, while the scaled counting process converges to unity a.s. due to Lemma 4.1.

An analogous result can be obtained for the second summand of (45) which yields almost surely for \(n\rightarrow \infty \)

$$\begin{aligned}&\frac{\mu _{ri}(h,t)}{nh}\sum _{i=N_{nt}^{(n)}+2}^{N_{n(t+h)}^{(n)}}\left( \xi _i^{(n)}-\hat{\mu }_{ri}(nh,nt)\right) ^2 \longrightarrow \left( \frac{(c-t)\mu _{ri}(h,t)}{h\mu _1}\,\,\left[ \sigma _1^2 + (\mu _1-\mu _{ri}(h,t))^2\right] \right) \nonumber \\&\quad +\left( \frac{(t+h-c)\mu _{ri}(h,t)}{h\mu _2}\,\, \left[ \sigma _2^2 + (\mu _2-\mu _{ri}(h,t))^2\right] \right) ,\nonumber \end{aligned}$$
(46)

and elementary calculations yield equality to \(\sigma _{ri}^2(h,t)\). The convergence in (44) can be concluded using an analogous discretization argument as in the proofs of Lemmas 4.2 and 4.1, such that the assertion (42) can be concluded. \(\square \)

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Messer, M., Schneider, G. The shark fin function: asymptotic behavior of the filtered derivative for point processes in case of change points. Stat Inference Stoch Process 20, 253–272 (2017). https://doi.org/10.1007/s11203-016-9138-0

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