The Slepian zero set, and Brownian bridge embedded in Brownian motion by a spacetime shift

This paper is concerned with various aspects of the Slepian process $(B_{t+1} - B_t, t \ge 0)$ derived from a one-dimensional Brownian motion $(B_t, t \ge 0 )$. In particular, we offer an analysis of the local structure of the Slepian zero set $\{t : B_{t+1} = B_t \}$, including a path decomposition of the Slepian process for $0 \le t \le 1$. We also establish the existence of a random time $T$ such that $T$ falls in the the Slepian zero set almost surely and the process $(B_{T+u} - B_T, 0 \le u \le 1)$ is standard Brownian bridge.


Introduction and main result
In a recent work [57], we were interested in continuous paths of length 1 in Brownian motion (B t ; t ≥ 0). We proved that Brownian meander m and the three-dimensional Bessel process R can be embedded into Brownian motion by a random translation of origin in spacetime, while it is not the case for either normalized Brownian excursion e or reflected Brownian bridge |b 0 |. The following question was left: Question 1.1. Can we find a random time T ≥ 0 such that (B T +u − B T ; 0 ≤ u ≤ 1) has the same distribution as standard Brownian bridge (b 0 u ; 0 ≤ u ≤ 1)?
As a natural candidate, the bridge-like process as below was considered: where F := inf{t ≥ 0; B t+1 − B t = 0}. (1.2) This bridge-like process bears some resemblance to Brownian bridge. At least, it starts and ends at 0, and is some part of a Brownian path in between. This leads us to the following question: Question 1.2.
(1) Is the bridge-like process defined as in (1.1) standard Brownian bridge? (2) If not, is the distribution of standard Brownian bridge absolutely continuous with respect to that of the bridge-like process?
To provide a context for these questions, we observe that where X t := (B t+u − B t ; 0 ≤ u ≤ 1) for t ≥ 0, (1.4) is the moving-window process associated to Brownian motion (B t ; t ≥ 0), and BR 0 := {w ∈ C[0, 1]; w(0) = w(1) = 0} is the set of bridges with endpoint 0. Let B be the Borel σ−field of C[0, 1], the set of continuous functions on [0, 1] starting at 0. Note that the moving-window process X is a stationary Markov process, with transition kernel P t : (C[0, 1], B) → (C[0, 1], B) for t ≥ 0 given by where P W is Wiener measure on C[0, 1], invariant with respect to (P t ; t ≥ 0). So it is not hard to see that X t+l and X t are independent for all t ≥ 0 and l ≥ 1.
For a suitably nice continuous-time Markov process (Z t ; t ≥ 0), there have been extensive studies on the post-T process (Z T +t ; t ≥ 0) with some random time T which is • a stopping time, see e.g. Hunt [30] for Brownian motion, Blumenthal [13], and Dynkin and Jushkevich [18] for general Markov processes; • an honest time, that is the time of last exit from a predictable set, see e.g. Meyer et al [50], Pittenger and Shih [59,60], Getoor and Sharpe [26,27,28], Maisonneuve [48] and Getoor [25]; • the time at which X reaches its ultimate minimum, see e.g. Williams [77] and Jacobsen [32] for diffusions, Pitman [54] for conditioned Brownian motion and Millar [51,52] for general Markov processes.
The problem is related to decomposition/splitting theorems of Markov processes. We refer readers to the survey of Millar [53], which contains a unified approach to most if not all of the above cases. See also Pitman [55] for a presentation in terms of point processes and further references. Moreover if Z is semi-martingale and T is honest, the semi-martingale decomposition of the post-T process was investigated in the context of progressive enlargement of filtrations, by Barlow [4], Yor [78], Jeulin and Yor [33] and in the monograph of Jeulin [34]. The monograph of Mansuy and Yor [49] consists of a user-friendly survey of this theory.
The study of the bridge-like process seems to be challenging, since the random time F as in (1.2) does not fit into any of the above classes. We even do not know whether this bridge-like process is Markov, and whether it enjoys the semi-martingale property. Note that if the answer to (2) of Question 1.2 is positive, then we can apply Rost's filling scheme [16,63] as in Pitman and Tang [57,Section 3.3] to sample Brownian bridge from a sequence of i.i.d. bridge-like processes in Brownian motion by iteration of the construction (1.1). While we are unable to answer either of the above questions about the bridge-like process, we are able to settle Question 1.1. Theorem 1.3. There exists a random time T ≥ 0 such that (B T +u − B T ; 0 ≤ u ≤ 1) has the same distribution as (b 0 u ; 0 ≤ u ≤ 1)).
In terms of the moving-window process, it is equivalent to find a random time T ≥ 0 such that X T has the same distribution as Brownian bridge b 0 . As mentioned in Pitman and Tang [57], this is a generalization of the Skorokhod embedding problem for C[0, 1]-valued process X. The proof relies on Last and Thorisson [40]'s construction of the Palm measure of some local times of the moving-window process. The idea of embedding Palm/Revuz measures arose earlier in the work of Bertoin and Le Jan [6], and the connection between Palm measures and Markovian bridges was made by Fitzsimmons et al [20]. The existence of local times stems from the Brownian structure of the zero set of the Slepian process S t := B t+1 − B t for t ≥ 0, which is introduced in Section 3. See e.g. Theorem 3.1 and Lemma 4.3.
We conclude this introductory part by reviewing related literature. Question 1.1 is closely related to the notion of shift-coupling, initiated by Aldous and Thorisson [2], and Thorisson [71]. Two continuous processes (Z u ; u ≥ 0) and (Z u ; u ≥ 0) are said to be shift-coupled if there are random times T, T ≥ 0 such that (Z T +u ; u ≥ 0) has the same distribution as (Z T +u ; u ≥ 0). From Theorem 1.3, we know that Z := X, the moving-window process can be shift-coupled with some C[0, 1]-valued process Z starting at Z 0 := b 0 for random times T ≥ 0 and T = 0.
More recently, Last et al [43] studied unbiased shifts of Brownian motion, those are random times T ∈ R such that (B T +u −B T ; u ≥ 0) is two-sided Brownian motion, independent of B T . In the paper, these unbiased shifts were characterized by allocation rules balancing mixtures of Brownian local times. In a study of forward Brownian motion, Burdzy and Scheutzow [15] asked whether a concatenation of independent pieces of Brownian paths truncated at stopping times forms Brownian motion. They showed that if these Brownian pieces are i.i.d. and the expected stopping times are finite, then Brownian motion is achieved by patchwork. The general case where the Brownian pieces are not identically distributed, is left open.
Organization of the paper: The rest of this paper is organized as follows.
• In Section 2, we present some analysis of random walks related to Question 1.2. • In Section 3, after recalling some results for the Slepian process due to Slepian [68] and Shepp [67], we provide a path decomposition for the Slepian process on [0, 1], Theorem 3.1. • In Section 4, we explore the local structure of the Slepian zero set {t ∈ [0, 1]; S t = 0}, or {t ∈ [0, 1]; X t ∈ BR 0 }. In particular, Theorem 3.1 is proved in Section 4.2. • In Section 5, after presenting essential background on Palm theory of stationary random measures, we give a proof of Theorem 1.3.

Acknowledgement:
The authors would express their gratitude to Krzysztof Burdzy for pointing out the relevance of the recent work of Last and Thorisson [44], and Last et al. [43] to Question 1.1.

Random walk approximation
In this section, we consider the discrete analog of the bridge-like process. Namely, for an even positive integer n, we run a simple symmetric random walk (RW k ) k∈N until the first level bridge of length n appears. That is, we consider the process (2.1) The following invariance principle is proved in Pitman and Tang [57,Proposition 2.4].
Proposition 2.1. [57] The distribution of the process where the walk is defined by linear interpolation between integer times, converges weakly to the distribution of the bridge-like process as in (1.1).
It is natural to ask whether there may be some almost sure version of this approximation, using Knight's [36,37] embedding of random walks in Brownian motion. The answer to this question does not seem obvious however. The problem is the lack of continuity of F n as a function of n, for some sample paths. Consider for instance a simple walk with first 1000 steps +1, then 2000 steps −1, and then 1000 steps +1. We obtain a bridge of length 4000, which contains no sub-bridges of length greater than 2000. So if the path starts out as this, the waiting time until the first bridge of length 4000 is 4000. But if the path continues e.g. with steps +1, the waiting time until the first bridge of length 3998 could be 8000 or more. Now we focus on the discrete bridge defined as in (2.1). Note that the support of the first level bridge is all bridge paths since the first n steps starting from 0 can be any path. For n = 2, the bridge (RW F 2 +k − RW F 2 ) 0≤k≤2 obviously has uniform distribution on the two possible paths, one positive and one negative. However, the first level bridge of length n is not uniform for n > 2. Using the Markov chain matrix method, we can compute the exact distribution of this first level bridge for n = 4 and 6. By up-down symmetry, we only need to be concerned with those paths whose first step is +1. The numerical results in Table 1 and 2 give us that the first level bridge fails to be uniform, at least, for n = 4 and 6. By elementary algebraic computation, it is not hard to check that this is true for all n > 2. Now it is natural to ask whether the first level bridge could be asymptotically uniform. To this end, we compute the ratio of extremal probabilities of the first level bridge for some small n's. In Table 3, the ratios max/min of hitting probabilities suggest that the first level bridge might not be asymptotically uniform. Thus, the answer to (1) of Question 1.2 may be negative, i.e. the bridge-like process defined as in (1.1) is not standard Brownian bridge. This is further confirmed by the following simulations, which show that as n grows, the empirical distribution of the maximum of the first level bridge does not appear to converge to the Kolmogorov-Smirnov distribution [38,69], that is the distribution of the supremum of Brownian bridge, see e.g. Billingsley [9,Section 13].

The Slepian process: Old and New
Let us turn back to the random time F defined as in (1.2). We rewrite it as where S t := B t+1 − B t for t ≥ 0 is a stationary Gaussian process with mean 0 and covariance ES t 1 S t 2 = max(1 − |t 1 − t 2 |, 0). This process was first studied by Slepian [68]. Later, Shepp [67] gave an explicit formula for as a t−fold integral when t is an integer and as a 2[t] + 2−fold integral when t is not an integer. Shepp's results are as follows. Let When t = n is an integer, where y 0 = 0, y 1 = |x| and D := {|x| < y 2 < · · · < y n+1 }. When t = n + θ where 0 < θ < 1, where x 0 = 0, x 1 = |x| and D := {|x| < x 2 < · · · < x n+1 and y 0 < · · · < y n+1 }. The distribution of the first passage time F is characterized by where I(t|x)φ(x) is given as (3.2) when t is integer and given as (3.3) when it is not. In particular, where Φ(x) := x −∞ φ(z)dz is the cumulative distribution function of the standard normal distribution.
In this paper, we study the local structure of the Slepian zero set, i.e. {t ∈ [0, 1]; S t = 0}, by showing that it is mutually absolutely continuous relative to that of Brownian motion with normally distributed starting point. The main result, which provides a path decomposition of the Slepian process on [0, 1], is stated as below.
Theorem 3.1. Let F := inf{t ≥ 0; S t = 0} and G := sup{t ≤ 1; S t = 0}. Given the quadruple (S 0 , S 1 , F, G) with 0 < F < G < 1, the Slepian process (S t ; 0 ≤ t ≤ 1) is decomposed into three conditionally independent pieces: In addition, the distribution of (S 0 , S 1 , F, G) with 0 < F < G < 1 is given by On the event {0 < F < G < 1}, the Slepian process is achieved by first creating the quadrivariate (S 0 , S 1 , F, G) and then filling in with usual Brownian components. Similarly, conditioned not to hit 0. Though it is irrelevant to the study of the Slepian zero set on [0, 1], it may carry forward to understand the interaction with zeros on [1,2].
The proof of Theorem 3.1 is deferred to Section 4.2. Our method relies on Shepp [66]'s result of the absolute continuity between Gaussian measures, where the Slepian process was proved to be mutually absolutely continuous with respect to some modified Brownian motion on [0, 1]. As pointed out by Shepp [67], the absolute continuity fails beyond the unit interval. This is the reason why we only restrict the study of the Slepian zero set on intervals of length 1. Nevertheless, we have the following conjecture:

The Slepian zero set and path decomposition
In this section, we study the Slepian zero set on [0, 1], that is {u ∈ [0, 1]; S u = 0}. The problem here has some affinity to level crossings of stationary Gaussian processes. We refer readers to the surveys of Blake and Lindsey [12], Abrahams [1], Kratz [39], as well as the books of Cramér and Leadbetter [17, Chapter 10], Azaïs and Wschebor [3, Chapter 3] for further development.
Berman [5] studied general criteria for stationary Gaussian processes to have local times. In particular, he proved that if (Z t ; t ≥ 0) is a stationary Gaussian process with covariance R Z (t) and 1 − R Z (t) ∼ t→0 |t| α for some 0 < α < 2, then Z has local times (L x t ; x ∈ R, t ≥ 0) such that for any Borel measurable set C ⊂ R and t ≥ 0, As discussed in the introduction, the Slepian process has covariance R S (t) := max(1 − |t|, 0), which obviously fits into the above category. See also the survey of Geman and Horowitz [22] for further development on Gaussian occupation measures. Below is the plan for this section: In Section 4.1, we deal with the local absolute continuity between the distribution of the Slepian process and that of Brownian motion with random starting point. This follows some general discussion on the absolute continuity between Gaussian measures by Shepp [66].
In Section 4.2, we give two different proofs for the path decomposition of the Slepian process (S t ; 0 ≤ t ≤ 1), Theorem 3.1.
In Section 4.3, we study a Palm-Itô measure associated to the gaps between Slepian zeros, with comparison to the well-known Itô's excursion law [31].

4.1.
Local absolute continuity between Slepian zeros and Brownian zeros. As proved by Shepp [66], for each fixed t ≤ 1, the distribution of the Slepian process (S u ; 0 ≤ u ≤ t) is mutually absolutely continuous with respect to that of where ξ ∼ N (0, 1). The Radon-Nikodym derivative is given by where P S (resp. P W ) is the distribution of the Slepian process S (resp. the modified Brownian motion B defined as in (4.1)) on C[0, 1]. As a first application, we compute the density of the first passage time F , defined as in (3.1) or (1.3), on the unit interval.

3)
Proof: Fix t ≤ 1. By the change of measures formula (4.2), Injecting (4.5) into (4.4), we obtain Remark 4.2. As a check, from (4.3), which agrees with the formula (3.5) derived from the determinental expressions (3.2), (3.3) and (3.4). Since the absolute continuity relation does not hold when t > 1, we are not able to derive a simple formula for the density of F on (1, ∞).
Next we deal with the local absolute continuity between the distribution of Slepian zeros and that of Brownian motion with normally distributed starting point. The result enables us to prove Proposition 2.1, that is the weak convergence of the discrete first level bridges to the bridge-like process as in (1.1). Lemma 4.3. For any fixed t ≥ 0, the distribution of (S u ; t ≤ u ≤ t+1) is mutually absolutely continuous with respect to that of ( B u ; t ≤ u ≤ t+1) defined as in (4.1). The Radon-Nikodym derivative is given by where P W t is the distribution of B on [t, t + 1]. In particular, the distribution of the Slepian zero set restricted to [t, t + 1], i.e. {u ∈ [t, t + 1]; S u = 0} is mutually absolutely continuous with respect to that of {u ∈ [t, t + 1]; ξ + B u = 0}, the zero set of Brownian motion starting at ξ ∼ N (0, 1).
Proof: It suffices to prove the first part of this lemma. By stationarity of the Slepian process, the distribution of (S u ; t ≤ u ≤ t + 1) is the same as that of (S u ; 0 ≤ u ≤ 1), which is mutually absolutely continuous relative to ( B u ; 0 ≤ u ≤ 1) with density given by (4.2). Now we conclude by noting that the distribution of ( B u ; t ≤ u ≤ t + 1) and that of ( B u ; 0 ≤ u ≤ 1) are mutually absolutely continuous, with Radon-Nikodym derivative As a consequence, all local properties of the Slepian zero set mimic closely those of Brownian motion with normally distributed starting point. In particular, with positive probability, the Slepian process visits the origin on the unit interval. And immediately thereafter, it returns to the origin infinitely often, as does Brownian motion. In addition, it is easy to see that the Radon-Nikodym derivative between the distribution of {u ∈ [0, 1]; S u = 0} and that of {u ∈ [0, 1]; ξ + B u = 0} is given by where Proj(w) := {u ∈ [0, 1]; w u = 0} is the zero set of w ∈ C[0, 1]. In the next paragraph, we will see how this conditional expectation, as the Radon-Nikodym derivative, can be made explicit by some sufficient statistics.  where Z := Z(w) is a random element and f (Z) is the Radon-Nikodym derivative of Q with respect to P. Futhermore, (1). Let A ∈ σ(Z) be an event determined by Z, with P(A) > 0 and Q(A) > 0; (2). Let Y be another random element such that under P, Y is independent of Z given A (such random element Y need only be defined conditional on A). Then the Q-distribution of Y given A is the same as the P-distribution of Y given A. And under Q, Y is independent of Z given A.
Proof: Take g, h : (Ω, F) → (R, B(R)) two measurable functions. First, we note that where (4.8) is due to the P-conditional independence of Y and Z given A. In addition, where (4.10) is again due to the P-conditional independence of Y and Z given A, and (4.11) follows readily from (4.9).
For w ∈ C[0, 1], let F := inf{t ≥ 0; w t = 0} be the time of first hit to 0, and G := sup{t ≤ 1; w t = 0} be the time of last exit from 0 on the unit interval. The following lemma provides the P S -joint distribution of the quadrivariate (w 0 , w 1 , F, G) on the event {0 < F < G < 1}, which is a refinement of Proposition 4.1.
Lemma 4.5. For x, y ∈ R and 0 < a < b < 1, Proof: We first compute the P W -joint distribution of (w 0 , w 1 , F, G), where P W is the distribution of B on [0, 1] defined as in (4.1).
Moreover, these two processes are conditionally independent given (w 0 , w 1 , F, G). It suffices to apply again Lemma 4.4 to conclude.
Then (Z t ; t ≥ 0) has the same distribution as the Slepian process (S t ; t ≥ 0).
In particular, the Slepian process on [0, 1] can be constructed by first picking independently S 0 , S 1 ∼ N (0, 1), and then filling in a √ 2-Brownian bridge from S 0 to S 1 . This bridge construction provides another proof of Theorem 3.1.
Alternative proof of Theorem 3.1: The first part of the statement is now quite straightforward from Proposition 4.6. Observe that P S (w 0 ∈ dx, w 1 ∈ dy, F ∈ da, G ∈ db) = dxdy 2π exp − x 2 + y 2 2 P Wx→y (F ∈ da, G ∈ db), where P Wx→y is the distribution of B conditioned to start at x and end at y. In addition, Injection (4.16) into (4.15), we recover the formula (4.12).
In view of the Brownian characteristics,it would be interesting to find a construction of the conditioned Slepian process (S t / √ 2; 0 ≤ t ≤ 1) with {0 < F < G < 1} by some path transformation of standard Brownian motion/bridge. We leave the interpretation open for future investigation. where ζ( ) := inf{t > 0; t = 0} is the lifetime of the excursion ∈ E. Following Pitman [56], the gaps between zeros of a process (Z t ; t ≥ 0) with σ-finite invariant measure can be described by a Palm-Itô measure n Z , defined on the space of excursions E as n Z (d ) := E#{0 < t < 1; Z t = 0 and e(t) ∈ d }, where e(t) is the excursion starting at time t > 0 in the process Z. The following result of last exit decomposition for stationary processes is read from Pitman [56, Theorem 1(iii)].
Theorem 4.7. [56] Let P Z governs a process (Z t ; t ≥ 0) with σ-finite invariant measure. For w ∈ C[0, 1], let G t := sup{u ≤ t; w u = 0}, be the last exit time from 0 before time t, and e(G t ) be the excursion straddling time t > 0 in the path. Then (4.17) Remark 4.8.
(1) Theorem 4.7 extends a result of Bismut [10], where Z is Brownian motion with invariant Lebesgue measure. In this case, the Palm-Itô measure n Z is just Itô's excursion law n. (2) There is an analog of last exit decomposition (4.17) for standard Brownian motion.
Let P W be Wiener measure on C[0, ∞), then where n is Itô's excursion law. The result is deduced from Getoor and Sharpe [29], who gave last exit decomposition for general Markov processes. Now we apply Theorem 4.7 to the Slepian process (S t ; t ≥ 0). Let P S be the distribution of the Slepian process, we have: where n S (d ) := E#{0 < t < 1; S t = 0 and e(t) ∈ d }.
(4.19) Recall that X t := (B t+u −B t ; 0 ≤ u ≤ 1) for t ≥ 0 is the moving-window process associated to Brownian motion and BR 0 := {w ∈ C[0, 1]; w(0) = w(1) = 0} is the set of bridges. As shown in the following lemma, the last exit time G t is closely related to the first passage time F defined as in (3.1). Lemma 4.9. Let t > 0. Under P S , t − G t has the same distribution as that of F · 1(F ≤ t) + t · 1(F > t), where G t (resp. F ) is defined as in (4.18) (resp. (3.1) or (1.3)).
Proof: Observe that under P S , t − G t has the same distribution as where for 0 ≤ u ≤ t, X u := X t−u is the time reversal of the moving-window process X, and inf ∅ := t. Consider the bijection Φ : It is easy to see that Φ preserves the bridge set BR 0 , and (Φ( X u ); 0 ≤ u ≤ t) has the same distribution as (X u ; 0 ≤ u ≤ t). Thus, t − G t and inf{u ≤ t; X u ∈ BR 0 } are the same in distribution, from which the result follows.
Remark 4.10. For t = 1, the fact that on the event {0 < F < G 1 < 1}, 1 − G 1 and F have the same distribution, follows readily from the formula (4.12) in view of the symmetry of F ∈ da and 1 − G 1 ∈ 1 − db. By integrating over S 0 ∈ dx and S 1 ∈ dy, we get (4.20) and integrating further (4.20) over G 1 ∈ db, we obtain, which agrees with the formula (4.3) found in Proposition 4.1. where n S is defined as in (4.19) and F is defined as in (3.1) or (1.3). In particular, n S (ζ > a) = 1 π 2 − a a for 0 < a < 1. Proof: It follows from (4.17) that According to Lemma 4.9, for t > a. Then (4.21) is a direct consequence of the above two observations. Combining with the formula (4.3), we derive further (4.22).
From the Palm-Lévy measure (4.22), we can see how the Slepian zero set restricted to [0, 1] differs from a plain Brownian zero set, where Itô's excursion law is given by n(ζ > a) = 2 πa for a > 0.
As expected, n S (ζ > a) and n(ζ > a) have asymptotically equivalent tails a − 1 2 when a → 0 + . Observe a constant factor √ π between them. This is because the invariant Lebesgue measure of reflected Brownian motion is σ-finite and there is no canonical normalization. We also refer readers to Pitman and Yor [58,Section 2] for the Palm-Lévy measure of the gaps between zeros of squared Ornstein-Uhlenbeck processes.

Brownian bridge embedded in Brownian motion
In this section, we prove Theorem 1.3, that is embedding Brownian bridge (b 0 u ; 0 ≤ u ≤ 1) into Brownian motion (B t ; t ≥ 0) by a random translation of origin in spacetime. The problem of embedding continuous paths into Brownian motion was broadly discussed in Pitman and Tang [57], to which we refer readers for a bird's-eye view.
Recall that (X t ; t ≥ 0) is the moving-window process associated to Brownian motion defined as in (1.4). We aim to find a random time T ≥ 0 such that X T has the same distribution as b 0 . A general result of Rost [64] implies that such a randomized stopping time T ≥ 0 exists (relative to the filtration of the moving window process, so T + 1 would be a randomized stopping time in the Brownian filtration) if and only if where P W is Wiener measure on C[0, 1] and P W 0 is Wiener measure pinned to 0 at time 1, that is the distribution of Brownian bridge b 0 . For α > 0, S α is the set of α−excessive functions, see e.g. the book of Sharpe [65] for background. However, the criterion (5.1) is difficult to check since P W 0 is singular with respect to P W . We work around the problem in another way, which relies heavily on Palm theory of stationary random measures. Such theory has been successfully developed by the Scandinavian probability school in the last few decades. The book of Thorisson [75] records much of this important work. For technical purposes, we introduce two-sided Brownian motion ( B t ; t ∈ R), and let be the moving-window process associated to two-sided Brownian motion B. Note that ( X t ; t ∈ R) is a stationary Markov process with state space (C[0, 1], B). Alternatively, X can be viewed as a random element in the space C[0, 1] R , to which we assign the following topology. We say that a sequence (x n ) n∈N converges in C[0, 1] R to x if and only if for each compact set K ⊂ R, where C[0, 1] is equipped with the sup-norm || · || ∞ . Below is the plan for this section: In Section 5.1, we provide background on Palm theory of stationary random measures. We define a notion of local times of the C[0, 1]−valued process X by weak approximation. Furthermore, we show that the 0-marginal of the Palm measure of local times is Brownian bridge.
In Section 5.2, we derive from a result of Last and Thorisson [40] that the Palm probability measure of jointly stationary random measure associated to X can be obtained by a random time-shift of X itself. In particular, there exists a random time T ∈ R such that X T has the same distribution as (b 0 u ; 0 ≤ u ≤ 1). In Section 5.3, we prove that if some distribution on C[0, 1] can be achieved in the movingwindow process X associated to two-sided Brownian motion, then we are able to construct a random time T ≥ 0 such that X T has that desired distribution. Theorem 1.3 follows immediately from the above observations. 5.1. Local times of X and its Palm measure. In this part, we present background on Palm theory of stationary random measures. To begin with, (Ω, F, P) is a generic probability space on which random elements are defined. Let (Z t ; t ∈ R) ∈ E R be a continuous-time process with a measurable state space (E, E). We further assume that the process Z is path-measurable, that is E R ×R (Z, t) → Z t ∈ E is measurable for all t ∈ R. See e.g. Appendix of Thorisson [70] for more on path-measurability. Let ξ be a random σ−finite measure on R.
Assume that the pair (Z, ξ) is jointly stationary, that is where θ s Z := (Z s+t ; t ∈ R) and θ s ξ(·) := ξ(· + s) are usual time-shift operations. Then the Palm measure P ξ of the stationary random measure ξ is defined as: for f : E R → R bounded measurable, Thus P ξ is a σ−finite measure on E R . If P ξ 1 = Eξ[0, 1) < ∞, then the normalized measure P ξ /P ξ 1 is called the Palm probability measure of ξ. We refer readers to Kallenberg [35,Chapter 11], Thorisson [75,Chapter 8], Last [41,42] and the thesis of Gentner [23] for further development on Palm versions of stationary random measures.
In the sequel, we adapt our problem setting to the above abstract framework. We take the state space E := C[0, 1] equipped with its Borel σ−field B. Recall that ( X t ; t ∈ R), the moving-window process defined as in (5.2), is a random element in the metric space (C[0, 1] R , ρ) with the distance ρ defined by (5.3).
For a Borel measurable set C ⊂ R, let be the set of bridge paths with endpoint in C. By Lemma 4.3, for any fixed t ∈ R, the distribution of {u ∈ [t, t + 1]; X u ∈ BR 0 } is mutually absolutely continuous relative to that of {u ∈ [0, 1]; B u = 0} where B is modified Brownian motion as in (4.1). Inspired from the notion of Brownian local times, we define a random σ−finite measure Γ on R as follows: for n ∈ N and C ⊂ R, where P W 0 is Wiener measure pinned to 0 at time 1, that is the distribution of standard Brownian bridge.
Proof: Take f : C[0, 1] → R bounded continuous. By injecting (5.5) into (5.4), we obtain: where P W is Wiener measure on C[0, 1]. The equality (5.6) is due to stationarity of the moving-window process X, and the equality (5.7) follows the weak convergence to Brownian bridge of Brownian motion, see e.g. Billingsley [9,Section 11].
Remark 5.2. The measure P Γ • Π −1 0 defined in Proposition 5.1 is closely related to the notion of Revuz measure of Markov additive functionals. Note that for s ∈ R and t ≥ 0, Γ[s, s + t] = Γ[0, t] • θ s , i.e. Γ induces a continuous additive functional of the moving-window process X. 1 Since ( X t ; t ∈ R) is stationary with respect to P W , can be viewed as Revuz measure of Γ in the two-sided setting. For further discussions on Revuz measure of additive functionals, we refer readers to Revuz [61], Fukushima [21], and Fitzsimmons and Getoor [19] among others.

5.2.
Brownian bridge in two-sided Brownian motion. In this paragraph, we show that there exists a random time T ∈ R such that ( B T +u − B T ; 0 ≤ u ≤ 1) has the same distribution as Brownian bridge b 0 . In terms of the moving-window process X, we show that Proposition 5.3. There exists a random time T ∈ R such that X T has the same distribution as (b 0 u ; 0 ≤ u ≤ 1) As mentioned in the introduction, our proof relies on a recent result of Last and Thorisson [40], which establishes a dual relation between stationary random measures and massstationary ones in the Euclidian space. We refer readers to Last and Thorisson [44,45] for the notion of mass-stationarity, which is an analog to point-stationarity of random point processes.
Before proceeding further, we need the following notations. Recall that (Z t ; t ∈ R) is a path-measurable process with a state space (E, E) such that (E R , E R ) is time-invariant, and ξ is a random σ−finite measure on R. Let N ξ be a simple point process on Z such that i ∈ N ξ ⇐⇒ ξ(i + [0, 1)) > 0.
Next we associate each j ∈ Z to the point of N ξ that is closest to j, choosing the smaller one when there are two such points. Then we obtain a countable number of sets, each of which contains exactly one point of N ξ . Let D be the set that contains 0, and S be the vector from N ξ −point in the set D to 0. Theorem 5.4. [40] Assume that (1). the pair (Z, ξ) is stationary, (2). Eξ[0, 1) < ∞ and (3). conv supp ξ = R a.s. where conv supp ξ is the convex hull of the support of ξ. Define where the conditional distribution of T ∈ [0, 1) given Z is θ −S ξ(·|[0, 1)). Define Then Z 0 under P 0 is the Palm probability measure of ξ.
Thorisson [72,74] presented a duality between stationary point processes and pointstationary ones in the Euclidian space. In particular, the stationary point process and its (modified) Palm version are the same with some random time-shift. Thus Theorem 5.4 is regarded as a generalization of those results in the random diffuse measure setting. Now we apply Theorem 5.4 to Z := X the moving-window process as in (5.2), and ξ := Γ the local times as in (5.5). It is straightforward that all assumptions in Theorem 5.4 are satisfied. This leads to: Corollary 5.5. There exists a random time T ∈ R such that the Palm probability measure of Γ, i.e. P Γ /P Γ 1 is absolutely continuous with respect to the distribution of θ T X.
By Proposition 5.1, the 0-marginal of the Palm probability measure of Γ is Brownian bridge. If we can show that the Palm probability measure P Γ /P Γ 1 is achieved by θ T X for a random time T ∈ R, then Proposition 5.3 follows as a consequence. To this end, we state a general result, the proof of which is deferred.
Theorem 5.6. Let (Z t ; t ∈ R) be a path-measurable process with a state space (E, E). Assume that (1) Z is ergodic under the time-shift group (θ t ; t ∈ R), that is P(Z ∈ H) = 0 or 1 for all H ∈ I := {H ∈ E R ; θ t H = H }; (2) µ is a probability measure on (E R , E R ) absolutely continuous with respect to the distribution of θ T Z for a random time T ∈ R.
Then there exists a random time T ∈ R such that θ T Z is distributed as µ.
Proof of Proposition 5.3: We apply Theorem 5.6 to Z := X the moving-window process and µ := P Γ /P Γ 1 the Palm probability measure of Γ. Observe that the invariant σ-field I ⊂ ∩ n∈N θ −1 n E R , the tail σ-field of ( X n ; n ∈ N) which are i.i.d. copies of Brownian motion on [0, 1]. By Kolmogorov's zero-one law, I is trivial under the distribution of Z and the assumption (1) is satisfied. The assumption (2) follows from Corollary 5.5. Combining Theorem 5.6 and Proposition 5.1, we obtain the desired result.
Remark 5.7. In ergodic theory, the process Z is said to be θ-mixing if where F t := σ(Z u ; u ≤ t) and F t+s := σ(Z u ; u ≥ t + s). See e.g. Bradley [14] for a survey on strong mixing conditions. It is quite straightforward that the moving-window process X is θ-mixing, since X t+l and X t are independent for all t ≥ 0 and l ≥ 1. Consequently, X is ergodic under time-shift (θ t ; t ∈ R). In Section 5.3, this notion of θ-mixing plays an important role in one-sided embedding out of two-sided processes.
In the rest of this part, we aim to prove Theorem 5.6. We need the following result of Thorisson [73], which provides a necessary and sufficient condition for two continuous-time processes being transformed from one to the other by a random time-shift.
Theorem 5.8. [73] Let (Z t ; t ∈ R) and (Z t ; t ∈ R) be two path-measurable processes with a state space (E, E). Then there exists a random time T ∈ R such that θ T Z has the same distribution as Z if and only if the distributions of Z and Z agree on the invariant σ−field I.
Proof of Theorem 5.6: We apply Theorem 5.8 to Z distributed as µ. Since (θ t ; t ∈ R) is ergodic under the distribution of Z, P(Z ∈ H) = 0 or 1 for all H ∈ I. If P(Z ∈ H) = 0 for H ∈ I, then P(θ T Z ∈ H) = P(Z ∈ θ −T H) = P(Z ∈ H) = 0. By the absolute continuity between the distribution µ and that of θ T Z, we have µ(H) = 0. By applying the same argument to the complement of H, P(Z ∈ H) = 1 for H ∈ I implies µ(H) = 1. Thus, the probability distribution µ and that of Z agree on the invariant σ−field I. Theorem 5.8 permits to conclude.

5.3.
From two-sided embedding to one-sided embedding. We explain here how to achieve a certain distribution on C[0, 1] in Brownian motion by a random spacetime shift, once this has been done in two-sided Brownian motion. We aim to prove that: Proposition 5.9. Assume that µ is a probability measure on (C[0, 1], B). If X T is distributed as µ for some random time T ∈ R, then there exists a random time T ≥ 0 such that X T is distributed as µ.
It is not hard to see that Theorem 1.3 follows readily from Corollary 5.3 and Proposition 5.9. In the sequel, we assume path-measurability for any involved continuous-time process. Let L(X ) be the distribution of any random element X . To prove Proposition 5.9, we begin with a general statement.
Proposition 5.10. Let (Z t ; t ∈ R) be a stationary process and θ-mixing as in Remark 5.7. Assume that Z T is distributed as µ for some random time T ∈ R. Given > 0 and N ∈ N, there exist random times 0 ≤ T 1 < · · · < T N on some event E N of probability larger than 1 − such that where || · || T V is the total variation norm of a measure.
Before proceeding the proof, we need the following lemma known as Blackwell-Dubins' merging of opinions [11]. In that paper, they only proved the result for discrete chains. But the argument can be easily adapted to the continuous setting. We rewrite Blackwell-Dubins' theorem for our own purpose, and leave full details to careful readers.
Lemma 5.11. [11] Let (Z t ; t ∈ R) and (Z t ; t ∈ R) be two path-measurable processes with a state space (E, E). If the distribution of Z is absolutely continuous with respect to Z, then where F t := σ(Z u ; u ≤ t) and F t := σ(Z u ; u ≤ t).
Proof of Proposition 5.10: We proceed by induction over N ∈ N. By stationarity of Z, let T θs be a random time such that (θ s Z) T θs is distributed as µ, and it is obvious As a consequence, on the event E 1 := {T 1 ≥ 0} of probability 1 − 2 > 1 − , Suppose that there exist 0 ≤ T 1 < · · · < T N on some event E N of probability larger than 1 − 4 such that Note that the distribution of the conditioned moving-window process (Z s ; s ∈ R|E N ) is absolutely continuous with respect to that of the original one (Z s ; s ∈ R). By Lemma 5.11, ||L(Z t+s ; s ≥ 0|E N , F t ) − L(Z t+s ; s ≥ 0|F t )|| T V → 0 as t → ∞. → 0 as t → ∞.
By applying Lemma 5.11 in the first step, we deduce from Proposition 5.10 that: Corollary 5.12. Let (Z t ; t ∈ R) be a stationary process and θ-mixing as in Remark 5.7. Assume that Z T is distributed as µ for some random time T ∈ R. Given > 0, N ∈ N and E 0 an event of positive probability, there exist random times 0 ≤ T 1 < · · · < T N on some event E N of probability larger than 1 − such that Now let us recall some elements of von Neumann's acceptance-rejection algorithm [76]. Assume that µ and ν are two probability measures such that the Radon-Nikodym derivative f := dν dµ is essentially bounded under µ. Let (Z n ) n∈N ∼ µ ⊗N be a sequence of i.i.d. random variables distributed as µ. Then where (U n ) n∈N is a sequence of i.i.d. uniform-[0, 1] random variables independent of (Z n ) n∈N . It is well-known that the total variation between the N th updated distribution and the target one is of geometric decay, i.e.

||L(Z
If the sample size N is large enough, a good portion of the target distribution ν can be sampled from (Z 1 , · · · , Z N ) ∼ µ ⊗Nà la von Neumann. The following lemma is a slight extension of the above result to the quasi-i.i.d. case. The proof is quite standard, and thus is omitted. Proof of Proposition 5.9: We use the same notation as in the proof of Proposition 5.10. Let t ∈ R be the 1 2 −quantile of T , and define T 1 := T θ −t . By taking T 1 ≥ 0 as the stopping rule, we obtain a 1 2 portion of µ. The idea now is to get the remaining 1 2 portion of µ by filling-type argument. Note that the target distribution L( X T 1 |T 1 < 0) is absolutely continuous with respect to µ with the Radon-Nikodym density which is bounded by 2. As indicated in Remark 5.7, the moving-window process X is stationary and θ-mixing. We apply Corollary 5.12 to ( X t ; t ∈ R|T 1 < 0): for any fixed N ∈ N, there exist random times 0 ≤ T 1 < · · · < T N on some event E N of probability larger than 3 4 such that ||L( X T 1 , · · · , X T N |T 1 < 0, E N ) − µ ⊗N || T V ≤ 1 4 .