Let $W$ be a standard Brownian motion and $\mathcal F\_t$ its natural filtration.


Suppose $\theta, A$ are positive $L^1$ random variables independent of $\mathcal F\_t$.


Let $Y\_t$ be the process


$$Y\_t := A \sin \, (\theta t) + W\_t $$


and denote by $\mathcal Y\_t$ its natural filtration.


**Question:** Is it true that $\mathbb E[A| \mathcal Y\_t] \to A$, and $\mathbb E[\theta| \mathcal Y\_t] \to \theta$ almost surely?



Filtering the period and amplitude of a sine wave corrupted by noise
By the Levy zero-one law, the question is equivalent to deciding whether $A$ and $\theta$ are measurable with respect to $\mathcal F\_\infty$, the common refinement of all the $\mathcal F\_t$. The answer is positive.


For $\alpha>0,\beta\ge 0$, consider
$$Z\_n=Z\_n(\alpha,\beta):=Y\_{\beta+(2n+1)\alpha }-Y\_{\beta+(2n-1)\alpha }=U\_n+X\_n$$
where, for $n=0,1,2,\ldots$, the sequence
$$U\_n:=A\sin\Bigl(\theta (\beta+(2n+1)\alpha)\Bigr)-A\sin\Bigl(\theta (\beta+(2n-1)\alpha)\Bigr)=2A\sin(\alpha \theta) \cdot \cos(\theta\beta+2n\theta\alpha)$$
is an almost periodic sequence obtained as a function of the (zero entropy) rotation by angle $2\alpha\theta$, and
$$X\_n:=W\_{\beta+(2n+1)\alpha }-W\_{\beta+(2n-1)\alpha }$$
is a sequence of i.i.d. Gaussian variables with mean 0 and variance $2\alpha$.


In [1], H. Furstenberg defined disjointness of dynamical systems and proved in Theorem I.2 that i.i.d. processes and zero-entropy processes are disjoint.
In Theorem 1.5 of the same paper, he showed that if $(U\_n)\_{n \ge 1}$ and $(X\_n)\_{n \ge 1}$ are sequences
of integrable real random variables which define two disjoint stationary
processes, then the sum sequence $(U\_n+X\_n)\_{n \ge 1}$ determines $(U\_n)\_{n \ge 1}$.
A generalization is in Theorem 7 of [2].


In particular, from the sequence $\{Z\_n(\alpha,\beta)\}\_{n \ge 1}$ we can obtain
$(U\_n)\_{n \ge 1}$ =$(U\_n(\alpha,\beta))\_{n \ge 1}$ for every $\alpha$ and a.e. $\beta$, and using continuity, for all $\beta$. Then $\theta=\pi/(4\alpha\_\*)$, where
$\alpha\_\*=\min\{\alpha>0: U\_1(\alpha,0)=0\}$ and then $A$ is easy to determine.


Theorem 1.6 in [3] gives another proof.


For an elementary direct argument,
define the deterministic function $f\_0(\alpha)=\lim\_n \frac{1}{n}\sum\_{k=1}^n Z\_n(\alpha,0)^2$
and observe that the smallest positive $\alpha$ where $f\_0(\alpha)\alpha^{-2}$ is minimized
is $\alpha\_{min}=\pi/\theta$. Moreover, $$f\_0(\alpha\_{min}/2)=4A^2+\alpha\_{min}$$
which yields $A$.


[1] H. Furstenberg (1967), Disjointness in ergodic theory, minimal sets, and a problem in
Diophantine approximation, Math. systems theory 1, pp. 1-49.
[https://mathweb.ucsd.edu/~asalehig/F\_Disjointness.pdf](https://mathweb.ucsd.edu/%7Easalehig/F_Disjointness.pdf)


[2] Furstenberg, Hillel, Yuval Peres, and Benjamin Weiss. "Perfect filtering and double disjointness." In Annales de l'IHP Probabilités et statistiques, vol. 31, no. 3, pp. 453-465. 1995.http://www.numdam.org/article/AIHPB\_1995\_\_31\_3\_453\_0.pdf


[3] Lev, Nir, Ron Peled, and Yuval Peres. "Separating signal from noise." Proceedings of the London Mathematical Society 110, no. 4 (2015): 883-931.
<http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.749.2820&rep=rep1&type=pdf>


