Hi: I have a question about how to show the independence of two random variables of form It$\hat{o}$ integral.
suppose the sequence {$w_{k}$}=$\frac{\pi}{2},\frac{3\pi}{2},\frac{5\pi}{2},\cdots$. Given a standard Brownian motion $W_{[0,T]}$.Define random variables $Y_{k} = \int_{0}^{T}sin(w_{k}t)dW_{t}$.
And I want to show that these $Y_{k}$'s are independent variables (I hope they are).
I start with standard way, so to show two random variable $X,Y$ are independent , it's equivalent to show $\mathbf{P}[X|Y]=\mathbf{P}[X],\mathbf{P}[Y|X]=\mathbf{P}[Y]$ namely knowing the other one won't help in predicting this one. But then I am stuck here because I don't know how to continue to calculate the $\mathbf{P}$ there since there are It$\hat{o}$ integral involved...
So could someone help to take a look, if $Y_{k}$'s are independent , how can I possibly show that? (My way starting out is not so good and convenient I think)
Thanks in advance for your help~