Re: Variance of SBM Powers

In article <emca8f$mtc$1@xxxxxxxxxxxxxxxx>,
Daniel Mayost <mayost@xxxxxxxxx> wrote:
In article <32960506.1166626211575.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
Peter <pck711@xxxxxxxxxxx> wrote:
Hi,

What is the variance of integral_0^T_(W(t)^2*dt), where W(t) is a Weiner process, of course.

Thanks.

Here is the method I would use from first principles based on Riemann
sums:

Divide the interval [0,T] into n equal-sized subintervals. We have:

W(T/n)^2 = N1^2

W(2T/n)^2 = N1^2 + N2^2 + 2 N1 N2

W(3T/n)^2 = N1^2 + N2^2 + N3^2 + 2 N1 N2 + 2 N1 N3 + 2 N2 N3 ....

where N1, N2, ... Nn are independent Normal(0,T/n) random variables.
The Riemann sum for the integral is:

T/n [N1 ... Nn] S [N1 ... Nn]^t

where S is the symmetric matrix:

[ n n-1 n-2 .... 1 ]
[ ]
[ n-1 n-1 n-2 .... 1 ]
[ ]
[ n-2 n-2 n-2 .... 1 ]
[ ]
[ n-3 n-3 n-3 .... 1 ]
[ ]
[ ..... ]
[ ]
[ 1 1 1 ..... 1 ]

Since S is symmetric, there exists an orthogonal matrix O and a diagonal
matrix D such that S = O D O^T, where D contains the eigenvalues of S.
Since O is orthogonal, the vector [N1 ... Nn] O is still a vector of n
independent N(0,T/n) random variables, and the Riemann sum is now the
sum of multiples of squares of independent normals.

To get the expected value of the integral we have to know the sum of the
eigenvalues of S (easy), but to get the variance of the integral we have
to know the sum of the squares of the eigenvalues of S (not so easy, at
least for me). In any case, taking the limit as n goes to infinity will
give the answer.

--
Daniel Mayost


I now realize that the sum of the squares of the eigenvalues of S is
just Tr(S^2), which is n^4 / 6 plus lower-order terms. I am thus
beginning to worry that the variance of this integral may in fact be
infinite....

--
Daniel Mayost

.

Relevant Pages

  • Re: Variance of SBM Powers
    ... Daniel Mayost wrote: ... What is the variance of integral_0^T_^2*dt), where Wis a Weiner process, of course. ... The Riemann sum for the integral is: ... sum of multiples of squares of independent normals. ...
    (sci.math)
  • Re: Variance of SBM Powers
    ... What is the variance of integral_0^T_^2*dt), where Wis a Weiner process, of course. ... The Riemann sum for the integral is: ... eigenvalues of S, but to get the variance of the integral we have ...
    (sci.math)
  • Re: z test how
    ... variance 2 is variance squared ... sum of V2/n = 0.0000145545 ... at the probability that is my probability that this 2 groups are ... It doesn't mean you need to square it. ...
    (sci.stat.edu)
  • Re: z test how
    ... variance 2 is variance squared ... sum of V2/n = 0.0000145545 ... at the probability that is my probability that this 2 groups are ... P value and statistical significance: ...
    (sci.stat.edu)
  • Re: N-th moment of the sum of two normally distributed variables
    ... where he decomposes a histogram into two gaussian distributions ... If there is no general way of calculating the moment from the sum ... central momemnts, from the variance alone). ...
    (sci.math)