Search Results

Just to slightly expand on the existing answer: you can see that it is indeed true as an immediate consequence of Bochner's theorem. Also, Gaussian measures on infinite-dimensional spaces are actually …

Yes, provided that you know a priori that $\int_{(0,1]} x^{-\kappa} d\mu(x) < \infty$ (and the same for $\nu$) for some $\kappa > 0$. By taking the limit $a \to \infty$, you can detect the size of the …

Yes. As you mentioned, this is really about positive $L^1$ functions. First, approximate your function by a compactly supported Lipschitz function, which we know are dense in $L^1$ (and its easy to sh …

As Bjørn already mentioned, the CLT shows that the limit is indeed normal with variance $3 T$. A more interesting case is that of $$ I = \lim_{\Delta \to 0} \sum_{i=0}^n f(W_{i\Delta},B_{i\Delta})\De …

You could define a distance on probability measures by the smallest $c$ such that there exists a coupling giving mass $1$ to a $c$-neighbourhood of the diagonal. Many pairs would be infinite distance …

As Nate already pointed out, the Brownian motion conditioned to stay inside $D$ up to time $1$ (which is what the limit in "Added later" gives you) will not satisfy (2). I suspect (and this what there …

This is a cute problem... Let's normalise your multiset $M$ to a probability measure $\mu$ by setting $\mu = {1\over |M|} \sum_{x \in M} \delta_x$ (repeated elements are repeated in the sum). Write al …