Limits of Brownian Motions

Question

Let $A_t$ and $B_t$ be independent Brownian motions.

Is $A_tB_t$ a Brownian motion?

This part of the question is not asked to be a proof, more of a "show." I went through the axioms which are:

1. $A_0B_0 = 0$ (obvious)
2. $\forall t >0, A_tB_t \in \mathcal{N(0,t)}.$ I reasoned this as since $A_t$ and $B_T$ are both Brownian motions, their variance increases linearly over time, and so does their product.
3. Stationary and IID: The first part of stationary seems simple as $E[A_tB_t]=E[A_t]E[B_t]$ and so $E[A_tB_t]=E[A_{t+s}B_{t+s}]=E[A_{t+s}]E[B_{t+s}]$. However, I'm not sure about the covariance equality.
4. The product of two continuous functions is continuous, so this step is done.

However, I am asked to calculate

1. $lim_{t\to\infty} \frac{B_t}{t}$
2. $lim_{t\to\infty} \frac{A_tB_t}{t}$

How can we calculate these limits when Brownian motion has a mean of zero, and are not differentiable?

Answer 1

This is not a Brownian motion indeed, as noted above. The pdf of $Z_t=A_tB_t$ is

$$ f(z)=\int\frac{dadb}{2\pi t}\delta(z-ab)e^{-\frac12\frac{a^2+b^2}{t}} $$

where the integral is over the whole plane. You can integrate this directly to obtain the Bessel functions of second kind of order zero and with imaginary argument (from Maple)

$$ f(z)=-\frac{1}{4t}\left(Y_0(iz/t)+Y_0(-iz/t)\right) $$

Or, you can look at the characteristic function (Fourier transform) of the above pdf

$$ C(u)=\int dz\,e^{iuz}f(z)=\frac{1}{\sqrt{1+t^2u^2}} $$

This is clearly not a Gaussian as we expect from a Brownian motion.

Regarding the scaled random variables I think you have to look at the limit in distribution. The pdf of $Z_t=B_t/t$ is

$$ g(z)=\frac{t}{\sqrt{2\pi t}}e^{-\frac12 \frac{(zt)^2}{t}} $$

which goes to zero uniformly as $t\to\infty$. Of course, for all $t>0$ it integrates to 1 so the probability is conserved, but the probability weight spreads out to infinity and is not localised anymore.

The other one, $Z_t=A_tB_t/t$, seems more interesting and I think it has a finite limit distribution as $t\to\infty$ and the pdf reads

$$ g(z)=-\frac{1}{4}\left(Y_0(iz)+Y_0(-iz)\right) $$

I hope it helps. There might be errors though, I put this together quickly with Maple. Cheers.