For a Gaussian Random walk where $x_n$ is the sum of $n$ normal random variables, what is $P(x_1 >0, x_2 >0)$?

Question

I know that the events $x_1 >0$ and $x_2 >0$ are not independent, but I can't think of a way to find a conditional probability so I can solve this.

Thanks!

Answer 1

Presumably (but this ought to be in the question), $x_2=x_1+y$ where $(x_1,y)$ is i.i.d. and standard normal. In particular, one assumes the increments are centered and with the same variance. Then, a picture in the $(x_1,y)$ plane reveals that $(x_1\gt0,y+x_1\gt0)$ is the angular sector going from the North to the South-East through North-East and East.

Thus, this sector is $\frac38$ of the whole. The distribution of $(x_1,y)$ is invariant by every rotation centered at $(0,0)$ hence $P[x_1\gt0,x_2\gt0]=\frac38$.

Answer 2

What you have is:

$$ \begin{align} x_1&=\epsilon_1\\ x_2&=\epsilon_1+\epsilon_2 \end{align} $$ where $\epsilon_i \sim N(0, \sigma^2)$. The probability you want is:

$$ Pr(x_1 >0, x_2>0)=Pr(x_2>0\mid x_1>0)Pr(x_1>0) $$

The second term is 1/2, but the first one still needs some work. Are you able to continue by yourself?