I can simulate first order autocorrelation as

restart;
with(Statistics);
randomize();
n := 1000;
p := 0.9;
r := Sample(RandomVariable(Normal(0, 1)), n);
for i from 2 to n do
x[1] := 0;
x[i] := p*x[i-1]+r[i] end do;
rr := [seq(x[i], i = 1 .. n)];
Correlation(rr[1 .. nops(rr)-1], rr[2 .. nops(rr)])

I can simulate second order autocorrelation as

How can I get random drawings from a list with data in the range of mean + - 1 standard deviation?

I know how to do it for a theoretical normal distribution

with(Statistics);
S := Sample(RandomVariable(Normal(0, 1)), 1000)

but not for an empirical distribution....

If we run the below code we will get a plot of a pure unit root and the serial correlation parameter for a pure unit root

My question is why do we have such a large amount of serial correlation in a process that is supposed to be random?

I ran a simulation and found that the average serial correlation for a 100 pure unit roots was 0.941

It does not seam right.....am I doing something wrong here?

I thought the basic definition of a random variable was serial correlation=zero.

ok,

my first quations is: How can I get  V(F(x), G(x)) in the last expression in the below picture evaluated at the end point t and T. 

The solution should be a vertical line to the right-hand side of the expression

My second quation is: Is there any way I can sequentialy solve the problem without having to manually rewrite the equations

Maybe this is a stupid question but if I have a function

how do I differentiate V with respect to t1 diff( V , t1) ?