Time series question - prove by induction Can anyone help with this question..
Consider the process {X(1), X(2), ...} satisfying
$X(t)=αX(t−1)+ε(t)$ for all integers $t≥2$, (1)
where the random variable $X(1)$ satisfies $E(X(1)) = 0$ and has finite variance.
Also suppose that:
(i) For all integers $t ≥ 2$, $ε(t)$ and $(X(t−1),...,X(1))(X(t−1),...,X(1))$ are independent.
(ii) The ε(t) are independent and identically distributed with $E(ε(t))=0$ and $E(ε^2(t))=σ^2>0$.
$X(t)=α^{t−1}X(1)+ε(t)+αε(t−1)+⋅⋅⋅+α^{t−2}ε(2)$    (2)
(a) Check that the formula (2) is true for $t = 2$.
(b) Suppose that the formula (2) is true for given $t ≥ 2$. Using (1) prove that (2) is true for t replaced by t + 1
What I started with for (a) was:
$αX(t−1)+ε(t)=α^{t−1}X(1)+ε(t)+αε(t−1)+⋅⋅⋅+α^{t−2}ε(2)$
= $αX(2−1)+ε(2)=α^{2−1}X(1)+ε(2)+αε(2−1)+⋅⋅⋅+α^{2−2}ε(2)$
= $αX(1)+ε(2)=α^{1}X(1)+ε(2)+αε(1)+⋅⋅⋅+α^{0}ε(2)$
but i don't know where to go from there (or if it's correct)...
What I started with for (b) was:
subbed t+1 into t
$αX(t+1−1)+ε(t+1)=α^{t+1−1}X(1)+ε(t+1)+αε(t+1−1)+⋅⋅⋅+α^{t+1−2}ε(2)$
= $αX(t)+ε(t+1)=α^{t}X(1)+ε(t+1)+αε(t)+⋅⋅⋅+α^{t-1}ε(2)$
and also didn't know where to go from there?
 A: You should add a [self-study] tag.
First of all there is sort of a mistake in formula (2). It should be written like this:
$X(t)= \alpha^{t−1} X(1) + \sum_{i=2}^t \alpha^{t-i} \epsilon (i)$
I hope you are ok with summation notation. If not let me know.
So for a) what you do is this:
$X(2) = \alpha X(1) + \epsilon(2)$. 
This we know is true since this formula is given to us in your description of the time series process. Right?
Now notice, formula (2) (written the way I wrote it) for $t=2$ looks like this:
$X(2)= \alpha^{1} X(1) + \sum_{i=2}^2 \alpha^{t-i} \epsilon (i) = \alpha^{1} X(1) +  \alpha^{0} \epsilon (2)$. This is equal to what we said we know is true for $X(2)$ up above. Check! we proved part a).
Now for b)
We are assuming that formula (2) is true for some arbitrary $t$ and we need to show it is true for $t+1$. We does that mean that it is true for $t$ ?
Let's write the formula:
We are assuming that $X(t) = \alpha^{t−1} X(1) + \sum_{i=2}^t \alpha^{t-i} \epsilon (i)$.
Now what do we know about $X(t+1)$? we know its relationship to $X(t)$ from the description of the process of the time series. well from here you should think a bit and I can help further if you need.
