I'm studying of the Grimmett some problems : Probability and random processes from Geoffrey Grimmett. This book is findable on the internet btw.
At some point, the author is talking about translation invariance. Let me restate briefly the problem.
You want to know when an event $H$ will occur. So what is $P(H_n)$, the probability, that $H$ occurs at time $n$.
We set : $$X_1 = \min (n : H \text{ occurs at n} )$$
$$ X_k = n_k - n_{k-1} $$ so mainly it is the time elapsed between occurences of $H$.
We assume that the $ \{X_i\} $ are independant, with value in integers. Moreover, apart from $X_1$, they are all i.i.d..
Now in the book, there is the formula (refered to as translation invariance) :
$$ P(H_n | X_1 = i ) = P(H_{n-i+1} | X_1 = 1 ) $$
I guess that formula is right. However, I don't understand why this would hold mathematicaly speaking, and secondly, why is it true when we consider the time of $X_1$ ? I would thought that this would have more sense, because of the i.i.d of all the $X_i$ apart from the first one.
$$ P(H_n | X_1 = i, X_2 = i+1+k ) = P(H_{n-k+1} | X_1 = i, X_2 = i+1 ) $$
I found a way to solve this problem. If anyone sees a mistake, please correct me. Studying the markov property gave me that idea.
Since, for the particular case of $ n=5, i = 2$, in the assumption that $X_1=2$, the following equality of events holds $$ H_5 = \{ X_2 = 3 \} \cup \{ X_2 = 2 \cap X_3 = 1 \} \cup \{ X_2 = 1 \cap X_3 = 2 \} \cup \{ X_{1,2,3} = 1 \} $$
we have that in fact : $$ P(H_5 | X_1 = 2 ) = P(H_{4} | X_1 = 1 ) $$
and generalized : $$ P(H_n | X_1 = i ) = P(H_{n-i+1} | X_1 = 1 ) $$
the only thing that makes me think I might be wrong is that, the only assumption I used is that $X_1$ is independant to all the others $X_i$. I used it when replacing $H_5$ by the union of all the other events, that are independant to the value of $X_1$ so one changes it how he wants as long it makes sense in the equalities of events.
However, in the book, they mentionne other requirements (independance of all of them, and identitical distribution).