By definition we have that $X_1, \ldots , X_n $ are exchangeable if $X_{i_1}, \ldots, X_{i_n}$ has the same joint distribution as $X_1, \ldots , X_n $ whenever $i_1, \ldots,i_n$ is a permutation of $1,2,\ldots,n$. Basically, if these are exchangeable, the joint distribution function of $X_1, \ldots , X_n $ is a symmetric function of $(x_1,\ldots,x_n)$. If I define $X_1,\ldots,X_n$ to be interarrival times of a renewal process, how can I show that conditional on $N(t)=n$, $X_1,\ldots,X_n$ are exchangeable.

My approach has been to write out the integrals and show that they can be moved around by something like Fubini's theorem. Does anyone have any idea how I can approach this problem? Thanks!


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