Let $\{X_n\}_{n=0}^\infty \space$ be a sequence of i.i.d non-lattice R.V. with $X_0=0,\space \space 0\lt E[X_1]\lt\infty $
Let a partial sum $S_n = X_1 + X_2 + ... +X_n. \space(S_0 = 0)$ and
first positive partial sum, $S_T$ where $T = min\{n;S_n\gt0\}$ and continuing $T_k = min\{n;n\gt T_{k-1}, S_n\gt S_{T_{k-1}}\} $,
Let $Z_k = S_{T_k} - S_{T_{k-1}}$ (aka Ladder R.V.) and a partial sum of $Z_k, \space\space S_{Z_n} = Z_1 + Z_2 + ... +Z_n. \space(Z_0 = 0)$
For any real number $ x, h(\gt0)$, $$ p\{x\le first \space\space S_n \lt x+h\} = p\{x\le first \space\space S_{Z_n} \lt x+h\}, \space as \space\space x\to\infty $$
I can't get this identity at all.
Eventually, I've just found out and understood that (by using an extended version of Renewal Theorem)
$$\lim_{x\to\infty}\sum_{k=1}^\infty p\{x\le first \space\space S_{Z_k} \lt x+h\} = \frac{h}{E[Z_1]} = \frac{h}{E[X_1]E[T]} = \lim_{x\to\infty}\sum_{n=1}^\infty p\{x\le first \space\space S_n \lt x+h\}$$
Nonetheless, that does NOT necessarily mean that the two probabilities have to be identical $for \space n$?