I have some trouble understanding the record value for a sequence of i.i.d. random variables of geometric distribution. Following quotation is from Univariate discrete distributions By Norman Lloyd Johnson, Adrienne W. Kemp, Samuel Kotz.
The lack-of-memory property of the geometric distribution gives it a role comparable to that of the exponential distribution. There are a number of characterizations of the geometric distribution based on record values.
For the record time $T_n$,
If $X_j$ is observed at time $j$ , then the record time sequence $\{T_n ,n \geq 0\}$ is defined as $T_0 = 1$ with probability $1$ and $T_n = \min\{j : X_j > X_{t_{n−1}} \}$ for $n\geq 1$.
Is there a typo in $T_n = \min \{j : X_j > X_{t_{n−1}} \}$? Should it be instead $T_n = \min\{j : X_j > X_{T_{n−1}} \}$?
For the record value $R_n$,
The record value sequence $\{R_n \}$ is defined as $R_n =X_{T_n} , n = 0, 1, 2, ...$. Suppose that the $X_j$ ’s are iid geometric variables with pmf $p_x = p(1 − p)^{x−1}, x = 1, 2, ...$ Then $R_n =X_{T_n} = \sum_{j=0}^{n} X_j$ is distributed as the sum of $n + 1$ iid geometric variables.
why does the second equality in "$R_n =X_{T_n} = \sum_{j=0}^{n} X_j$ " hold?
For the process of the record values $\{ R_n, n \in \mathbb{N}\}$,
Each of the following properties characterizes the geometric distribution:
(i) Independence: The rv’s $R_0 , R_1 - R_0 , ... , R_{n+1} - R_n ,...$ are independent.
(ii) Same Distribution: $R_{n+1} - R_n$ has the same distribution as $R_0$ .
(iii) Constant Regression: $E[R_{n+1}- R_n |R_n ]$ is constant.
How to show that the three properties hold? Are they derived from the memoryless property of geometric distribution?
What else can we say about the process based on the memoryless property of geometric distribution?
Thanks for your advice!