The intensity function is defined as

$$\lambda^*(t)=\frac{f(t|H_{t_n})}{1-F(t|H_{t_n})}$$ where $f$ is the density function and $F$ is the distribution function, and $H_{t_n}$ is the history of all the previous points of $t$ up to $t_n$. Moreover, there is proven that $F(t|H_{t_n})$ is also given as:

$$F(t|H_{t_n})=1-e^{-\int_t^{t_n}\lambda^*(s)ds}$$.

An assumption is then made, saying that

1. $\lambda^*(t)$ is non-negative and is integrable on every interval after $t_n$.

2. $\int_t^{t_n}\lambda^*(s)ds \to 1$ for $t \to \infty$.

It is then said that hence the three points follows:

1. $0 \leq F(t|H_{t_n}) \leq 1$
2. $F(t|H_{t_n})$ is a non-decreasing function of $t$.
3. $F(t|H_{t_n}) \to 1$ for $t \to \infty$

Can someone explain to me how these three points follow given the two assumptions above? Thank you.

The intensity function is defined as

$$\lambda^*(t)=\frac{f(t|H_{t_n})}{1-F(t|H_{t_n})}$$ where $f$ is the density function and $F$ is the distribution function, and $H_{t_n}$ is the history of all the previous points of $t$ up to $t_n$. Moreover, there is proven that $F(t|H_{t_n})$ is also given as:

$$F(t|H_{t_n})=1-e^{-\int_{t_n}^t\lambda^*(s)ds}$$.

An assumption is then made, saying that

1. $\lambda^*(t)$ is non-negative and is integrable on every interval after $t_n$.

2. $\int_{t_n}^t\lambda^*(s)ds \to 1$ for $t \to \infty$.

It is then said that hence the three points follows:

1. $0 \leq F(t|H_{t_n}) \leq 1$
2. $F(t|H_{t_n})$ is a non-decreasing function of $t$.
3. $F(t|H_{t_n}) \to 1$ for $t \to \infty$

Can someone explain to me how these three points follow given the two assumptions above? Thank you.

The intensity function is defined as

$\lambda*(t)=\frac{f(t|H_{t_n})}{1-F(t|H_{t_n})}$

where f is the density function and F is the distribution function, and H_{t_n} is the history of all the previous points of t up to t_n. Moreover, there is proven that F(t|H_{t_n}) is also given as:

$F(t|H_{t_n})=1-e^{-\int_t^{t_n}\lambda*(s)ds}$.

An assumption is then made, saying that

1. $\lambda*(t)$ is none-negative and is integrable on every interval after $t_n$.

2. $\int_t^{t_n}\lambda*(s)ds -> 1$ for $t -> \infty$.

It is then said that hence the three points follows:

1. $0 \leq F(t|H_{t_n}) \leq 1$
2. $F(t|H_{t_n})$ is a non-decreasing function of $t$.
3. $F(t|H_{t_n)} -> 1$ for $t -> \infty$

Can someone explain to me how these three points follow given the two assumptions above? Thank you.

The intensity function is defined as

$$\lambda^*(t)=\frac{f(t|H_{t_n})}{1-F(t|H_{t_n})}$$ where $f$ is the density function and $F$ is the distribution function, and $H_{t_n}$ is the history of all the previous points of $t$ up to $t_n$. Moreover, there is proven that $F(t|H_{t_n})$ is also given as:

$$F(t|H_{t_n})=1-e^{-\int_t^{t_n}\lambda^*(s)ds}$$.

An assumption is then made, saying that

1. $\lambda^*(t)$ is non-negative and is integrable on every interval after $t_n$.

2. $\int_t^{t_n}\lambda^*(s)ds \to 1$ for $t \to \infty$.

It is then said that hence the three points follows:

1. $0 \leq F(t|H_{t_n}) \leq 1$
2. $F(t|H_{t_n})$ is a non-decreasing function of $t$.
3. $F(t|H_{t_n}) \to 1$ for $t \to \infty$

Can someone explain to me how these three points follow given the two assumptions above? Thank you.