Let $X_{1}, X_{2}, \ldots$, be a sequence of independent, identically distributed positive random variables, with a common probability density function $f(x), x>0$. Call $X_{n}$ a record value if $X_{n}>\max \left\{X_{1}, \ldots, X_{n-1}\right\}$. Consider the sequence of record values

$$V_{0}=0, V_{1}=X_{1}, \ldots, V_{n}=X_{i_{n}},$$

where

$$i_{n}=\min \left\{i \geqslant 1: X_{i}>V_{n-1}\right\}, n>1 .$$

Define the record process $\left(R_{t}\right)_{t \geqslant 0}$ by $R_{0}=0$ and

$$R_{t}=\max \left\{n \geqslant 1: V_{n}<t\right\}, \quad t>0$$

(a) By induction on $n$, or otherwise, show that the joint probability density function of the random variables $V_{1}, \ldots, V_{n}$ is given by:

$$f_{V_{1}, \ldots, V_{n}}\left(x_{1}, \ldots, x_{n}\right)=f\left(x_{1}\right) \frac{f\left(x_{2}\right)}{1-F\left(x_{1}\right)} \times \ldots \times \frac{f\left(x_{n}\right)}{1-F\left(x_{n-1}\right)},$$

where $F(x)=\int_{0}^{x} f(y) \mathrm{d} y$ is the cumulative distribution function for $f(x)$.

(b) Prove that the random variable $R_{t}$ has a Poisson distribution with parameter $\Lambda(t)$ of the form

$$\Lambda(t)=\int_{0}^{t} \lambda(s) \mathrm{d} s,$$

and determine the 'instantaneous rate' $\lambda(s)$.

[Hint: You may use the formula

$$\begin{aligned} &\mathbb{P}\left(R_{t}=k\right)=\mathbb{P}\left(V_{k} \leqslant t<V_{k+1}\right) \\ &=\int_{0}^{t} \ldots \int_{0}^{t} \mathbf{1}_{\left\{t_{1}<\ldots<t_{k}\right\}} f_{V_{1}, \ldots, V_{k}}\left(t_{1}, \ldots, t_{k}\right) \\ &\quad \times \mathbb{P}\left(V_{k+1}>t \mid V_{1}=t_{1}, \ldots, V_{k}=t_{k}\right) \prod_{j=1}^{k} \mathrm{~d} t_{j}, \end{aligned}$$

for any $k \geqslant 1 .]$