Random measures and point processes

Explain why the function $Z(B, \cdot )$ defined according to (14) is, for each $B \in \mathscr {B}(\mathrm{S})$, a random variable with values in the space $\left(\mathbb {Z}_{+}, \mathscr {A}\right)$, where $\mathbb {Z}_{+} = \left\{ 0,1, \ldots \right\} = \left\{ 0\right\} \cup \mathbb {N}, \mathscr {A}$ is the $\sigma$-algebra of all subsets of $\mathbb {Z}_{+}$. (It follows that $Z = \left\{ Z(B), B \in \mathscr {B}(\mathrm{S})\right\}$ is a real-valued stochastic process indexed by the sets $\mathscr {B}(\mathrm{S})$.)

Example 6. Let $\lambda$ be a finite measure ($\lambda \not\equiv 0$) defined on some measurable space $(\mathrm{S}, \mathscr {B})$. Let $Y, X_{1}, X_{2}, \ldots$ be independent random elements on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ such that $Y$ is a real Poisson random variable with parameter $\lambda (\mathrm{S})$, and $X_{1}, X_{2}, \ldots$ are independent identically distributed $\mathscr {F} \mid \mathscr {B}$-measurable variables for which $\mathbb {P}_{X_{1}}\left(B\right) = \lambda (B) / \lambda (\mathrm{S})$ for $B \in \mathscr {B}$. Define on $\mathscr {B} \times \Omega$ the function $Z(B, \omega )$ by setting

($Z(B, \omega ) = 0$ when $Y(\omega ) = 0$). Formula (14) defines the so-called Poisson random measure with intensity measure (or leading measure) $\lambda$.

Example 6. Let $\lambda$ be a finite measure ($\lambda \not\equiv 0$) defined on some measurable space $(\mathrm{S}, \mathscr {B})$. Let $Y, X_{1}, X_{2}, \ldots$ be independent random elements on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ such that $Y$ is a real Poisson random variable with parameter $\lambda (\mathrm{S})$, and $X_{1}, X_{2}, \ldots$ are independent identically distributed $\mathscr {F} \mid \mathscr {B}$-measurable variables for which $\mathbb {P}_{X_{1}}\left(B\right) = \lambda (B) / \lambda (\mathrm{S})$ for $B \in \mathscr {B}$. Define on $\mathscr {B} \times \Omega$ the function $Z(B, \omega )$ by setting

($Z(B, \omega ) = 0$ when $Y(\omega ) = 0$). Formula (14) defines the so-called Poisson random measure with intensity measure (or leading measure) $\lambda$.

Let $\lambda$ be a $\sigma$-finite measure on some measurable space ( $\mathrm{S}, \mathscr {B}$ ) $\left(\mathrm{S} = \bigcup_{m = 1}^{\infty } \mathrm{S}_{m}\right.$, where $\mathrm{S}_{m} \in \mathscr {B}, \mathrm{S}_{m} \cap \mathrm{S}_{k} = \varnothing$ for $m \neq k$ and $0 < \lambda \left(\mathrm{S}_{m}\right) < \infty$, $m, k \in \mathbb {N}$ ). In this case the construction of the Poisson random measure (14) is easily modified as follows.

On some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ construct a countable collection of independent variables

such that $Y^{(m)}$ has the Poisson distribution with parameter $\lambda_{m} = \lambda \left(\mathrm{S}_{m}\right)$ and $\operatorname {Law}\left(X_{j}^{(m)}\right) = \lambda (\cdot ) / \lambda \left(\mathrm{S}_{m}\right)$ on $\mathscr {B}_{m} = \mathscr {B} \cap \mathrm{S}_{m} ; m, j \in \mathbb {N}$. Let $Z_{m}(B, \omega )$ for $B \in \mathscr {B}_{m}$ be defined according to (14) with $Y$ replaced by $Y^{(m)}$ and $X_{j}$ by $X_{j}^{(m)}$. Set

Note that $Z(B, \omega )$ may also take infinite values.

Prove that if $\lambda (B) = \infty$ for some $B \in \mathscr {B}$, then $Z(B) = \infty$ a.s., and if $\lambda (B) < \infty$, then $Z(B) < \infty$ a.s.

Example 6. Let $\lambda$ be a finite measure ($\lambda \not\equiv 0$) defined on some measurable space $(\mathrm{S}, \mathscr {B})$. Let $Y, X_{1}, X_{2}, \ldots$ be independent random elements on a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ such that $Y$ is a real Poisson random variable with parameter $\lambda (\mathrm{S})$, and $X_{1}, X_{2}, \ldots$ are independent identically distributed $\mathscr {F} \mid \mathscr {B}$-measurable variables for which $\mathbb {P}_{X_{1}}\left(B\right) = \lambda (B) / \lambda (\mathrm{S})$ for $B \in \mathscr {B}$. Define on $\mathscr {B} \times \Omega$ the function $Z(B, \omega )$ by setting

($Z(B, \omega ) = 0$ when $Y(\omega ) = 0$). Formula (14) defines the so-called Poisson random measure with intensity measure (or leading measure) $\lambda$.

Let $\lambda$ be a $\sigma$-finite measure on some measurable space ( $\mathrm{S}, \mathscr {B}$ ) $\left(\mathrm{S} = \bigcup_{m = 1}^{\infty } \mathrm{S}_{m}\right.$, where $\mathrm{S}_{m} \in \mathscr {B}, \mathrm{S}_{m} \cap \mathrm{S}_{k} = \varnothing$ for $m \neq k$ and $0 < \lambda \left(\mathrm{S}_{m}\right) < \infty$, $m, k \in \mathbb {N}$ ). In this case the construction of the Poisson random measure (14) is easily modified as follows.

On some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ construct a countable collection of independent variables

such that $Y^{(m)}$ has the Poisson distribution with parameter $\lambda_{m} = \lambda \left(\mathrm{S}_{m}\right)$ and $\operatorname {Law}\left(X_{j}^{(m)}\right) = \lambda (\cdot ) / \lambda \left(\mathrm{S}_{m}\right)$ on $\mathscr {B}_{m} = \mathscr {B} \cap \mathrm{S}_{m} ; m, j \in \mathbb {N}$. Let $Z_{m}(B, \omega )$ for $B \in \mathscr {B}_{m}$ be defined according to (14) with $Y$ replaced by $Y^{(m)}$ and $X_{j}$ by $X_{j}^{(m)}$. Set

Note that $Z(B, \omega )$ may also take infinite values.

(see [32; p. 370]). Let $B \in \mathscr {B}$ be such that $0 < \lambda (B) < \infty$, where $\lambda$ is a $\sigma$-finite measure on $(\mathrm{S}, \mathscr {B})$. Let $B = \bigcup_{q = 1}^{n} B_{q}$, where $B_{q} \in \mathscr {B}, B_{j} \cap B_{q} = \varnothing$ for $q \neq j(q, j = 1, \ldots , n)$. Prove that for any $k_{1}, \ldots , k_{n} \in \mathbb {Z}_{+}$ and $k = \sum_{q = 1}^{n} k_{q}$ the following equality holds

Let S be a locally compact Hausdorff space satisfying the second axiom of countability and $M(\mathrm{S})$ the space of locally finite measures $\mu = \mu (\cdot )$ on the Borel $\sigma$-algebra $\mathscr {B}(\mathrm{S})$ (i.e., the space of measures that are finite on bounded sets of the space S), equipped with the topology in which the mappings

are continuous for functions $f$ from the class $C_{K}^{+}$ of continuous nonnegative functions with compact support.

Prove that $M(\mathrm{S})$ is a Polish space. To do this, note that if $f_{1}, f_{2}, \ldots$ is a set dense in $C_{K}^{+}$, then the function

metrizes the topology introduced above. Moreover, $\mathscr {B}(M(\mathrm{S}))$ is generated both by the mappings $\pi_{f}, f \in C_{K}^{+}$ (i.e., it is the smallest $\sigma$-algebra with respect to which all these mappings are measurable) and by the mappings $\pi_{B}: \mu \mapsto \mu (B)$ for $B \in \mathrm{S}_{\mu }$, where $\mu \in M(\mathrm{S}), \mathrm{S}_{\mu } = \left\{ B \in \mathscr {M}: \mu (\partial B) = 0\right\}$, and $\mathscr {M}$ is the collection of bounded subsets of the space $S$.

Let S be a locally compact Hausdorff space satisfying the second axiom of countability and $M(\mathrm{S})$ the space of locally finite measures $\mu = \mu (\cdot )$ on the Borel $\sigma$-algebra $\mathscr {B}(\mathrm{S})$ (i.e., the space of measures that are finite on bounded sets of the space S), equipped with the topology in which the mappings

are continuous for functions $f$ from the class $C_{K}^{+}$ of continuous nonnegative functions with compact support.

Definition 10. Let a probability space $(\Omega , \mathscr {F}, \mathbb {P})$ and the space of locally finite measures $M(\mathrm{S})$ introduced above be given. A random measure is an $\mathscr {F} \mid \mathscr {B}(M(\mathrm{S}))$-measurable mapping $\mu : \Omega \rightarrow M(\mathrm{S})$.

The collection of bounded subsets of the space $\mathrm{S}$ is denoted by $\mathscr {M}$.

Let $\mu , \mu_{1}, \mu_{2}, \ldots$ be random measures on $S$ — a locally compact Hausdorff space satisfying the second axiom of countability. Prove that the following three conditions are equivalent:

1. $\mu_{n} \stackrel{\mathscr {D}}{\rightarrow } \mu$ as $n \rightarrow \infty$,

2. $\left\langle f, \mu_{n} \right\rangle \stackrel{\mathscr {D}}{\rightarrow } \left\langle f, \mu \right\rangle$ for $f \in C_{K}^{+}(n \rightarrow \infty )$,

3. $\left(\mu_{n}\left(B_{1}\right), \ldots , \mu_{n}\left(B_{k}\right)\right) \stackrel{\mathscr {D}}{\rightarrow }\left(\mu \left(B_{1}\right), \ldots , \mu \left(B_{k}\right)\right)$ for $B_{1}, \ldots , B_{k} \in \mathrm{S}_{\mu }, k \in \mathbb {N}$, where $\mathrm{S}_{\mu } = \left\{ B \in \mathscr {M}: \mu (\partial B) = 0 \text{ a.s.}\right\}$.

Define $m(B) = \mathbb {E}\left[\mu (B, \cdot )\right]$ for all $B \in \mathscr {B}\left(\mathbb {R}_{+}\right)$, where $\mu (B, \cdot )$ appears in (85).

Suppose that on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ we are given a sequence of random variables $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ such that for $\omega \in \Omega_{0}$, where $\mathbb {P}\left(\Omega_{0}\right) = 1$,

Definition 11. The random point measure $\mu : \mathscr {B}\left(\mathbb {R}_{+}\right) \times \Omega \rightarrow \overline{\mathbb {Z}}_{+} = \mathbb {Z}_{+} \cup \left\{ \infty \right\}$ is given by the formula

where the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ is described above, and $\delta_{x}$ is the Dirac measure.

Will the function $m(\cdot )$ be a measure on $\mathscr {B}\left(\mathbb {R}_{+}\right)$? Can a Poisson process with a locally finite intensity measure $m$ be represented in the form (86)?

Definition 12. The random point process $Y = \left\{ Y(t), t \geqslant 0\right\}$, where $Y(0) = 0$, is introduced for $t > 0$ as follows:

where the random point measure $\mu$ is given by formula (85).

(compare with Theorem 14 of Chapter V). Prove that the random point process (86) is a Poisson process if $Y$ has independent increments and $Y$ is stochastically continuous on $\mathbb {R}_{+}$ (see (I.56)).

Theorem 14 of Chapter V (Doob). Let a process $X = \left\{ X_{t}, t \geqslant 0\right\}$ take values in $\mathbb {R}^{m}$ for each $t$ and have independent increments. If the sample paths of the process are continuous a.s., then the process $Y = \left\{ Y_{t} = X_{t}-X_{0}, t \geqslant 0\right\}$ is Gaussian. Moreover, for $0 \leqslant s \leqslant t < \infty$

where $M: \mathbb {R}_{+} \rightarrow \mathbb {R}^{m}$ and $G: \mathbb {R}_{+} \rightarrow \mathbb {R}^{m^2}$ are continuous functions.

Suppose that on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ we are given a sequence of random variables $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ such that for $\omega \in \Omega_{0}$, where $\mathbb {P}\left(\Omega_{0}\right) = 1$,

Definition 11. The random point measure $\mu : \mathscr {B}\left(\mathbb {R}_{+}\right) \times \Omega \rightarrow \overline{\mathbb {Z}}_{+} = \mathbb {Z}_{+} \cup \left\{ \infty \right\}$ is given by the formula

where the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ is described above, and $\delta_{x}$ is the Dirac measure.

Definition 12. The random point process $Y = \left\{ Y(t), t \geqslant 0\right\}$, where $Y(0) = 0$, is introduced for $t > 0$ as follows:

where the random point measure $\mu$ is given by formula (85).

Formula (I.56): a process $X = \left\{ X_{t}, t \in T\right\}$ with values in a metric space $(\mathrm{S}, \rho )$ is stochastically continuous on $T$ if for each $t \in T$ and any $\varepsilon > 0$

By Theorem 3, the random variables $\tau_{k} = \xi_{1}+\cdots +\xi_{k}, k \in \mathbb {N}$, form a Markov chain if the summands are independent real random variables.

Theorem 3. Let $X = \left\{ X_{t}, t \in T\right\}$ be a process with independent increments taking values in $\mathbb {R}^{d}$, $d \geqslant 1$. Then $X$ is a Markov process (with respect to the natural filtration).

Will the process (86) be Markov if $\left\{ \tau_{k}, k \in \mathbb {N}\right\}$ is an arbitrary Markov chain satisfying condition (84)?

Suppose that on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ we are given a sequence of random variables $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ such that for $\omega \in \Omega_{0}$, where $\mathbb {P}\left(\Omega_{0}\right) = 1$,

Definition 11. The random point measure $\mu : \mathscr {B}\left(\mathbb {R}_{+}\right) \times \Omega \rightarrow \overline{\mathbb {Z}}_{+} = \mathbb {Z}_{+} \cup \left\{ \infty \right\}$ is given by the formula

where the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ is described above, and $\delta_{x}$ is the Dirac measure.

Definition 12. The random point process $Y = \left\{ Y(t), t \geqslant 0\right\}$, where $Y(0) = 0$, is introduced for $t > 0$ as follows:

where the random point measure $\mu$ is given by formula (85).

Definition 13. Suppose that on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ we are given a Poisson process $N = \left\{ N(t), t \geqslant 0\right\}$ with intensity $\lambda = 1$ and a process $\Lambda = \left\{ \Lambda (t), t \geqslant 0\right\}$ independent of it, with a.s. nondecreasing, finite and right-continuous sample paths starting from zero. The Cox process is the process $Z = \left\{ Z(t), t \geqslant 0\right\}$, where $Z(t, \omega ) = N(\Lambda (t, \omega ), \omega ), t \geqslant 0, \omega \in \Omega$.

Is the Cox process Markov?

Find the covariance function of the process (87) if the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ defines a Poisson point process with intensity $\lambda$.

Suppose that on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ we are given a sequence of random variables $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ such that for $\omega \in \Omega_{0}$, where $\mathbb {P}\left(\Omega_{0}\right) = 1$,

Definition 14. We say that a stochastic process $X = \left\{ X(t), t \geqslant 0\right\}$ is "shot noise" if

where the bounded Borel function $\phi : \mathbb {R} \rightarrow \mathbb {R}$ has compact support, and the random variables $\tau_{k}$ ($k \in \mathbb {N}$) satisfy condition (84) (which ensures that the series (87) converges for each $t \geqslant 0$ on the set $\Omega_{0}$ of probability one).

Is it sufficient, for the convergence of the series (87) with probability one for each $t \geqslant 0$, to require that the bounded function $\phi$ be Lebesgue integrable?

Suppose that on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ we are given a sequence of random variables $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ such that for $\omega \in \Omega_{0}$, where $\mathbb {P}\left(\Omega_{0}\right) = 1$,

Suppose that, besides the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ of random variables satisfying condition (84), on the same probability space $(\Omega , \mathscr {F}, \mathbb {P})$ there is some sequence $\left\{ \eta_{k}\right\}_{k \geqslant 1}$ of real random variables "attached" to the times $\tau_{k}, k \geqslant 1$.

Definition 15. A marked random point process is a process (indexed by sets $C$) of the following form

The function

is called the counting function.

Draw the graph of the sample path $Y(t, D), t \geqslant 0$, for fixed $D \in \mathscr {B}(\mathbb {R})$.

Let $N = \left\{ N(t), t \geqslant 0\right\}$ be a Poisson process with mean measure $m = m(\cdot )$, determined by the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ that specifies the jump points of $N$. Let $\left\{ \eta_{k}\right\}_{k \geqslant 1}$ be a sequence of independent identically distributed real random variables with $\operatorname {Law}\left(\eta_{k}\right) = \widetilde{\mathbb {P}}$. Take $D \in \mathscr {B}(\mathbb {R})$ such that $\widetilde{\mathbb {P}}(D) > 0$. Prove that the "thinned process" $Y(t, D), t \geqslant 0$, defined by formula (88), is a Poisson process with mean measure $m(\cdot ) \widetilde{\mathbb {P}}(D)$.

Definition 11. The random point measure $\mu : \mathscr {B}\left(\mathbb {R}_{+}\right) \times \Omega \rightarrow \overline{\mathbb {Z}}_{+} = \mathbb {Z}_{+} \cup \left\{ \infty \right\}$ is given by the formula

where the sequence $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ is described above, and $\delta_{x}$ is the Dirac measure.

Definition 12. The random point process $Y = \left\{ Y(t), t \geqslant 0\right\}$, where $Y(0) = 0$, is introduced for $t > 0$ as follows:

where the random point measure $\mu$ is given by formula (85).

Formula (88):

where the process $Y$ is given by formula (86).

Definition 1. A family $Z = \left\{ Z(B), B \in \mathscr {K}\right\}$ in $L^{2}(\Omega )$ indexed by the sets of a semiring $\mathscr {K}$ is called an orthogonal random measure if

1. $Z(B) - Z(C)$ for $B, C \in \mathscr {K}, B \cap C = \varnothing$, i.e. $(Z(B), Z(C)) = 0$, where $(\cdot , \cdot )$ is the inner product in $L^{2}(\Omega )$;

2. for every $B = \bigcup_{k = 1}^{\infty } B_{k}$, where $B, B_{1}, B_{2}, \ldots \in \mathscr {K}$ and $B_{n} \cap B_{m} = \varnothing$ ($n \neq m$),

   $$Z(B) = \sum _{k = 1}^{\infty } Z\left(B_{k}\right) \text{ a.s., }$$

   and the series converges in mean square. An orthogonal random measure $\mu$ is called centered if $\mathbb {E}\left[Z(B)\right] = 0$ for every $B \in \mathscr {K}$.

Show that the Poisson random measure (see §9 of Chapter I), defined on the semiring of bounded Borel sets in $\mathbb {R}^{n}$, is an orthogonal (noncentered) random measure.

For the shot noise process (see Definition 14 of Chapter VI) with covariance function $r = r(s, t)$, prove that asymptotic stationarity arises in the sense that the limit $\lim_{s \rightarrow \infty } r(s, s+t)$ exists.

Definition 14 of Chapter VI. We say that a stochastic process $X = \left\{ X(t), t \geqslant 0\right\}$ is “shot noise" if

where the bounded Borel function $\phi : \mathbb {R} \rightarrow \mathbb {R}$ has compact support, and the random variables $\tau_{k}$ ($k \in \mathbb {N}$) satisfy condition (VI.84) (which ensures that the series (VI.87) converges for each $t \geqslant 0$ on the set $\Omega_{0}$ of probability one).

Condition (VI.84): on some probability space $(\Omega , \mathscr {F}, \mathbb {P})$ we are given a sequence of random variables $\left\{ \tau_{k}\right\}_{k \geqslant 1}$ such that for $\omega \in \Omega_{0}$, where $\mathbb {P}\left(\Omega_{0}\right) = 1$,