> Li Chen, Wang Guifa. Poisson point process and its properties [J]. Journal of Xinxiang University, 2012, 29 (6): 483-484. doi:10.3969/j.issn.1674-3326.2012.06.002.
* * The following article for the rearrangement of citations **## 1. Preliminary Knowledge-# # # 1: Set $ (X,{\re _x}) $ is a measurable space, if $ {d_p} \subset (0,\infty) $ is a maximum number of sets, then the map $ p:{d_p} \to x $ is a point function on $ x $.
If $ P $ is a point function on $ X $, then $ {n_p}\left ({\left ({0,t} \right] \times U} \right) \buildrel \delta \over = \left\{{s|s \in { D_p},p (s) \in U} \right\} $ where $ T \in \left ({0,\infty} \right), U \in {\re _x} $ This defines $ \left ({0,\infty} \right) \tim A count measure on es X $ is $ n_p (DTDX) $.
Make $ {\pi _x} $ represents a collection of $ x $ on all-point functions, denoted by $ \re ({\pi _x}) = \sigma \left ({\left\{} {n_p}\left ({\left} 0,t) \right U } \right): {\pi _x} \to {\mathbb{z}^ +} \cup \left\{\infty \right\}|t \in \left ({0,\infty} \right), U \in {\Re _X}} \ri Ght\}} \right) $.-# # # definition 2: A random variable that takes a value of $ \left ({{_x},\re \left}} \pi)} _x) $ is called a point procedure on X.
Set $ P $ is $ X $ on the dot function, and $ t \geqslant 0 $, make $ {d_{\theta TP}} = \left\{{s|s \in \left ({0,\infty} \right), S + t \in {D_ P}} \right\} $, definition: $ \theta TP (s) = P (t + s) $ for $q TP (s) = P (t +s) $. where $ s \in d_{\theta TP} $. Set $ P $ is $ X $ on the dot process, if for any $ t \geqslant 0 $, $ \theta TP $ with $ p $ with the same distribution, then called $ P $ for smooth.
-# # # # Definition 3: If $ n_p (DTDX) $ is $ \left ({0,\infty} \right) \times X $ on the $ Poisson $ random measure, then the point procedure is $ P $ for $ possion $ for the point procedure.
Obviously, a $ Poisson $ point procedure $ p $ is smooth when and only if its strength $ n_p (DTDX) $ has the following form: $ n_p (DTDX) =DTN (DX) $, here $ N (DX) $ is $ (x,{\re _x}) on the test degree, called the characteristic measure of $ P $.
-# # # 4: Set $ \{y_t\} $ is a real numeric procedure if for any $ n \in N $, $ t_1 < T_2 < \cdots < T_N $, if there is $ y_{t_2}-y_{t_1}, \cdots , y_{t_n}-y_{t_{n-1}} $ is independent of each other, then called $ \{y_t\} $ for an independent incremental process; If for any $ s < T $, the distribution of $ y_t-y_s $ is only related to $ T-S $, then the independent incremental process $ \{y_ T\} $ is smooth. The right continuous smooth independent increment process is known as the $ Levy $ process.
-# # # Definition 5: For $ \{\omega, f\} $ on the function $ T: \omega \to \left[{0, \infty} \right) $ if for each $ t \geqslant 0 $, there are $ \{t \ge Qslant t \} \in f_t $, then called $ T $ for stop.
# # 2. $ possion $ point Process Nature-# # # theorem 1: if $ n (DX) $ is $ (x,{\re _x}) $ on the $ \SIGMA $ finite measure, then there is a smooth Poisson point process on $ X $, making its feature measure to $ n (DX) $. Proof: For the probability space $ (\omega, F, p) $ and the random measure defined on $ (\omega, F, p) $ n_p (DTDX) $ is $ \left ({0, \infty} \right) \times X $ on the $ Poisson $ random measure, and its strength is $ DTN (DX) $. Take a column collection of $ u_n (n=1,2, \cdots) $, making $0 < N (u_n) < \INFTY $, and $ u_n $ is monotonically incrementing, $ \cup _{n-1}^{\infty} u_n=x $. For each $ N $, it is easy to see that the procedure $ x_{t}^{(n)} =n \left ({\left ({0,t} \right] \times U} \right) $ is the right continuous $ Poisson $ process with a parameter of $ n (u_n) $, therefore, event $ {\lambda _n} = \left\{{\omega |\exists t \in \left ({0,\infty} \right) \to x_t^{(N)}-x_{t-}^{(n)} \GEQSL The probability of Ant 2} \right\} $ is 0. Make $ \lambda = \cup _n^\infty {\lambda _n} $, then $ P (\LAMBDA) = 0 $, and $ \lambda =\left\ {{\omega |\exists T \in \left ({0,\infty} \right), N\left ({\left\{t \right\} \times X} \right) \ge 2} \right\} $. Take a $ X $ on the dot function $ p_0:d_{p_0} \to X $, Make $ d_{p_0} (\omega) $ as follows: $ {\omega \notin when $ \omega0}}} (\omega) = \left\{{s|\exists x \in x,n\left ({\left\{{\left ({s,x} \right)} \right\}} \right) > 0} \right\} $; $ d_{p_0}=d_p $ \omega \in \omega $. The form of $ P (\omega) (s) $ is as follows: if $ s \in d_p (\omega) $, $ N ({(S, x)}) > 0 $, $ \omega \notin \omega $, then there is $ p (w) (s) = x \in X $; if $ s \in d_{p_0} $, $ \omega \in \omega $, then there is $ p (w) (s) = P_0 (s) $; Obviously, $ P $ is $ X $ on the dot process, and for any $ T > 0 $, $ U \subset \re_x $, there is $ {n_p}\left ({\left ({0,t} \right) \times U} \right) \buildrel \delta \over = \left\{{s|\left ({s), P (s)} \right) \in \left ({0,t} \right) \times U} \right\} = \left\{{(s,x) | ( s,x) \in \left ({0,t} \right) \times u} \right\} = N\left ({\left ({0,t} \right) \times u} \right) $. Therefore, $ P $ is $ X $ on the smooth $ Poisson $ point process, and it has a characteristic measure of $ n (DX) $.-# # # # theorem 2: Set \{y_t\} is defined in a probability space of $ (\omega,\psi , p) $ on $ The Levy $ process, $ \sigma_t (T \geqslant 0) $ is $ \omega $ on the shift operator, making $ y_t \circ \sigma_t = y_{t+s} $. Where any $ T $, $ s \geqslant 0 $, then $ \{y_t\} $ is adapted to filter $ \{\psi_t \} $ The strong $ Markov $ process, whereas $ \psi_t = \kappa \vee \sigma \left ({\left\{{{y_s}|s \ge 0} \right\}} \right) $, $ \kappa = \ {a | \exists b \subset \re, a \subset B, P (b) = 0\} $, $ t \geqslant 0 $. Proof: by $ Kolmogorov 0-1 $ law, filter $ \{y_t\} The $ \{y_t\} $ procedure is a strong $ Markov $ process that is adapted to the filter $ \{\psi_t\} $, which is right continuous and meets the usual conditions.-# # # # # # theorem 3: For any $ U \in \re_x $, $ {N_p}\lef T ({\left ({0,t} \right] \times U} \right) $ The $ \CDOT $ distribution that follows the parameter for $ t Poisson N (U) $, then $ p $ is $ \mathbb{r} on $ Poisson The $ point procedure whose characteristic measure is $ n (\cdot) $. Prove the process withheld. $ [$ NOTE 1: $ (\mathbb{r},{\re _\mathbb{r}}) on the measure $ n (\cdot) $ is called the $ levy $ procedure for the $ levy $ measure. NOTE 2: Set $ n (DX) $ Yes $ (X, \re_x $ \SIGMA $ limited measure on $ (\omega, \PSI, p) $ is a complete probability space, $ P $ is defined in $ (\omega, \PSI, p) $ on a smooth point $ Poisson $ process, its characteristic measure is $ n (d x) $ $] $.-# # # # Theorem 4: If $ f (\cdot) $ is $ (x $ re_x) $ A non-negative measurable function, $ \int_x {f (x) n (DX)} < \infty $, then $ {x_t } = \sum\limits_{s \in {d_p},s \leqslant T} {f (P)} = \int_{\left ({0,t} \right]} {\int_x {f (X) N (DSDX)}} < \infty $, $ (T \geqslant 0) $ is the $ Levy $ process, and on any $ t \geqslant 0 $ there is $ e\{x_t\} = T \int_x {f (X) n (DX)} $, so $ \kappa = \{a| A \in F, P (a) = 0 \} $, for any $ t \geqslant 0 $, to $ {f_t} = \kappa \vee \sigma \left ({\left\{{n_p}\left ({\left 0,s} \right] \times U} \right) |s \leqslant t,u \in {\re _x}} \right\}} \right) $, by $ Kolmogorov 0-1 $ law, filter $ \{f_t\} $ right Continuous, satisfies the usual conditions.-# # # # # # # # # # # # # # # # # # 5: $ \CDOT $ ($ \re_x) $ (X $ $) $ \{z_t\}_{t \geqslant 0} $ is suitable for $ \{f_t \} $ for non-negative Material process, there is $$ e\left\{{\sum\limits_{s \in d,s \leqslant T} {{z_s}f\left ({p (s)} \right)}} \right\} = e\left\{{\int_0^t {{Z _s}ds\int_x {f (X) n (DX)}} \right\} $$ proof: because $ x_t = {\sum\limits_{s \in d,s \leqslant T} {f (P)}} $, $ t \geqslant 0 $ is a right continuous independent increment process, and for any $ T > s \geqslant 0 $, due to $ e\left\{{{x_t}-t\int_x {f (X) n (DX) |{ f_s}}} \right\} = e\left\{{x_t}-{x_s}|{ f_s}} \right\}-t\int_x {f (X) n (DX)} + {x_s} = e\left\{{{x_{t-s}}} \right\}-t\int_x {f (X) n (DX)} + {x_s} = (T-s) \int_x {f (x) n (DX)} -t\int_x {f (x) n (DX)} + X = {x_s}-s\int_x {f (x) n (DX)} $, so, $ \left\{{ {x_t}-t\int_x {f (X) n (DX)}} \right\} $ is the right consecutive $ \{f_t \} $. For arbitrary bounded stops $ S $, $ T $, and $ s < T $, then there are $ e\left\{{{x_t }-t\int_x {f (x) n (DX)}} \right\} = e\left\{{x_s}-s\int_x {f (x) n (DX)}} \right\} $, ie $ e\left\{{\sum\limits_{s \in D,s \leqslant T} {f (P)}} \right\} = e\left\{{{x_t}-{x_s}} \right\} = e\left\{{\int_{\left ({s,t} \right]} {\int_x {f ( x) n (ds)}} \right\} $, which means $ e\left\{{\sum\limits_{s \in d,s \leqslant T} {{z_s}f\left ({p (s)} \right)}} \right\} = E \left\{{\int_0^t {{z_s}ds\int_x {f (X) n (DX)}}} \right\} $, for the material process $ \left\{{{z_t} = {I_{\left ({s,t} \right]}} (t)} \rig Ht\} $ is established, that is, the above formula for any material process is established.
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Poisson point process and its properties
