### 统计代写|随机过程代写stochastic process代考|Processes with a semi-Markov chance interference

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## 统计代写|随机过程代写stochastic process代考|Processes with a semi-Markov chance interference

We construct a measure which corresponds to this process on $\left{\mathscr{F}^, \mathscr{N}^\right}$, where $\mathscr{F}^$ is the set of functions $x(t)$ defined on the sum of the intervals $\cup\left[t_k, t_{k+1}\right)$, $0=t_0$ is the $\sigma$-algebra generated by the cylinders on $\mathscr{F}^*$. Such a measure will depend on $x$ and $y$ which are initial values of the Markov and semi-Markov processes, respectively.

Let $\mathscr{F}y$ be the set of piecewise constant functions $y(t)$ with values in $\mathscr{Y}$. Let $y(t)=y_k$ if $\zeta_k \leqslant t<\zeta{k+1}, 0=\zeta_0<\zeta_1<\cdots<\zeta_n<\cdots, \zeta_n \uparrow \infty$. We associate a measure $\mathrm{P}x^{y(\cdot)}$ on $\left{\mathscr{F}^, \mathscr{N}^\right}$ with the function $y(\cdot)$ in the following manner: if $$A=\bigcap{k=0}^n \theta_{\zeta_k} A_k$$
where $A_k$ is a cylindrical set on $\mathscr{F}^*$ defined by the values of $x(t)$ for $0 \leqslant t<\zeta_{k+1}-\zeta_k$, then
\begin{aligned} \mathrm{P}x^{y(\cdot)}(A)= & \int \mathrm{P}_x^{y_0}\left{A_0 \cap C{\zeta_1}\left(d x_1\right)\right} \int \mathrm{P}{x_1}^{y_1}\left{A_1 \cap C{\zeta_2-\zeta_1}\left(d x_2\right)\right} \times \cdots \ & \times \int \mathrm{P}{x{n-1}}^{y_{n-1}}\left{A_{n-1} \cap C_{\zeta_n-\zeta_{n-1}}\left(d x_n\right)\right} \mathrm{P}{x_n}^{y_n}\left(A_n\right), \quad C_t(B)={x(\cdot): x(t) \in B} . \end{aligned} It is easy to verify that $\mathrm{P}_x^{y(\cdot)}(A)$ is an $\mathscr{N}{y y}$-measurable function on $\mathscr{F}y$, where $\mathscr{N}{\mathscr{X}}$ is a $\sigma$-algebra generated by the cylincers in $\mathscr{F}{a y}$. Therefore the integral $$\mathrm{P}{y, x}(A)=\int \mathrm{P}x^{y(\cdot)}(A) \mu{y y}(d y(\cdot)),$$
where $\mu_{x y}(\cdot)$ is a measure on $\left{\mathscr{F}{y y}, \mathscr{N}{q y}\right}$ corresponding to the component of the semi-Markov process $(y(t) ; \xi(t))$, is meaningful.

## 统计代写|随机过程代写stochastic process代考|The ergodic theorem for processes with a discrete chance interference

The ergodic theorem for processes with a discrete chance interference. Let $(x(t) ; \xi(t))$ be a process with a discrete chance interference in the phase space ${\mathscr{X}, \mathfrak{B}}$. We assume that the process is regular, i.e. that $\xi(t)$ possesses a finite number of jumps on each finite interval. We shall be concerned with the behavior of
$$\frac{1}{T} \int_0^T f(x(t), \xi(t)) d t$$

as $T \rightarrow \infty$, where $f(x, s)$ is a $\mathfrak{B} \times \mathfrak{B}{+}$-measurable function. Below we shall derive sufficient conditions for expression (42) to have with probability 1 the limit of the form $$S(f)=\iiint_0^{\infty} Q{x, 0}(t, d y, \mathscr{X}) f(y, t) d t \pi(d x) / \iint_0^{\infty} Q_{x, 0}(t, \mathscr{X}, \mathscr{X}) d t \pi(d x)$$
as $T \rightarrow \infty$, where $Q_{x, 0}\left(t, B_1, B_2\right)$ is the function associated with a process with discrete chance interference by relation (38) and $\pi(d x)$ is the stationary distribution of the Markov chain in $(\mathscr{X}, \mathfrak{B})$ with the transition probability
$$\pi(x, B)=Q_{x, 0}(0, \mathscr{X}, B),$$
i. e. for all $B \in \mathfrak{B}$
$$\pi(B)=\int \pi(d x) \pi(x, B)$$
In order that expression (43) be meaningful it is necessary that there exist a measure $\pi$ which satisfies (45) and the function $f(y, t)$ be such that
$$\iiint_0^{\infty} Q_{x, 0}(t, d y, \mathscr{X})|f(y, t)| d t \pi(d x)<\infty$$
Recall that the Markov chain $\left{x_n\right}$ in ${\mathscr{X}, \mathfrak{B}}$ is called ergodic if there exists for this chain a stationary distribution $\pi(d x)$ and for all $x \in \mathscr{X}$ and $\mathfrak{B}$-measurable function $f$ such that $\int|f(x)| \pi(d x)<\infty$ we have, with probability $\mathrm{P}{x, 0}=1$, $$\lim {n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^n f\left(x_k\right)=\int f(x) \pi(d x) .$$

# 随机过程代考

## 统计代写|随机过程代写stochastic process代考|Processes with a semi-Markov chance interference

\begin{aligned} \mathrm{P}x^{y(\cdot)}(A)= & \int \mathrm{P}x^{y_0}\left{A_0 \cap C{\zeta_1}\left(d x_1\right)\right} \int \mathrm{P}{x_1}^{y_1}\left{A_1 \cap C{\zeta_2-\zeta_1}\left(d x_2\right)\right} \times \cdots \ & \times \int \mathrm{P}{x{n-1}}^{y{n-1}}\left{A_{n-1} \cap C_{\zeta_n-\zeta_{n-1}}\left(d x_n\right)\right} \mathrm{P}{x_n}^{y_n}\left(A_n\right), \quad C_t(B)={x(\cdot): x(t) \in B} . \end{aligned}很容易验证$\mathrm{P}x^{y(\cdot)}(A)$是$\mathscr{F}y$上的一个$\mathscr{N}{y y}$ -可测量函数，其中$\mathscr{N}{\mathscr{X}}$是$\mathscr{F}{a y}$中的圆柱体生成的一个$\sigma$ -代数。因此积分$$\mathrm{P}{y, x}(A)=\int \mathrm{P}x^{y(\cdot)}(A) \mu{y y}(d y(\cdot)),$$ 其中$\mu{x y}(\cdot)$是$\left{\mathscr{F}{y y}, \mathscr{N}{q y}\right}$上对应于半马尔可夫过程$(y(t) ; \xi(t))$分量的测度，是有意义的。

## 统计代写|随机过程代写stochastic process代考|The ergodic theorem for processes with a discrete chance interference

$$\frac{1}{T} \int_0^T f(x(t), \xi(t)) d t$$

$$\pi(x, B)=Q_{x, 0}(0, \mathscr{X}, B),$$

$$\pi(B)=\int \pi(d x) \pi(x, B)$$

$$\iiint_0^{\infty} Q_{x, 0}(t, d y, \mathscr{X})|f(y, t)| d t \pi(d x)<\infty$$

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