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# 数学代写|随机过程Stochastic Porcesses代考|STAT507 Stationarity

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## 数学代写|随机过程Stochastic Porcesses代考|Stationarity

Definition 2.2.1. We say that the stochastic process ${X(t), t \in T}$ is stationary, or strict-sense stationary (SSS), if its distribution function of order $n$ is invariant under any change of origin:
$$F\left(x_1, \ldots, x_n ; t_1, \ldots, t_n\right)=F\left(x_1, \ldots, x_n ; t_1+s, \ldots, t_n+s\right)$$
for all $s, n$, and $t_1, \ldots, t_n$.
Remark. The value of $s$ in the preceding definition must be chosen so that $t_k+s \in T$, for $k=1, \ldots, n$. So, if $T=[0, \infty)$, for instance, then $t_k+s$ must be nonnegative for all $k$.

In practice, it is difficult to show that a given stochastic process is stationary in the strict sense (except in the case of Gaussian processes, as will be seen in Section 2.4). Consequently, we often satisfy ourselves with a weaker version of the notion of stationarity, by considering only the cases where $n=$ 1 and $n=2$ in Definition 2.2.1.
If ${X(t), t \in T}$ is a (continuous) SSS process, then we may write that
$$f(x ; t)=f(x ; t+s) \quad \forall s, t$$
and
$$f\left(x_1, x_2 ; t_1, t_2\right)=f\left(x_1, x_2 ; t_1+s, t_2+s\right) \quad \forall s, t_1, t_2$$

## 数学代写|随机过程Stochastic Porcesses代考|Ergodicity

In statistics, to estimate an unknown parameter of a distribution function, for example, the parameter $\lambda$ of an r.v. $X$ having a $\operatorname{Poi}(\lambda)$ distribution, we draw a random sample of $X$. That is, we take $n$ observations, $X_1, \ldots, X_n$, of $X$ and we assume that the $X_k$ ‘s have the same distribution function as $X$ and are independent. Next, we write that the estimator $\hat{\lambda}$ of $\lambda$ (which is the mean of the distribution) is the arithmetic mean of the observations. Similarly, to estimate the mean $m_X(t)$ of a stochastic process ${X(t), t \in T}$ at time $t$, we must first take observations $X\left(t, s_k\right)$ of the process. Next, we define
$$\widehat{m_X(t)}=\frac{1}{n} \sum_{k=1}^n X\left(t, s_k\right)$$
Thus, we estimate the mean $m_X(t)$ of the s.p. by the mean of a random sample taken at time $t$. Of course, the more observations of the process at time $t$ we have, the more precise the estimator $m_X(t)$ should be. Suppose, however, that we only have a single observation, $X\left(t, s_1\right)$, of $X(t)$. Since we cannot estimate $m_X(t)$ in a reasonable way from a single observation, we would like to use the values of the process for the other values of $t$ to estimate $m_X(t)$. For this to be possible, it is necessary (but not sufficient) that the mean $m_X(t)$ be independent of $t$.

Definition 2.3.1. The temporal mean of the s.p. ${X(t), t \in \mathbb{R}}$ is defined by
$$\langle X(t)\rangle_S=\frac{1}{2 S} \int_{-S}^S X(t, s) d t$$

## 数学代写|随机过程随机过程代考|平稳性

$$F\left(x_1, \ldots, x_n ; t_1, \ldots, t_n\right)=F\left(x_1, \ldots, x_n ; t_1+s, \ldots, t_n+s\right)$$

。在前面的定义中，必须选择$s$的值，以便$t_k+s \in T$表示$k=1, \ldots, n$。因此，例如，如果$T=[0, \infty)$，那么$t_k+s$对于所有$k$必须是非负的 在实践中，很难证明给定的随机过程在严格意义上是平稳的(高斯过程除外，如第2.4节所述)。因此，我们通常只考虑定义2.2.1中$n=$ 1和$n=2$的情况，从而满足于平稳性概念的较弱版本。如果${X(t), t \in T}$是一个(连续的)SSS进程，那么我们可以写
$$f(x ; t)=f(x ; t+s) \quad \forall s, t$$

$$f\left(x_1, x_2 ; t_1, t_2\right)=f\left(x_1, x_2 ; t_1+s, t_2+s\right) \quad \forall s, t_1, t_2$$

## 数学代写|随机过程Stochastic processes代考|Ergodicity

.

$$\widehat{m_X(t)}=\frac{1}{n} \sum_{k=1}^n X\left(t, s_k\right)$$

$$\langle X(t)\rangle_S=\frac{1}{2 S} \int_{-S}^S X(t, s) d t$$

## MATLAB代写

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