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# 数学代写|信息论代写INFORMATION THEORY代写|EE625 Entropy of a stationary sequence. Gaussian sequence

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## 数学代写|信息论代写INFORMATION THEORY代写|Entropy of a stationary sequence. Gaussian sequence

In Section $5.1$ we considered the entropy of a segment of stationary process $\left{\xi_{k}\right}$ in discrete time, i.e. of a stationary sequence. There we assumed that each element of the sequence was a random variable itself. The generalization of the notion of entropy given in Section $1.6$ allows us to consider the entropy of a stationary sequence that consists of arbitrary random variables (including continuous random variables), and therefore generalizing the results of Section 5.1.

If the auxiliary measure $v$ satisfied the multiplicativity condition (1.7.9), then (as it was shown before) conditional entropies in the generalized version possess all the properties that conditional entropies in the discrete version possess. The specified properties (and essentially only them) were used in the material of Section 5.1. That is why all the aforesaid in Section $5.1$ can be related to arbitrary random variables and entropy in the generalized case.
Measure $v$ is assumed to be multiplicative
$$v\left(d \xi_{k_{1}}, \ldots, d \xi_{k_{r}}\right)=\prod_{i=1}^{r} v_{k_{i}}\left(d \xi_{k_{i}}\right)$$
At the same time ‘elementary’ measures $v_{k}$ are assumed to be identical in view of stationarity. We also assume that the condition of absolute continuity of probability measure $P\left(d \xi_{k}\right)$ with respect to $v_{k}\left(d \xi_{k}\right)$ is satisfied as well. Process $\left{\xi_{k}\right}$ appears to be stationary with respect to distribution $P$, i.e. the condition of type (5.1.1) is valid for all $k_{1}, \ldots, k_{r}, a$.

The entropy rate $H_{1}$ is introduced by formula (5.1.3). However, $H_{\xi_{k} \mid \xi_{k-l}, \ldots, \xi_{k-1}}$ in this case should be understood as entropy (1.7.13). Hence, the mentioned definition corresponds to the formula
$$H_{1}=-\int \ln \frac{P\left(d \xi_{k} \mid \xi_{k-1}, \xi_{k-2}, \ldots\right)}{v_{k}\left(d \xi_{k}\right)} P\left(d \xi_{k} \mid \xi_{k-1}, \xi_{k-2}, \ldots\right) .$$
Theorem $5.1$ is also valid in the generalized version. Luckily, that theorem can be proven in the same way. Now it means the equality
$$H_{1}=-\lim {l \rightarrow \infty} \frac{1}{l} \int \ldots \int \ln \frac{P\left(d \xi{1}, \ldots, d \xi_{l}\right)}{v_{1}\left(d \xi_{1}\right) \ldots v_{l}\left(d \xi_{l}\right)} P\left(d \xi_{1}, \ldots, d \xi_{l}\right)$$

## 数学代写|信息论代写INFORMATION THEORY代写|Entropy of stochastic processes in continuous time. General concepts and relations

We assume that process $\left{\xi_{i}\right}$ is given on some interval $a \leqslant t \leqslant b$. Consider an arbitrary subinterval $\alpha \leqslant t \leqslant \beta$ lying within the feasible interval of the process. We use notation $\xi_{\alpha}^{\beta}=\left{\xi_{t}, \alpha \leqslant t \leqslant \beta\right}$ for it. Therefore, $\xi_{\alpha}^{\beta}$ denotes the value set of process $\left{\xi_{t}\right}$ on subinterval $[\alpha, \beta]$.

The initial process $\left{\xi_{t}\right}$ is described by probability measure $P$. According to the definition of entropy given in Section 1.6, in order to determine entropy $H_{\xi_{\alpha} \beta}$ for any distinct intervals $[\alpha, \beta]$ we need to introduce an auxiliary non-normalized measure $v$ or the corresponding probability measure $Q$. Measure $v$ (or $Q$ ) has to be defined on the same measurable space, i.e. on the same field of events related to the behaviour of process $\xi(t)$ on the entire interval $[a, b]$, i.e. process ${\xi(t)}$ having probabilities $Q$ can be interpreted as a new auxiliary stochastic process ${\eta(t)}$ different from the original process ${\xi(t)}$.

Measure $P$ has to be absolutely continuous with respect to measure $Q$ (or $v$ ) for the entire field of events pertaining to the behaviour of process ${\xi(t)}$ on the whole feasible interval $[a, b]$. Consequently, the condition of absolute continuity will be satisfied also for any of its subinterval $[\alpha, \beta]$.

Applying formula (1.6.17) to values of the stochastic process on some chosen subinterval $[\alpha, \beta]$ we obtain the following definition of entropy of this interval:
$$H_{\xi_{\alpha}^{\beta}}^{P / Q}=\int \ln \frac{P\left(d \xi_{\alpha}^{\beta}\right)}{Q\left(d \xi_{\alpha}^{\beta}\right)} P\left(d \xi_{\alpha}^{\beta}\right) .$$
Furthermore, according to the contents of Section $1.7$ (see (1.7.17)) we can introduce the conditional entropy
$$H_{\xi_{\alpha}^{\beta} \mid \xi_{\gamma}^{\delta}}^{P / Q}=\int \ln \frac{P\left(d \xi_{\alpha}^{\rho} \mid \xi_{\gamma}^{\delta}\right)}{Q\left(d \xi_{\alpha}^{\beta}\right)} P\left(d \xi_{\alpha}^{\beta} d \xi_{\gamma}^{\delta}\right)$$
where $[\gamma, \delta]$ is another subinterval not overlapping with $[\alpha, \beta]$.

## 数学代写|信息论代写INFORMATION THEORY代写|Entropy of a stationary sequence. Gaussian sequence

$$v\left(d \xi_{k_{1}}, \ldots, d \xi_{k_{r}}\right)=\prod_{i=1}^{r} v_{k_{i}}\left(d \xi_{k_{i}}\right)$$

$$H_{1}=-\int \ln \frac{P\left(d \xi_{k} \mid \xi_{k-1}, \xi_{k-2}, \ldots\right)}{v_{k}\left(d \xi_{k}\right)} P\left(d \xi_{k} \mid \xi_{k-1}, \xi_{k-2}, \ldots\right) .$$

$\$ \$$\mathrm{H}{-}{1}=-\lim {1 \mid \mathrm{~ | r i g h t a r r o w ~ l i n f t y } ~ | f r a c { 1 } { 1 } ~ | i n t | / d o t s |} \mathrm{~ V d o t s , ~ d ~ | x i { l } | r i g h t ) } { v _ { 1 }} |xi_{1}, Vdots, d |xi_{1}}right) \ \$$

## 数学代写|信息论代写INFORMATION THEORY代写|Entropy of stochastic processes in continuous time. General concepts and relations

$\begin{array}{ll}\text { \left 的分隔符缺失或无法识别 } & \text { 为了它。所以， } \xi_{\alpha}^{\beta} \text { 表示过程的值集 } \ \text { \eft 的分隔符缺失或无法识别 } & \text { 在子区间 }[\alpha, \beta] .\end{array}$ \left 的分隔符缺失或无法识别 $\quad$ 在子区间 $[\alpha, \beta]$.

$$H_{\xi_{\alpha}^{\beta}}^{P / Q}=\int \ln \frac{P\left(d \xi_{\alpha}^{\beta}\right)}{Q\left(d \xi_{\alpha}^{\beta}\right)} P\left(d \xi_{\alpha}^{\beta}\right)$$

$$H_{\xi_{\alpha \alpha}^{\beta} \xi_{\gamma}^{\delta}}^{P / Q}=\int \ln \frac{P\left(d \xi_{\alpha}^{\rho} \mid \xi_{\gamma}^{\delta}\right)}{Q\left(d \xi_{\alpha}^{\beta}\right)} P\left(d \xi_{\alpha}^{\beta} d \xi_{\gamma}^{\delta}\right)$$

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