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## 澳洲代写|信息论代写information theory代写|Asymptotic equiprobability and entropic stability

The ideas of preceding section concerning asymptotic equivalence of nonequiprobable and equiprobable outcomes can be extended to essentially more general cases of random sequences and processes. It is not necessary for random variables $\xi_{j}$ forming the sequence $\eta^{n}=\left(\xi_{1}, \ldots, \xi_{n}\right)$ to take only one of two values and to have the same distribution law $P\left(\xi_{j}\right)$. There is also no need for $\xi_{j}$ to be statistically independent and even for $\eta^{n}$ to be the sequence $\left(\xi_{1}, \ldots, \xi_{n}\right)$. So what is really necessary, the asymptotic equivalence?

In order to state the property of asymptotic equivalence of non-equiprobable and equiprobable outcomes in general terms we should use the notion of entropic stability of family of random variables.

A family of random variables $\left{\eta^{n}\right}$ is entropically stable if the ratio $H\left(\eta^{n}\right) / H_{\eta^{n}}$ converges in probability to one as $n \rightarrow \infty$. This means that whatever $\varepsilon>0, \eta>0$ are, there exists $N(\varepsilon, \eta)$ such that the inequality
$$P\left{\left|H\left(\eta^{n}\right) / H_{\eta^{n}}-1\right| \geqslant \varepsilon\right}<\eta$$
is satisfied for every $n \geqslant N(\varepsilon, \eta)$.
The above definition implies that $0<H_{\eta^{n}}<\infty$ and $H_{\eta^{n}}$ does not decrease with $n$. Usually $H_{\eta^{n}} \rightarrow \infty$.

Asymptotic equiprobability can be expressed in terms of entropic stability in the form of the following general theorem.

## 澳洲代写|信息论代写information theory代写|Definition of entropy of a continuous random variable

Up to now we have assumed that a random variable $\xi$, with entropy $H_{\xi}$, can take values from some discrete space consisting of either a finite or a countable number of elements, for instance, messages, symbols, etc. However, continuous variables are also widespread in engineering, i.e. variables (scalar or vector), which can take values from a continuous space $X$, most often from the space of real numbers. Such a random variable $\xi$ is described by the probability density function $p(\xi)$ that assigns the probability
$$\Delta P=\int_{\xi \varepsilon \Delta X} p(\xi) d \xi \approx p(A) \Delta V \quad(A \in \Delta X)$$
of $\xi$ appearing in region $\Delta X$ of the specified space $X$ with volume $\Delta V(d \xi=d V$ is a differential of the volume).

## 澳洲代写|信息论代写INFORMATION THEORY代写|Properties of entropy in the generalized version. Conditional entropy

Entropy (1.6.13), (1.6.16) defined in the previous section possesses a set of properties, which are analogous to the properties of an entropy of a discrete random variable considered earlier. Such an analogy is quite natural if we take into account the interpretation of entropy (1.6.13) (provided in Section 1.6) as an asymptotic case (for large $N$ ) of entropy (1.6.1) of a discrete random variable.

The non-negativity property of entropy, which was discussed in Theorem $1.1$, is not always satisfied for entropy (1.6.13), (1.6.16) but holds true for sufficiently large $N$. The constraint
$$H_{\xi}^{P / Q} \leqslant \ln N$$
results in non-negativity of entropy $H_{\xi}$.
Now we move on to Theorem $1.2$, which considered the maximum value of entropy. In the case of entropy (1.6.13), when comparing different distributions $P$ we need to keep measure $v$ fixed. As it was mentioned, quantity (1.6.17) is non-negative and, thus, (1.6.16) entails the inequality
$$H_{\xi} \leqslant \ln N$$
At the same time, if we suppose $P=Q$, then, evidently, we will have
$$H_{\xi}=\ln N$$

## 澳洲代写|信息论代写INFORMATION THEORY代写|ASYMPTOTIC EQUIPROBABILITY AND ENTROPIC STABILITY

P\left{\left|H\left(\eta^{n}\right) / H_{\eta^{n}}-1\right| \geqslant \varepsilon\right}<\etaP\left{\left|H\left(\eta^{n}\right) / H_{\eta^{n}}-1\right| \geqslant \varepsilon\right}<\eta

## 澳洲代写|信息论代写INFORMATION THEORY代写|DEFINITION OF ENTROPY OF A CONTINUOUS RANDOM VARIABLE

Δ磷=∫XeΔXp(X)dX≈p(一个)Δ在(一个∈ΔX)

HX磷/问⩽ln⁡ñ

HX⩽ln⁡ñ

HX=ln⁡ñ

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