Strong Law of Large Numbers for Iterates of Some Random-Valued Functions

Assume $$ (\Omega , {\mathscr {A}}, P) $$ ( Ω , A , P ) is a probability space, X is a compact metric space with the $$ \sigma $$ σ -algebra $$ {\mathscr {B}} $$ B of all its Borel subsets and $$ f: X \times \Omega \rightarrow X $$ f : X × Ω → X is $$ {\mathscr {B}} \otimes {\mathscr {A}} $$ B ⊗ A -measurable and contractive in mean. We consider the sequence of iterates of f defined on $$ X \times \Omega ^{{\mathbb {N}}}$$ X × Ω N by $$f^0(x, \omega ) = x$$ f 0 ( x , ω ) = x and $$ f^n(x, \omega ) = f\big (f^{n-1}(x, \omega ), \omega _n\big )$$ f n ( x , ω ) = f ( f n - 1 ( x , ω ) , ω n ) for $$n \in {\mathbb {N}}$$ n ∈ N , and its weak limit $$\pi $$ π . We show that if $$\psi :X \rightarrow {\mathbb {R}}$$ ψ : X → R is continuous, then for every $$ x \in X $$ x ∈ X the sequence $$\left( \frac{1}{n}\sum _{k=1}^n \psi \big (f^k(x,\cdot )\big )\right) _{n \in {\mathbb {N}}}$$ 1 n ∑ k = 1 n ψ ( f k ( x , · ) ) n ∈ N converges almost surely to $$\int _X\psi d\pi $$ ∫ X ψ d π . In fact, we are focusing on the case where the metric space is complete and separable.

Legal Comparison against the Death Penalty Sanctions regulated in the Positive Laws of Indonesia and China

Corruption is a severe problem worldwide, so it takes a strict rule of law and strong law enforcement efforts to eradicate it. China's legal system has proven to be effective in reducing corruption among state officials. One of China's anti-corruption efforts is to impose harsh penalties on perpetrators, including the death penalty. In light of this success, this study will conduct a legal comparison with the death penalty, which is regulated in Indonesian and Chinese positive law. The type of research used is normative juridical with a conceptual and statute approach. The data processed in this study include primary data and secondary data with data collection techniques and management using a literature review. The findings of this study highlight the threat of the Death Penalty, as outlined in Article 2 Paragraph (2) of the Corruption Crime Act, which focuses on corrupt acts committed under certain conditions. In Indonesia, no one has ever been sentenced to death for corruption. The People's Republic of China's Criminal Law of the death penalty threat has existed since 1900 AD. Article 383 of the Chinese Criminal Code stipulates that anyone who accepts bribes is subject to the death penalty. A significant difference from this Comparison lies in the classification of capital punishment with a corruption amount of more than 50,000 Yuan and for bribery cases in Chinese regulations. Meanwhile, there is no such regulation in Article 2 paragraph (2) of the Indonesian Corruption Laws.

Some strong limit theorems for weighted sums of measurable operators

The aim of this study is to provide some strong limit theorems for weighted sums of measurable operators. The almost uniform convergence and the bilateral almost uniform convergence are considered. As a result, we derive the strong law of large numbers for sequences of successively independent identically distributed measurable operators without using the noncommutative version of Kolmogorov’s inequality.

The Dynamics of Canal Colonies; Agricultural Development and Socio-Political Change in British Multan

Punjab was the last province which annexed with British India in 1949.British implemented many reforms in sub-continent, but Canal Colonies were excellent irrigation system introduced by the British, especially in the south west part of the Punjab province. These canal colonies proved very useful for the agriculture sector which boost up the not only the economic condition of the native people of the colonies areas but also increase revenue collection of the British government in India. Canal Colonies changed the arid/ barren land of the southern district of the province into a fertile land. British government also achieved their objectives through canal colonies. Therefore, government allotted the land of the canal colonies in different categories and rewarded some specific classes in native communities of the canal colonies. In canal colonies government reserved huge piece of land for their military objective because Indian Military was necessary for the strong law and order in British India and their strong influence in Middle East and other parts of the world. These canal colonies bring an agriculture revolution in the province which change the socio-economic condition of the people of the native districts. In Multan three major canal colonies were introduced, Sidhnai, Lower Bari Doab and Nili Bar canal colony. In these canal colonies British government also implemented same polices like the previous canal colonies of the province. Therefore, the in Multan district landed elites were loyal and cooperative with government with their wealth and men. These landed elites played important role at political canvas of the province during British rule in Multan.

Permutation Invariant Strong Law of Large Numbers for Exchangeable Sequences

We provide a permutation invariant version of the strong law of large numbers for exchangeable sequences of random variables. The proof consists of a combination of the Komlós–Berkes theorem, the usual strong law of large numbers for exchangeable sequences, and de Finetti’s theorem.

The Strong Law of Large Numbers

This chapter focuses largely on methods of proof of the strong law, building on the fundamental convergence lemma. It covers Kolmogorov's three‐series theorem, strong laws for martingales, and random weighting. Then a range of strong laws are proved for mixingales and for near‐epoch dependent and mixing processes.

Illegal land clearing and green economy campaign in Indonesia

Indonesia’s total area of forest and waters is around 130.68 million hectares, but it consistently decreases yearly. The illegal land clearing for agricultural activity is considered as the main cause of the forest reduction, as around 5.6 million hectares of land burned between 2015 and 201 and it costs of hundreds of trillions to the country. In order to lessen the negative impacts of land clearing, Indonesian government indeed has started to promote a green economy campaign, known as sustainable development with an environmental perspective. This study aims to examine the relationship between illegal land clearing and its effect on the Indonesian economy and its relationship with the green economy campaign initiated by the Indonesian government. We use the data from the Indonesian Ministry of Environment and Forestry, Central Bureau of Statistics, Greenpeace Indonesia, and World Bank to analyze this issue. We find that the governments’ poor regulation and monitoring have caused forest and land fires almost every year and it has huge economic and environmental costs that should be paid by the government. Therefore, a strong law enforcement is needed so that the green economy campaign in Indonesia can be executed properly.