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Limit Theorems

Limit Theorems in Probability are useful for solving the complex Probability and Statistics problems. Here we will discuss some important limit theorems used in probability and statistics:
There are following limit theorems in probability and statistics -
Theorem 1: If event E(Xi Xj) = EXi E Xj, where ‘I’ is not equals to ‘j’, then Set of random variables Xi with Exi2 < ∞ are not correlated.
Theorem 2: If P(∩j=1n ) Xj = πj=1n Xj,
Then set of random variables are independent.
Theorem 3: If Xi and Xj are independent, ‘I’ is not equals to ‘j’.
Then the set of random variables Xj is pair wise independent.
Theorem 4: If S = X1 + X2 +........... +Xn,
Then the set of random variables, X1, X2,...........Xn are not correlated.
Theorem 5: If a set of random variables X1, X2,...........Xn are sequence of integrable and independent Random Variable, then 1 / n . ∑m=1 n (Xm – E Xm) - > 0.
Theorem 6: If a set of random variables X1, X2,...........Xn, are taking values from Sample Space ‘S’, then Xn = X.
These are six basic and important limit theorems probability. 

Topics Covered in Limit Theorems

Weak Law of Large Numbers

The Law of Large Numbers is a theorem in the theory of Statistics and Probability. The weak law of large number gives us information about the change in result if we perform the same experiment a large number of time. In this we expect a value that should come which is called the expected value. The objective is to obtain the average of the results from a large...Read More

The Strong Law of Large Numbers

The Probability consists of different laws and various theorems known as the Laws of large Numbers. There are two categories of the laws of large numbers.
1.      Weak law of large number
2.      Strong law of large number
We are going to clarify relationship between weak and the strong law of large numbers. The laws of large numbers make statements ab...Read More

Bernoulli Process

The Bernoulli process is a process which is a finite or infinite sequence of binary random variables (binary implies that it can only have 0 or 1 as its value). It is a discrete time stochastic process. The discrete time stochastic process only takes binary values (two values which are 0 and 1). The Bernoulli variables are identical and also do not depend on other vari...Read More

converge in probability

The concept of Convergence in Probability is based on different methods of measuring distance between two random variables (means how close are Random Variable to each other).
The concept of convergence in probability can be defined as- “Two random variables are ‘close to each other’ if there is a high probability that the difference between them is very smal...Read More

Central Limit Theorem

The Probability Theory is a part of mathematics. The CLT that is Central Limit Theorem is given from the Probability theory. This theorem is first version of this theorem was given by Sir Abraham De Moivre.
This theorem tells us about the conditions under which a Mean is taken of the large number of random variables with variance; will be normally distributed. The...Read More

Markov and Chebyshev Inequalities

Markov and Chebyshev inequalities are topics covered in Probability. Let’s look at Markov and Chebyshev inequalities in detail. In Markov inequalities a non negative random number and its expected value is given, whereas in case of Chebyshev inequalities the random variables will be having an expected value and variance, Chebyshev also provides a region...Read More

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