In the previous post, we mentioned Markov chains and a closed communicating class. If we have a Markov chain with transition matrix P and fix n transitions from the overall N, then the Markov chain with transition matrix Pn has exactly the same closed communicating classes. Recall that each single-element set from the states is a closed communicating class and by definition a closed communicating class has all the states that the origin leads to. A general Markov chain can have an infinite number of communicating classes. This is something we see from sets and relations. When we visualize the topological structure of a Markov chain, we will see closed communicating classes that have no arrows for communication and arrows between two non-closed communicating classes that are in the same direction. If we remove the non-closed communicating classes, we can get disjoint closed communicating classes.
If we have an irreducible chain where all the states form a single closed communicating class, we can further simplify the behavior of the chain by noticing that if the chain is in one set it can move to only a few other sets. For example, if we take the set with four communicating classes, the diagonal states will transition to the opposite. This we say has a period of two. A triangular three communicating class will have a period of three. The numbers two and three are indicative of a divisor in the communicating classes. If an integer n divides m and the reflexive state transitions are possible with n transitions, then its possible with m transitions. So, whether it is possible to return to i in m steps can be decided by one of the integer divisors of m. The period of the state is defined as the greatest common divisor of all natural numbers n with such that it is possible to return to i in n steps. A state i is called aperiodic if it can be returned to only one step.

The topological structure of a Markov chain can be represented as a digraph and denoted by a pair (V,E ) with V being the vertices and E being the edges of the graph. The graph of a chain refers to the structure of the process that does not depend on the exact values of the transition probabilities but only on which of them are positive. We say that a state i leads to state j if, starting from i the chain will visit j at some finite time.
We denote this by i ~ j and it is equivalent to stating the probability for some final state j after n iterations starting from i is greater than zero. This relation is reflexive. In fact for all the states in the event space, we can have probabilities between pairs to be greater than zero.
This relation is also transitive. If we can reach state j from i and k from j then it implies that there is a chain possible from i to k .
If it's a symmetric relationship which means that there is a chain from i to j and j to i then we can say that i communicates with j.
The communication relationship is an equivalence relation which means it is symmetric, reflexive and transitive. Equivalence classes also called as communicating classes partition the space. The communicating class corresponding to state i is, by definition the set of all states that communicate with i.
Two communicating classes are either identical or completely disjoint.
A communicating class is said to be closed when all the states belong to the communicating class. Closed communicating classes are particularly important because they decompose the chain into smaller more manageable parts.The single element state i is essential if it belongs to closed communicating class. Otherwise the state is inessential. If all the states communicate with all others, then the chain is considered irreducible.

Today we will continue to discuss Skorokhod theorem. Recall that the theorem mentions the stopping time in terms of a probability from the distribution.  The simpler case was the symmetrical random walk but Skorokhod embedding theorem works even better for the distribution. Moreover we use it for describing a random walk with a new random variable that can take values above a value a and below b.

Today we will continue to discuss random walks. In the previous post, we were proving Skorokhod embedding theorem. We introduced a random variable for a simple random walk that takes a value x greater than the value taken by random variable A but not reaching the value taken by the random variable B and assuming that this walk is independent of both A and B. The claim was that this new random variable takes a probability pk from the distribution and thus leading to the said theorem that the stopping time takes a probability from the distribution. To make the claim, we looked at the initial values when k = 0, then Z = 0 if and only if A = 0 and B = 0 and this has probability p0 by definition. Then we take k > 0 and in this case we work out the probability that our random variable takes the value k.  We first expand this probability as the cumulative sum of the independent probabilities over all i < 0 and j > 0 . The independent probabilities are for the random variable to take a value k and for A to take value i and B to take value j.  We can eliminate j let B take value k for the probability we are calculating. Then we apply the theorem from the gambler's ruin that describes the probability to take value b > x before reaching a  < x as equals (x - a) / (b - a). Further we use the modified expression of that probability in terms of a random walk Ta ^ Tb  to reach level b as the same probability and equaling  (-a / (b-a)) or (-i/(k-i)) as in this case.  So we simplify the independent probabilities with this value and the normalization factor times (k-i) pi pk. Since the sum of this has a zero average component, it simplifies to the probability pk thus proving the theorem.

In the previous post, we mentioned the Skorokhod embedding theorem. This theorem gives the probability for the stopping time T as one of the probabilities in the distribution.  In this post, we try to prove it.  We define a pair of random variables (A,B) that takes values anywhere in the N x N grid with the following distribution:
the initial state with A=0 and B=0 to have a probability p0 and the
any other state with A=i and B=j to have a probability as a normalization factor times j-i times the combination of the independent probabilities for i and j from the distribution.
We can now cumulate all the other states to have a probability of 1 - p0. We can split the (j-1) to two separate terms.
Using the zero mean equation in the cumulation equation, we can apply the above two for all i < 0 < j, we get that the normalization factor is the inverse of this common value : sum of i and pi over all i >  0
Assuming the stopping time as an infinite series of Sn = i and (A,B) to be independent of Sn, we can take a new random variable Z in terms of the intersection of the random walk to reach value TA and before it reaches TB.
The claim is that the probability for this random variable to take a value k is the corresponding probability from the distribution.  For k = 0,  Z = 0 has a probability p0. For k > 0 we can use the theorem that computes the probability of the random variable we defined and we see that this has a value pk.

Today we will look at Skorokhod embedding theorem. We consider a simple symmetric random walk where the probability for an event with value v is 1/2 and that against is 1/2. When we sample, we want the probability that the gambler's fortune will reach level b > x before reaching level a < x  is given by (x-a)/(b-a). When we start the random walk from S0 = 0, the probability will reach level b > x before reaching level a < x equals -a / (b-a) and the probability that it will reach level a < x before b > x is b / (b-a).
These same probabilities can now be read as simple symmetric random walks  with distribution as the range between a and b to reach b and a respectively.
We view the random walk this way. Given a 2 point probability distribution (p, 1-p) such that p is rational, we can always find integers a,b a < 0 < b such that pa + (1-p)b  = 0.
This means we can simulate the distribution by generating a simple symmetric random walk starting from 0 and watching it till the first time it exits the interval [a,b] The probability it exits from the left point is p and the the probability it exits from the right point is 1-p. Having introduced the theorem, lets look at some more complicated probability distribution.
If the probabilities were to be many in a distribution, can we find a stopping time T such that the random walk has the given distribution ? This is where Skorokhod embedding theorem helps.
This theorem states that for a simple symmetric random walk and (pi, i belongs to Z) a given probability distribution on Z with zero mean which means satisfying the equation sum of i.pi = 0, then there exits a stopping time T where the probability for random walk to arrive at i  = pi

Today we will continue to discuss random walks. We discussed single dimension and two dimension random walks so far.We saw that the two dimensional random walk can be expressed in terms of random walks in single dimensions that are independent. We proved this by enumerating the probabilities of the transitions and by considering them relative to origin. We now look at another lemma that states the probability for the two dimensional random walk to return to zero after 2n times is equal to twice that of the simple symmetric random walk (ssrw) to return to zero after 2n times. This intuitively follows from the first argument we made where we described the two dimensional random walk to be expressed in terms of the two independent ssrw. Further more, both the ssrw have the same probabilities for the final state we are interested in. It follows that the probability for the two dimensional ssrw is twice that of the same for a ssrw which in this case is the count of number of paths from (0,0) to (2n,0) to the overall paths and works out to be 2n Choose n times 2 ^ -2n. The probability for the final state for the two dimensional random walk is double that.
We next look at recurrence
Remember that Markov chain has finitely many states and at least one state must be visited infinitely many times. Perhaps some states will be visited again and again infinitely. Such a state is called recurrent. If the chain returns to a state i infinitely many times starting from i, we say that the Pi(Xn = i for infinitely many n) = 1
In the two dimensional symmetric random walk, from the previous lemma we can say that the recurrence probability holds true for both sides as it tends to 1 for infinitely many series and given the probability we worked out and substituting for different n and approximating with Stirling's approximation to c/n for some constant c, we see that the recurrence holds true for two dimensional random walk.