 So, we are going to now start understanding what we mean by convergence of a sequence of random variable, right. So, we already known like when you have a given sequence what it converges to and the limits are very important because many of the times like integral differential functions are all defined in terms of the limits. And also when I have a process let us say I have modeled it as some stochastic process which are indexed by time, I want to understand how this process evolves over time, right. So, I would be interested in knowing as I let my t go to infinity how my process evolves and behaves. Suppose you are investing your money in a gamble every day you are going to win or lose and based on that every day you have some money left with you and you want to understand eventually if I continue to play the same game eventually as the number of place becomes large will I end up with positive amount with me or it will all go to 0 or it becomes negative. So, you want to understand as time evolves or in my stochastic processes as the index becomes large how my process looks like is there any limiting behavior in that. But then the question is fine we understand what we mean by convergence of a sequence of random variable sorry convergence of a sequence of numbers then what does we mean by convergence in random variables, right. So, we need to make that notion of convergence of a sequence of random variables precise and that is what we will focus on may be in the next two to three classes. So, let us say I have a sequence of random numbers random variables all random variables defined on the same way. So, another thing is I am going to only focus on our random processes here or the sequence of random variables which are indexed by discrete which are indexed by discrete numbers. So, here n is 1, 2, 3, 4 like that. Now, the first notion of sorry convergence in almost sure sense we are going to say the sequence of random variables xn they converge almost surely if probability of limit as tends to infinity of this x is going to take the value x. So, what I mean by this limit here I have already said that limit of xn is equals to x, right. So, what I mean by this here suppose you fix omega belongs to omega and then you are going to look at xn of omega is this a deterministic sequence xn of omega s, right. If I fix an omega this is because xn is a function of from capital omega to r, right. So, if you fix an omega this is some real number and I have a sequence of such real numbers and I know what I mean by convergence of this sequence of real numbers, right that is a standard convergence and whatever is the limit. Now, what we are saying is if the meaning of this is if limit of xn of omega. So, this is the short form for this I am going to look at omega and see whether the sequence of xn omega s converge to my limited random variable and then look if wherever that convergence happens I am going to look at all this omega s and then look at whether that probability is equals to 1. If this happens then I am going to call my xn convergence to x almost surely. So, let us look an example. So, let us say I am going to define a sequence of random variables like this, okay. So, before that I want to introduce this notion of unit interval probabilities. So, earlier we have already defined the notion of what is a event space, what is sigma algebra and what is what we mean by probability space and all right. So, now I want to like define a special probability space which is as following. Let omega is that omega is so omega is basically interval and then in this outcome omega is drawn from omega no preference towards subset, okay. In a way like I am saying that I am going to draw omega from this sample space in a way uniformly right without giving any preference to any of this subset. Then I am going to look I am going to now start constructing my event space on this let us say f to include all intervals. What I mean by this I am going to say that let I am going to take. So, these are all possible intervals right take a and b and which is a is between 0 1 and also b is between 0 1 this is going to define one interval and let us all possible intervals be contained in this event space, okay. Now that I have been drawing omega from this capital omega without any preference to any of this subset then one natural way I am going to assign probabilities to this interval says b by a, okay. Assume that b is going to be greater than or equals to a. Now I have defined omega I have defined my f partially here because I only said it includes closed I mean this intervals here. But if it has to be a sigma algebra then I know that all its it has to satisfy the properties of sigma algebra which says that complements should be there and their unions finitely many and countably many unions should all be there right. So, if intervals are there then the open sets are also there in the sigma algebra f right. I will have open intervals in this I will have closed intervals in this by taking their unions I will have sets which are open at one end and closed at other end and I will have all such kind of combinations if I have to look at a sigma f. Now when I have a such a sigma then to make this completely probability space I also need to say how I am going to define probability for each of the elements in my f right for intervals it is easy I have defined like that. But if I am going to look at all possible elements that are in f how I am going to define it. So, if you are going to say b equals to a then you have only one element and for which you are saying 0. I am saying you take any interval here for which we have defined like this and you can go like this and if you want this f to be sigma algebra it will have all possible subsets of my 0 1 interval right. So, the worst case what you can take one we can take this f to be a power set of this that means it includes all subsets of the interval 0 1. But here is some technicality here when you do this it so happens that we will end up with certain sets for which we will not be able to consistently assign probabilities I mean that comes from some complicated analysis or some better understanding of the real sets but we are not going to that. But what we are going to take is when this f to include what we are going to take this is f to be the smallest sigma algebra containing sub intervals. So, we take f to be the smallest sigma algebra. So, you can come up with many many sigma algebra which will contain all these intervals right. But we are going to take that sigma algebra to make this two to define this which is the one which contains the which contains all sub intervals of omega. Is this point clear? So, we are going to take all so the way we are defining f is let it include all the intervals to make it sigma algebra we also has to allow it to possibly include all open sets, their unions, their intersections so many combinations are there. So, you may end up with so many sigma algebras which so many f's which may it on its own satisfy the properties of sigma algebra. But among all them we will take the one to include the smallest one which includes all the sub intervals. So, this is just to make this bit more formal and then such a when we have such an f which contains all the smallest all these sub intervals we know how to assign probability to them and using that we can up try to define the probability for each of the element in that f and such probability space we are going to call it as unit interval probability. So, for all practical purpose what we mean by unit interval probability space is my sample space is unit interval and my sigma algebra is such that it contains all possible intervals and on each of the intervals there I am going to assign probability like this if I have an interval. So, we only need to take so our understanding of unit probability space will be just this. So, but it has something more to it in terms of how this f is defined in terms of the smallest sigma algebra containing all the sub intervals. Fine, this was just a detour and this is what our understanding of unit interval probability space. Now to understand our notion of convergence we will be looking at the examples or random variables defined on this unit interval probability space because this is going to be easier for us to understand. So, I am going to take let omega fp be what I call as unit interval probability space. Now I am going to define my random variables like this fix an n and that n is going to be such that for any omega that is coming from capital omega it is going to be defined like this. Now let us try to map this. So, let us say this is x1 and this is my omega. So, how does x1 look like? It is going to be linear curve right it is going to be linear and going hitting at 1. And how does x2 look like? It is like a quadratic right and it goes to 1 and how does x3 look like? So, it is curvature will open up this side or that side it is going to be right like this. Now let us try to apply this definition here and see where it will converge. So, let us before this as we saw that as we move from 1 to 3 the curvature is opening towards the right and you as n goes to infinity how does this curve look like? So, it is going to almost 0 till this point and at omega equals to 1 it is going to be 1. So, in this case in a way as n is standing to infinity we see that this sequence of graphs here are converging to a place where it is all 0 and then it suddenly shoots up to 1 at omega equals to 1. So, let us take that to be our limiting x. So, then let us try to analyze whether this property holds and in this case can we call xn converges to that x. So, the claim is we want to check whether xn converges to x in almost surely where x is at omega equals to 1 this is my x. So, now can you verify and see whether this guy satisfies this property whether this is true or wrong. So, let us take a omega which is not at. So, let us take omega which is not 0 not 1. If you take any omega not 1 that means it is strictly less than 1 what is going to happen xn of omega will converge to what? It is going to converge to 0 and x is also 0 in that range and if you take omega equals to 1 what is this going guys are going to converge xn of omega they are going to be 1 and what is this guy is this is going to be 1 right. And then it looks like so all omega which are between 0, 1 are going to satisfy this property. So, they are included in this set right and what is the probability of that set? What is the probability of that set? 1 right because every point in omega is omega has satisfied this and so probability of big omega is 1 that we already know. So, by this by this definition we already know that this example converges to x which is like this almost surely. Just let me check this at the point omega equals to 1 at omega equals to 1 these guys are all this guys are all 1, 1, 1 and so fine if I have defined my x equals to like this which is 1 only at omega equals to 1. So, let us say I am going to define my x to be in a slightly different fashion I am going to take my x to be 0 all the way even at omega equals to 1 it is not jumping at all here. Is it true that in this case let me call this as x prime here? Is it true that my xn converges to x prime almost surely? No. Why? Yeah, but I do not care about 1 points right what I care about this probability. So, as you said other than omega equals to 1 everywhere this holds only omega equals to 1 not included. So, what is the probability of this set where all omegas are included except omega equals to 1? It is going to be still 1 right because probability of that singleton 1 is going to be 0. Is it true that in that case my xn converges to x prime also here because that singleton values did not have any mass in our example here. So, both like both are valid like I can say that both converge to this random variables. So, that is fine. Another thing we will write away notice is when we have a deterministic sequence of random variables whether my limit was always unique was it like is it possible to let us say I have a deterministic sequence an can it have two limits. So, then limit is always unique right but when we are talking about converges of this random variable that is not the case right. So, here x is the when I said xn converges to x this is my limiting random variable and these are my sequence of random variables. So, it is fine that is not the case that I have only one unique random variable. So, but as you will see that this is only at the points which carry 0 mass. So, in that way this random variable and this random variable on this probability space they are identical because they only differ at points which are 0 mass.