Good advice: stop reading this section after c! Results b and c are proved in the appendix to this chapter. We shall see in the appendix that the more correct measurability condition on I is that I be 'Bn-measurable'.
Warning - for the over-enthusiastic only. It is the latter rather than the former which gives the appropriate type of f in d. Generally, we shall not use the theorem in the main text, preferring 'just to use bare hands'. However, for product measure in Chapter 8, it becomes indispensable.
Then il 1i contains the indicator function of every set in some 71"system I, then 1i contains every bounded a I -measurable function on S. For proof, see the appendix to this chapter. Chapter 4- Independence Let n,. Definitions of independence Note. We focus attention on the q-algebra formulation and describe the more familiar forms of independence in terms of it to acclimatize ourselves to thinking of IT-algebras as the natural means of summarizing information.
Section 4. We now use the Uniqueness Lemma 1. Let us concentrate on the case of two a-algebras. Suppose that 9 and 1i are sub-a-algebras of F, and that I and.
Then 9 and 1i are independent if and only if I and. Proof Suppose that I and. For fixed I in I, the measures check that they are measures! By Lemma 1. They therefore. Command: Do Exercise E4.
By combining in an appropriate way think about this! I have included in the appendix to this chapter the statement of a truly fantastic theorem about precise description of long-term behaviour: Sirassen's Law. Independence A number of exercises in Chapter E are now accessible to you. Equation 0. The trick answer based on the existence of Lebesgue measure given in the next section does settle the question.
A more satisfying answer is provided by the theory of product measure, a topic deferred to Chapter 8. The existence of two different expansions of a dyadic rational is not going to cause any problems because the set 0 say of dyadic rationals in [0,1] has Lebesgue measure 0 - it is a countable set! We now need a bit of common sense. Since the sequence has the same 'coin-tossing' properties as the full sequence w n is clear that Y1 has the uniform distribution on [0,1]; and similarly for the other Y's.
Now suppose that a sequence Fn : n E N of distribution functions is given. By the Skorokhod representation of Section 3. But because the V-variables are independent, the same is obviously true of the X -variables.
Satisfy yourself that you could if forced carry through these intuitive arguments rigorously. Obviously, this is again largely a case of utilizing the Uniqueness Lemma 1. Of course, we now know that for any given distribution function F, we can construct a triple n,F,p carrying a sequence of lID RVs with common distribution function F. In particular, we can construct a rigorous model for our branching process.
The fundamental question about existence of a stochastic process with prescribed joint distributions is to all intents and purposes settled by the famous Daniell-Kolmogorov theorem, which is just beyond the scope of this course. We think of Xn as the value of the process X at time n.
A very important example of a stochastic process is provided by a Markov chain. By a time-homogeneou3 Markov chain Z Exercise. Give a construction of such a chain Z expressing Zn w explicitly in terms of the values at w of a suitable family of independent random variables.
See the appendix to this chapter. Here is a silly example, to which we apply a silly method, but one which both illustrates very clearly the use of the monotonicity properties of measures in Lemma 1. See the 'Easy exercise' towards the end of this section for an instantaneous solution to the problem. Let us agree that correctly typing WS, the Collected Works of Shakespeare, amounts to typing a particular sequence of N symbols on a typewriter.
A monkey types symbols at random, one per unit time, producing an infinite sequence Xn of lID RVs with values in the set of all possible symbols. Let H be the event that the monkey produces infinitely many copies of WS.
Hence, P Hm,k P H. Hence, by Lemma 1. However, as k r 00, Hk! H, and so, by Lemma 1. Fortunately, it does not tell us which - and it therefore generates a lot of interesting problems! Easy exercise. Let El be the event that the monkey produces WS right away, that is, during time period [1, N]. Then P El 2': eN. Tricky exercise to which we shall return. The next three sections involve quite subtle topics which take time to assimilate.
They are not strictly necessary for subsequent chapters. The Kolmogorov law is used in one of our two proofs of the Strong Law for lID RVs, but by that stage a quick martingale proof of the law will have been provided. Perhaps the otherwise-wonderful 'lEX makes its T too like I. Below, I use K instead of I to avoid the confusion. Script X, X, is too like Greek chi, X, too; but we have to live with that. Independence 4. Prove that FI ,F2 and F3 are are T-measurable, that the event H in the monkey problem is a tail event, and that the various events of probability 0 and 1 in Section 4.
Hint - to be read only after you have already tried hard. Look at F3 for example. That Fin E Tn now follows from Lemmas 3. Proof of i.
Step 1: We claim that Xn and Tn are independent. Proof of claim. But the assumption that the sequence Xk is independent implies that K and:r are independent. Lemma 4. Step 2: Xn and T are independent. Moreover, Koo and T are independent, by Step 2. Step 4. So, suppose that c is finite. The examples in Section 4. The Three Series Theorem Theorem So, you can see that the law poses numerous interesting questions. In the branching-process example of Chapter 0, the variable is measurable on the tail u-algebra of the sequence Zn : n E N but need not be almost deterministic.
But then the variables Zn : n E N are not independent. Prove that Hint. The phenomenon illustrated by this example tripped up even Kolmogorov and Wiener. The very simple illustration given here was shown to me by Martin Barlow and Ed Perkins. Deciding when we can assert that for Y a u-algebra and Tn a decreasing sequence of u-algebras is a tantalizing problem in many probabilistic contexts.
Chapter 5 Integration 5. Notation, etc. It is worth mentioning now that we shall also use the equivalent notations for A E E: with a true definition on the extreme right!
Something else worth emphasizing now is that, of course, summation is a special type 01 integration. We then define b with 0. Checking all the properties just mentioned is a little messy, but it involves no point of substance, and in particular no analysis.
We skip this, and turn our attention to what matters: the Monotone-Convergence Theorem. Hence, using 1. We shall see that other key results such as the Fatou Lemma and the Dominated-Convergence Theorem follow trivially from it. The MON theorem is proved in the Appendix. Obviously, the theorem relates very closely to Lemma 1. The proof of MON is not at all difficult, and may be read once you have looked at the following definition of o: r.
Chapter 5: Integration 52 Often, we need to apply convergence theorems such as MON where the hypothesis fn i I in the case of MON» holds almost everywhere rather than everywhere.
Let us see how such adjustments may be made. Now let r i 00, and use MON. The result now follows from MON. We do not bother to spell out such extensions for the other convergence theorems, often stating results with 'almost everywhere' but proving them under the assumption that the exceptional null set is empty.
So f is Lebesgue measurable see Section Al. Chapter 5: ,Integration 5. This is a totally routine consequence of the result in Section 5. Chapter 6 Expectation 6. Expectation is just the integral relative to P. See Section 6. That our present definitions agree with those in terms of probability density function if it exists etc.
With the notation of Section 5. The fact that P u 9 ::; P u9 now yields o Proof of b. This we do. But then, if a r is our familiar staircase function, then where the sums are over finite parameter sets, and where for each i and j, Ai in o- X» is independent of Bj in o- Y».
This is not necessarily true when X and 71 7. It is important that independence obviates the need for such inequalities. Strong Law - first version The following result covers many cases of importance.
You should note that though it imposes a 'finite 4th moment' condition, it makes no assumption about identical distributions for the Xn sequence. It is remarkable that so fine a result has so simple a proof. Then 0 a. But we can also prove a immediately by the elementary inequality 6. Thus using Theorem 7.
Let Xk , Sn etc. Now f is bounded on [0,1], If y 1 ::; K, Vy E [0,1]. Then Zn ::; 8 implies that Yn 8 ::;! Earlier, we chose a fixed 8 at a. Except for the matter of infinite products, it is all a case of relentless use of either the standard machine or the Monotone-Class Theorem to prove intuitively obvious things made to look complicated by the notation.
When you do begin a serious study, it is important to appreciate when the more subtle Monotone-Class Theorem has to be used instead of the standard machine. Suppose now that SI and S2 are metrizable spaces and that! If SI and S2 are separable, then! However, if SI and S2 are not separable, then 8 S may be strictly larger than! It is perhaps as well to be warned of such things. It proves to be very advantageous to look at this idea in a new way.
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