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7. Convergencia

En esta sección discutimos varios tópicos que son un poco avanzados, pero son muy importantes. En particular los resultados obtenidos em esta seccoón serán esenciales para establecer

• propiedades de las funciones de distribución,
• la ley débil de los grandes números ,
• la ley fuerte de los grandes números.

Cotas y Límites

We first start with some concepts from real analysis that we will need. First, if A is a subset of R, recall that the infimum (or greatest lower bound) of A, denoted inf A is the number u satisfying

1. u x for any x in A (u is a lower bound for A).
2. if v x for any x in A then v u (u is the greatest lower bound).

Similarly, the supremum (or least upper bound) of A, denoted sup A is the number w satisfying

1. x w for any x in A (w is an upper bound for A).
2. if x z for any x in A then w z (w is the least upper bound).

The infimum and supremum of A always exist, either as real numbers or as infinity or negative infinity. Now suppose that a_n, n = 1, 2, ... is a sequence of real numbers.

1. Show that inf{a_k: k = n, n + 1, ...}, n = 1, 2, ... is an increasing sequence.

The limit of the sequence in the last exercise is the limit inferior of the original sequence a_n:

lim inf_n a_n = lim_n inf{a_k: k = n, n + 1, ...}.

2. Show that sup{a_k: k = n, n + 1, ...}, n = 1, 2, ... is an decreasing sequence.

The limit of the sequence in the last exercise is the limit superior of the original sequence a_n:

lim sup_n a_n = lim_n sup{a_k: k = n, n + 1, ...}

Recall that lim inf_n a_n lim sup_n a_n and equality holds if and only if lim_n a_n exists (and is the common value).

For the rest of this section, we assume that we have a random experiment with sample space S, and probability measure P. For notational convenience, we will write lim_n for the limit as n .

El Teorema de Continuidad para Eventos en aumentoThe Continuity Theorem for Increasing Events

Una secuencia de eventos A_n, n = 1, 2, ... se dice que es en aumento si A_n A_n+1 para cada n. La terminología es justificada considerando las variables indicadoras correspondientes.

3. Let I_n denote the indicator variable of an event A_n for n = 1, 2, ... Show that the sequence of events is increasing if and only if the sequence of indicator variables is increasing in the ordinary sense: I_n I_n+1 for each n.

Si  A_n, n = 1, 2, ...es una secuencia de eventos en aumento,nos referimos a la unión de los eventos como el  límite de los eventos:

lim_n A_n = _{n = 1, 2, ...} A_n.

Una vez más, la terminología se clarifica con las correspondientes variables indicadoras.

4. Supongamos que A_n, n = 1, 2, ... es una secuencia en aumento de eventos. Sean  I_n la variable indicador de A_n para n = 1, 2, ... y sea I la variable indicador de la unión de los eventos. Muestre que

lim_n I_n = I.

Generally speaking, a function is continuous if it preserves limits. Thus, the result in the following exercise is referred to as the continuity theorem for increasing events:

5. Suppose that A_n, n = 1, 2, ... is an increasing sequence of events. Show that

P(lim_n A_n) = lim_n P(A_n).

Hint: Let B₁ = A₁ and for i = 2, 3, ... let B_i = A_i A_i-1^c. Show that B₁, B₂, ... are pairwise disjoint and have the same union as A₁, A₂, .... Then use the additivity axiom of probability and the definition of an infinite series.

An arbitrary union of events can always be written as a union of increasing events, as the next exercise shows.

6. Suppose that A_n, n = 1, 2, ... is a sequence of events.

1. Show that _{i = 1, ..., n} A_i is increasing in n.
2. Show that lim_{n i = 1, ..., n} A_i = _{n = 1, 2, ...} A_n.
3. Show that lim_n P[_{i = 1, ..., n} A_i] = P[_{n = 1, 2, ...} A_n] .

7. Suppose that A is an event for a basic experiment with P(A) > 0. In the compound experiment that consists of independent replications of the basic experiment, show that the event "A eventually occurs" has probability 1.

The Continuity Theorem for Decreasing Events

A sequence of events A_n, n = 1, 2, ... is said to be decreasing if A_n+1 A_n for each n. Again, the terminology is justified by considering the corresponding indicator variables.

8. Let I_n denote the indicator variable of an event A_n for n = 1, 2, ... Show that the sequence of events is decreasing if and only if the sequence of indicator variables is decreasing in the ordinary sense: I_n+1 I_n for each n.

If A_n, n = 1, 2, ... is an decreasing sequence of events, we refer to the intersection of the events as the limit of the events:

lim_n A_n = _{n = 1, 2, ...} A_n.

Once again, the terminology is clarified by the corresponding indicator variables.

9. Suppose that A_n, n = 1, 2, ... is a decreasing sequence of events. Let I_j denote the indicator variable of A_j for j = 1, 2, ... and let I denote the indicator variable of the intersection of the events. Show that

lim_n I_n = I.

The following exercise gives the continuity theorem for decreasing events:

10. Suppose that A_n, n = 1, 2, ... is a decreasing sequence of events. Show that

P(lim_n A_n) = lim_n P(A_n).

Hint: Apply the continuity theorem for increasing events to the events A_n^c, n = 1, 2, ...

Any intersection can be written as a decreasing intersection, as the next exercise shows.

11. Suppose that A_n, n = 1, 2, ... are events for an experiment.

1. Show that _{i = 1, ..., n} A_i is decreasing sequence in n = 1, 2, ...
2. Show that lim_{n i = 1, ..., n} A_i = _{n = 1, 2, ...} A_n.
3. Show that lim_n P[_{i = 1, ..., n} A_i] = P[_{n = 1, 2, ...} A_n].

The First Borel-Cantelli Lemma

Suppose that A_n, n = 1, 2, ... is an arbitrary sequence of events.

12. Show that _{i = n, n + 1, ...} A_i is decreasing in n = 1, 2, ...

The limit (that is, the intersection) of the decreasing sequence in the previous exercise is called the limit superior of the original sequence A_n, n = 1, 2, ...

lim sup_n A_n = _{n = 1, 2, ... i = n, n + 1, ...} A_i.

13. Show that lim sup_n A_n is the event that occurs if and only if A_n occurs for infinitely many values of n.

Once again, the terminology is justified by the corresponding indicator variables:

14. Suppose that A_n, n = 1, 2, ... is a sequence of events. Let I_n denote the indicator variable of A_n for n = 1, 2, ... and let I denote the indicator variable of lim sup_n A_n. Show that

I = lim sup_n I_n.

15. Use the continuity theorem for decreasing events to show that

P(lim sup_n A_n) = lim_n P[_{i = n, n + 1, ...} A_i].

The result in the next exercise is the first Borel-Cantelli Lemma, named after Emil Borel and Francessco Cantelli. It gives a condition that is sufficient to conclude that infinitely many events occur with probability 0.

16. Suppose that A_n, n = 1, 2, ... is a sequence of events. Show that

_{n = 1, 2, ...} P(A_n) < implies P[lim sup_n A_n] = 0.

Hint: Use the result of the previous exercise and Boole's inequality.

The Second Borel-Cantelli Lemma

Suppose that A_n, n = 1, 2, ... is an arbitrary sequence of events. For n = 1, 2, ..., define

17. Show that _{i = n, n + 1, ...} A_i is increasing in n = 1, 2, ...

The limit (that is, the union) of the increasing sequence in the previous exercise is called the limit inferior of the original sequence A_n, n = 1, 2, ...

lim inf_n A_n = _{n = 1, 2, ... i = n, n + 1, ...} A_i.

18. Show that lim inf_n A_n is the event that occurs if and only if A_n occurs for all but finitely many values of n.

Once again, the terminology is justified by the corresponding indicator variables:

19. Suppose that A_n, n = 1, 2, ... is a sequence of events. Let I_j denote the indicator variable of A_j for j = 1, 2, ... and let I denote the indicator variable of lim inf_n A_n. Show that

I = lim inf_n I_n.

20. Use the continuity theorem for increasing events to show that

P[lim inf_n A_n] = lim_n P[_{i = n, n + 1, ...} A_i].

21. Show that lim inf_n A_n lim sup_n A_n.

22. Show that (lim sup_n A_n)^c =lim inf_n A_n^c. Hint: Use DeMorgan's law.

The result in the next exercise is the second Borel-Cantelli Lemma. It gives a condition that is sufficient to conclude that infinitely many events occur with probability 1.

23. Suppose that A_n, n = 1, 2, ... are (mutually) independent events. Show that

_{n = 1, 2, ...} P(A_n) = implies P(lim sup_n A_n) = 1.

Hint: Use the result of the previous exercise, independence, and the fact that 1 - P(A_k) exp[-P(A_k)], since 1 - x e^-x for any x.

24. Suppose that A is an event in a basic experiment with P(A) > 0. Show that in the compound experiment that consists of independent replications of the basic experiment, the event "A occurs infinitely often" has probability 1.

25. Suppose that we have an infinite sequence of coins labeled 1, 2, ... Moreover, coin n has probability of heads 1/n^a for each n, where a > 0 is a parameter. We toss each coin in sequence one time. In terms of a, find the probability that there will be

1. infinitely many heads.
2. infinitely many tails.

Convergence of Random Variables

Suppose that X_n, n = 1, 2, ... and X are real-valued random variables for an experiment. We will discuss two ways that the sequence X_n can "converge" to X as n increases. These are fundamentally important concepts, since some of the deepest results in probability theory are limit theorems.

First, we say that X_n X as n with probability 1 if

P(X_n X as n ) = 1.

The statement that an event has probability 1 is the strongest statement that we can make in probability theory. Thus, convergence with probability 1 is the strongest form of convergence. The phrases almost surely and almost everywhere are sometimes used instead of the phrase with probability 1.

Next we say that X_n X as n in probability if for each r > 0,

P(|X_n - X| > r ) 0 as n .

The phrase in probability sounds superficially like the phrase with probability 1. However, as we will see, convergence in probability is much weaker than convergence with probability 1. Indeed, convergence with probability 1 is often called strong convergence, while convergence in probability is often called weak convergence. The next sequence of exercises explores convergence with probability 1.

26. Show that the following events are equivalent:

1. X_n does not converge to X as n .
2. For some r > 0, |X_n - X| > r for infinitely many n.
3. For some rational r > 0, |X_n - X| > r for infinitely many n.

27. Use the result of the previous exercise to show that the following are equivalent

1. P(X_n X as n ) = 1
2. For every r > 0, P[|X_n - X| > r for infinitely many n] = 0.
3. For every r > 0, P(|X_k - X| > r for some k n) 0 as n .

28. Use the result of the previous exercise and the first Borel-Cantelli lemma to show that

_{n = 1, 2, ...} P(|X_n - X| > r) < for each r > 0 implies P(X_n X as n ) = 1.

Exercise 27 now leads to our main result: convergence with probability 1 implies convergence in probability.

29. Show that if X_n X as n with probability 1 then X_n X as n in probability.

The converse fails with a passion as the next exercise shows.

30. Suppose that X₁, X₂, X₃, ... is a sequence of independent random variables with

P(X_n = 1) = 1 / n, P(X_n = 0) = 1 - 1 / n for n = 1, 2, ...

1. Use the second Borel-Cantelli lemma to show that P(X_n = 0 for infinitely many n) = 1.
2. Use the second Borel-Cantelli lemma to show that P(X_n = 1 for infinitely many n) = 1.
3. Use (b) and (c) to show that P(X_n does not converge as n ) = 1.
4. Show that X_n 0 as n in probability.

There are two other modes of convergence that we will discuss later:

• convergence in k'th mean,
• convergence in distribution.

Tail Events

Let X₁, X₂, X₃, .... be a sequence of random variables. The tail sigma algebra of the sequence is

T = _{n = 1, 2, ...} sigma{X_k: k = n, n + 1, ...},

and an event B T is a tail event for the sequence X₁, X₂, X₃, .... Thus, a tail event is an event that can be defined in terms of X_n, X_{n + 1}, ... for each n.

The tail sigma algebra and tail events for a sequence of random variables A₁, A₂, A₃, .... are defined analogously (replace X_k with I_k, the indicator variable of A_k for each k).

31. Show that lim sup_n A_n and lim inf_n A_n are tail events for a sequence of events A₁, A₂, A₃, ....

32. Show that the event that X_n converges as n is a tail event for a sequence of random variables X₁, X₂, X₃, ....

The following exercise gives the Kolmogorov zero-one law, named for Andrey Kolmogorov.

33. Suppose that B is a tail event for a sequence of independent random variables X₁, X₂, X₃, .... Show that either P(B) = 0 or P(B) =1.

1. Argue that for each n, X₁, X₂, ..., X_n, B are independent.
2. From (a) argue that X₁, X₂, ..., B are independent.
3. From (b) argue that B is independent of itself.
4. From (c) show that P(B) = 0 or P(B) = 1.

From Exercises 31 and 33, note that if A₁, A₂, A₃, ... is sequence of independent events, then lim sup_n A_n must have probability 0 or 1. The second Borel-Cantelli lemma gives a condition under which the probability is in fact 1.

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