Lecture 2: Convergence of Random Variables

Transcription

1 Lecture 2: Convergence of Random Variables Hyang-Won Lee Dept. of Internet & Multimedia Eng. Konkuk University Lecture 2 Introduction to Stochastic Processes, Fall / 9

2 Convergence of Random Variables In many situations, it is necessary to characterize the asymptotic behavior of a sequence of random variables. For instance, consider a sequence of random variables X 1, X 2,... that are independent and identically distributed with mean µ and variance σ 2. The sample mean M n defined by M n = X 1 + X X n n is often used as an estimate of the true mean µ. One could ask here Does M n converge to µ? If yes, then In what sense? Our goal here is to understand what is means a sequence of random variables converges. There are some subtleties in the convergence of random variables, and it is conceivable that the convergence should be defined in a probabilistic sense. Lecture 2 Introduction to Stochastic Processes, Fall / 9

3 Almost Sure Convergence (Convergence with Probability 1) We start with Almost Sure Convergence which is the strongest in the sense that almost sure convergence implies other types of convergence (except for mean square convergence) that will be introduced later. Convergence with Probability 1 Consider a sequence of random variables X 1, X 2,..., and let a be a real number. We say that X n converges to a almost surely or with a.s. probability 1 (denoted as X n a) if ( ) P lim X n = a = 1. n To understand convergence with probability 1, recall that the probability of an event is the probability measure assigned to the set of samples leading to the event, that is, P(X = a) = P(ω : X(ω) = a). Lecture 2 Introduction to Stochastic Processes, Fall / 9

4 Almost Sure Convergence (contd.) An equivalent statement of almost sure convergence is ( ) P ω : lim X n(ω) = a = 1, n Almost sure convergence thus states that almost all samples ω lead to lim n X n (ω) = a, and other samples ω such that lim n X n (ω) a are extremely unlikely in the sense that their total probability is zero. Example(from Hajek Note): Let (X n : n 1) be a sequence of random variables distributed in the interval [0, 1] and defined by X n (ω) = ω n, ω [0, 1]. This sequence converges for all ω, with the limit lim X n(ω) = n { 0, if 0 ω < 1 1, if ω = 1. Lecture 2 Introduction to Stochastic Processes, Fall / 9

5 Almost Sure Convergence (contd.) Since {1} has probability zero, X n converges a.s. to zero. Example: Consider a discrete-time arrival process. Time is partitioned into consecutive intervals I 1, I 2,... such that an interval I k = {2 k, 2 k + 1,..., 2 k+1 1} and thus its length is 2 k. During each interval I k, there is exactly one arrival and the arrival time is uniformly distributed over the time instances in I k. The arrival times within different intervals are independent. Define X n = 1 if there is an arrival at time n and X n = 0 otherwise. Thus, we have P(X n 0) = 1 if n I 2 k k. Note that as n grows to infinity, k such that n I k goes to infinity as well. Consequently, it is true that lim P(X 1 n 0) = lim n k 2 k = 0. By this result, some may be tempted to conclude that X n a.s. 0. However, it is clear that X n becomes 1 infinitely often, and thus X n never converges to 0. Note that a sample ω is an infinite sequence of arrivals, and there is no ω such that lim n X n (ω) = 0. Therefore, X n does not converge to zero in the a.s. sense. This in fact motivates a weaker notion of convergence introduced next. Lecture 2 Introduction to Stochastic Processes, Fall / 9

6 Convergence in Probability In the previous example, we see that the probability P(X n 0) converges to 0 as n. This is what Convergence in Probability says. Convergence in Probability A sequence of random variables X 1, X 2,... is said to converge to a i.p. real number a (denoted as X n a) if for every ɛ > 0 lim P( X n a ɛ) = 0. n Hence, in the previous example, X n converges to 0 in probability. Convergence in probability can also be stated as follows: For every ɛ > 0 and δ > 0, there exists a number n 0 such that P( X n a ɛ) δ, n n 0. Lecture 2 Introduction to Stochastic Processes, Fall / 9

7 Convergence in Probability (contd.) Consider a sequence of independent random variables X n uniformly distributed in the interval [0, 1]. Let Y n = min{x 1,..., X n }. The sequence of values of Y n is non-increasing. For ɛ > 0, we have P( Y n 0 ɛ) = P(X 1 ɛ,..., X n ɛ)? = P(X 1 ɛ) P(X n ɛ) = (1 ɛ) n. Thus, Y n converges to zero in probability. Does it converge to zero in the a.s. sense as well? The answer is yes (can be shown using the Borel-Cantelli lemma which will not be covered in this course). Lecture 2 Introduction to Stochastic Processes, Fall / 9

8 Convergence in Distribution Next, we discuss Convergence in Distribution which is a weaker notion of convergence than the previous two definitions, in the sense that the previous two imply convergence in distribution. Convergence in Distribution Consider a sequence of random variables X n. We say that the sequence d converges to a random variable X in distribution (denoted as X n X) if lim n F X n (x) = F X (x) for every x R at which F X is continuous. Lecture 2 Introduction to Stochastic Processes, Fall / 9

9 Convergence in Distribution (contd.) Example: Consider a sequence of i.i.d. random variables X n, and let X be a random variable independent of X n s and identically distributed with X n. Then, it is clear that X n converges to X in distribution. Note, however, that it does not converge to X in probability (why?) Note: The three definitions have the following relationship: X n a.s. X X n i.p. X X n d X. There is another definition of convergence in mean square sense, but will not be discussed in this course. Lecture 2 Introduction to Stochastic Processes, Fall / 9