1 00:00:00,000 --> 00:00:01,460 2 00:00:01,460 --> 00:00:03,880 For part E and F of the problem, we'll be introducing 3 00:00:03,880 --> 00:00:07,160 a new notion of convergence, so-called the convergence E 4 00:00:07,160 --> 00:00:09,260 mean squared sense. 5 00:00:09,260 --> 00:00:16,820 We say that xn converges to a number c in mean squared, if 6 00:00:16,820 --> 00:00:24,090 as we take and go to infinity, the expected value of xn minus 7 00:00:24,090 --> 00:00:29,200 c squared goes to 0. 8 00:00:29,200 --> 00:00:33,130 To get a sense of what this looks like, let's say we let c 9 00:00:33,130 --> 00:00:38,740 equal to the expected value of xn, and let's say the expected 10 00:00:38,740 --> 00:00:40,940 value of xn is always the same. 11 00:00:40,940 --> 00:00:44,060 So the sequence of random variables has the same mean. 12 00:00:44,060 --> 00:00:48,010 Well, if that is true, then mean square convergence simply 13 00:00:48,010 --> 00:00:54,170 says the limit of the variance of xn is 0. 14 00:00:54,170 --> 00:00:58,820 So as you can imagine, somehow as xn becomes big, the 15 00:00:58,820 --> 00:01:03,130 variance of xn is very small, so xn is basically highly 16 00:01:03,130 --> 00:01:05,300 concentrated around c. 17 00:01:05,300 --> 00:01:10,080 And by this I mean, the density function for xn. 18 00:01:10,080 --> 00:01:11,400 So that's the notion of convergence 19 00:01:11,400 --> 00:01:12,940 we'll be working with. 20 00:01:12,940 --> 00:01:16,440 Our first task here is to show that the mean square 21 00:01:16,440 --> 00:01:20,000 convergence is in some sense stronger than the convergence 22 00:01:20,000 --> 00:01:23,410 in probability that we have been working with from part A 23 00:01:23,410 --> 00:01:28,290 to part D. That is, if I know that xn converged to some 24 00:01:28,290 --> 00:01:33,620 number c in mean squared, then this must imply that xn 25 00:01:33,620 --> 00:01:37,660 converges to c in probability. 26 00:01:37,660 --> 00:01:42,560 And now, we'll go show that for part E. 27 00:01:42,560 --> 00:01:45,400 Well, let's start with a definition of convergence in 28 00:01:45,400 --> 00:01:46,620 probability. 29 00:01:46,620 --> 00:01:49,990 We want to show that for a fixed constant epsilon the 30 00:01:49,990 --> 00:01:55,650 probability that xn minus c, greater than epsilon, 31 00:01:55,650 --> 00:02:00,990 essentially goes to 0 as n goes to infinity. 32 00:02:00,990 --> 00:02:03,830 To do so, we look at the value of this term. 33 00:02:03,830 --> 00:02:07,470 Well, the probability of absolute value xn minus c 34 00:02:07,470 --> 00:02:11,700 greater than epsilon is equal to the case if we were to 35 00:02:11,700 --> 00:02:15,150 square both sides of the inequality. 36 00:02:15,150 --> 00:02:19,000 So that is equal to the probability that xn minus c 37 00:02:19,000 --> 00:02:22,320 squared greater than epsilon squared. 38 00:02:22,320 --> 00:02:26,170 We can do this because both sides are positive, hence this 39 00:02:26,170 --> 00:02:28,590 goes through. 40 00:02:28,590 --> 00:02:33,630 Now, to bound this equality, we'll invoke the Markov's 41 00:02:33,630 --> 00:02:46,970 Inequality, which it says this probability of xn, some random 42 00:02:46,970 --> 00:02:51,420 variable greater than epsilon squared, is no more than is 43 00:02:51,420 --> 00:02:56,950 less equal to the expected value of the random variable. 44 00:02:56,950 --> 00:03:00,850 In this case, the expected value of x minus c squared 45 00:03:00,850 --> 00:03:04,890 divided by the threshold that we're trying to cross. 46 00:03:04,890 --> 00:03:08,530 So that is Markov's Inequality. 47 00:03:08,530 --> 00:03:12,430 Now, since we know xn converges to c in mean 48 00:03:12,430 --> 00:03:15,590 squared, and by definition, mean square we know this 49 00:03:15,590 --> 00:03:19,710 precise expectation right here goes to 0. 50 00:03:19,710 --> 00:03:25,040 And therefore, the whole expression goes to 0 as n goes 51 00:03:25,040 --> 00:03:26,140 to infinity. 52 00:03:26,140 --> 00:03:30,380 Because the denominator here is a constant and the top, the 53 00:03:30,380 --> 00:03:32,920 numerator here, goes to 0. 54 00:03:32,920 --> 00:03:34,190 So now we have it. 55 00:03:34,190 --> 00:03:39,760 We know that the probability of xn minus c absolute value 56 00:03:39,760 --> 00:03:43,630 greater than epsilon goes to 0 as n goes to infinity, for all 57 00:03:43,630 --> 00:03:47,660 fixed value of epsilons and this is the definition of 58 00:03:47,660 --> 00:03:48,910 convergence in probability. 59 00:03:48,910 --> 00:03:55,310 60 00:03:55,310 --> 00:03:58,740 Now that we know if xn converges to c mean squared, 61 00:03:58,740 --> 00:04:02,720 it implies that xn converges to c in probability. 62 00:04:02,720 --> 00:04:06,160 One might wonder whether the reverse is true. 63 00:04:06,160 --> 00:04:09,430 Namely, if we know something converges in probability to a 64 00:04:09,430 --> 00:04:12,080 constant, does the same sequence of random variables 65 00:04:12,080 --> 00:04:14,820 converge to the same constant in mean squared? 66 00:04:14,820 --> 00:04:17,279 It turns out that is not quite the case. 67 00:04:17,279 --> 00:04:19,980 The notion of probability converges in probability is 68 00:04:19,980 --> 00:04:25,030 not as strong as a notion of convergence in mean squared. 69 00:04:25,030 --> 00:04:27,810 Again, to look for a counter example, we do not have to go 70 00:04:27,810 --> 00:04:31,410 further than the yn's we have been working with. 71 00:04:31,410 --> 00:04:38,210 So here we know that yn converges to 0 in probability. 72 00:04:38,210 --> 00:04:40,160 But it turns out it does not converge to 73 00:04:40,160 --> 00:04:42,030 0 in the mean squared. 74 00:04:42,030 --> 00:04:45,520 And to see why this is the case, we can take the expected 75 00:04:45,520 --> 00:04:50,820 value of yn minus 0 squared, and see how that goes. 76 00:04:50,820 --> 00:04:54,780 Well, the value of this can be computed easily, which is 77 00:04:54,780 --> 00:05:01,090 simply 0, if yn is equal to 0, with probability 1 minus n 78 00:05:01,090 --> 00:05:06,800 plus n squared when yn takes a value of n, and this happens 79 00:05:06,800 --> 00:05:09,140 with probability 1 over n. 80 00:05:09,140 --> 00:05:13,490 The whole expression evaluates to n, which blows up to 81 00:05:13,490 --> 00:05:18,400 infinity as n going to infinity. 82 00:05:18,400 --> 00:05:26,490 As a result, the limit n going to infinity of E of yn minus 0 83 00:05:26,490 --> 00:05:32,200 squared is infinity and is not equal to 0. 84 00:05:32,200 --> 00:05:35,580 And there we have it, even though yn converges to 0 in 85 00:05:35,580 --> 00:05:39,290 probability, because the variance of yn, in some sense, 86 00:05:39,290 --> 00:05:42,660 is too big, it does not converge in a 87 00:05:42,660 --> 00:05:43,910 mean squared sense. 88 00:05:43,910 --> 00:05:45,167