TAILIEUCHUNG - Introduction to Probability phần 10

Năm 1962, cuốn sách của Edward Thorp Beat 10 Đại lý cung cấp cho người đọc với một chiến lược để chơi các trò chơi casino phổ biến của blackjack sẽ đảm bảo các cầu thủ tích cực dự kiến sẽ giành chiến thắng. Cuốn sách này | . FUNDAMENTAL LIMIT THEOREM 451 as n goes to TO. But by similar reasoning to that used above the difference between this last expression and P Xn j goes to 0 as n goes to TO. Therefore P Xn j Wj as n goes to TO. This completes the proof. In the above proof we have said nothing about the rate at which the distributions of the Xn s approach the fixed distribution w. In fact it can be shown that18 r P Xn j - Wj 2P T n . j i The left-hand side of this inequality can be viewed as the distance between the distribution of the Markov chain after n steps starting in state Sj and the limiting distribution w. Exercises 1 Define P and y by P .5 .5 1 V .25 .75 J y 0 . Compute Py P2y and P4y and show that the results are approaching a constant vector. What is this vector 2 Let P be a regular r X r transition matrix and y any r-component column vector. Show that the value of the limiting constant vector for Pny is wy. 3 Let 1 0 0 P I .25 0 .75 001 be a transition matrix of a Markov chain. Find two fixed vectors of P that are linearly independent. Does this show that the Markov chain is not regular 4 Describe the set of all fixed column vectors for the chain given in Exercise 3. 5 The theorem that Pn W was proved only for the case that P has no zero entries. Fill in the details of the following extension to the case that P is regular. Since P is regular for some N PN has no zeros. Thus the proof given shows that MnN mnN approaches 0 as n tends to infinity. However the difference Mn mn can never increase. Why Hence if we know that the differences obtained by looking at every Nth time tend to 0 then the entire sequence must also tend to 0. 6 Let P be a regular transition matrix and let w be the unique non-zero fixed vector of P. Show that no entry of w is 0. 18T. Lindvall Lectures on the Coupling Method New York Wiley 1992 . 452 CHAPTER 11. MARKOV CHAINS 7 Here is a trick to try on your friends. Shuffle a deck of cards and deal them out one at a time. Count the face cards each .

TỪ KHÓA LIÊN QUAN
Đã phát hiện trình chặn quảng cáo AdBlock
Trang web này phụ thuộc vào doanh thu từ số lần hiển thị quảng cáo để tồn tại. Vui lòng tắt trình chặn quảng cáo của bạn hoặc tạm dừng tính năng chặn quảng cáo cho trang web này.