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+/Limits [(section.3.2) (section.3.7)] >> endobj -1038 0 obj +1085 0 obj << -/Names [(subsection.1.1.4) 77 0 R (subsection.1.3.1) 89 0 R (subsection.1.3.2) 93 0 R (subsection.1.3.3) 97 0 R (subsection.1.4.1) 105 0 R (subsection.1.4.2) 109 0 R] -/Limits [(subsection.1.1.4) (subsection.1.4.2)] +/Names [(section.3.8) 333 0 R (section.3.9) 337 0 R (subsection.1.1.1) 65 0 R (subsection.1.1.2) 69 0 R (subsection.1.1.3) 73 0 R (subsection.1.1.4) 77 0 R] +/Limits [(section.3.8) (subsection.1.1.4)] >> endobj -1039 0 obj +1086 0 obj << -/Names [(subsection.1.4.3) 113 0 R (subsection.2.1.1) 129 0 R (subsection.2.1.2) 133 0 R (subsection.2.1.3) 137 0 R (subsection.2.2.1) 145 0 R (subsection.2.2.2) 149 0 R] -/Limits [(subsection.1.4.3) (subsection.2.2.2)] +/Names [(subsection.1.3.1) 89 0 R (subsection.1.3.2) 93 0 R (subsection.1.3.3) 97 0 R (subsection.1.4.1) 105 0 R (subsection.1.4.2) 109 0 R (subsection.1.4.3) 113 0 R] +/Limits [(subsection.1.3.1) (subsection.1.4.3)] >> endobj -1040 0 obj +1087 0 obj << -/Names [(subsection.2.2.3) 153 0 R (subsection.2.2.4) 157 0 R (subsection.2.2.5) 161 0 R (subsection.2.3.1) 169 0 R (subsection.2.3.2) 173 0 R (subsection.2.3.3) 177 0 R] -/Limits [(subsection.2.2.3) (subsection.2.3.3)] +/Names [(subsection.2.1.1) 129 0 R (subsection.2.1.2) 133 0 R (subsection.2.1.3) 137 0 R (subsection.2.2.1) 145 0 R (subsection.2.2.2) 149 0 R (subsection.2.2.3) 153 0 R] +/Limits [(subsection.2.1.1) (subsection.2.2.3)] >> endobj -1041 0 obj +1088 0 obj << -/Names [(subsection.2.3.4) 181 0 R (subsection.2.3.5) 185 0 R (subsection.2.5.1) 197 0 R (subsection.2.5.2) 201 0 R (subsection.2.5.3) 205 0 R (subsection.2.6.1) 213 0 R] -/Limits [(subsection.2.3.4) (subsection.2.6.1)] +/Names [(subsection.2.2.4) 157 0 R (subsection.2.2.5) 161 0 R (subsection.2.3.1) 169 0 R (subsection.2.3.2) 173 0 R (subsection.2.3.3) 177 0 R (subsection.2.3.4) 181 0 R] +/Limits [(subsection.2.2.4) (subsection.2.3.4)] >> endobj -1042 0 obj +1089 0 obj << -/Names [(subsection.2.6.2) 217 0 R (subsection.2.7.1) 225 0 R (subsection.2.7.2) 229 0 R (subsection.2.8.1) 237 0 R (subsection.2.8.2) 241 0 R (subsection.3.1.1) 253 0 R] -/Limits [(subsection.2.6.2) (subsection.3.1.1)] +/Names [(subsection.2.3.5) 185 0 R (subsection.2.5.1) 197 0 R (subsection.2.5.2) 201 0 R (subsection.2.5.3) 205 0 R (subsection.2.6.1) 213 0 R (subsection.2.6.2) 217 0 R] +/Limits [(subsection.2.3.5) (subsection.2.6.2)] >> endobj -1043 0 obj +1090 0 obj << -/Names [(subsection.3.1.2) 257 0 R (subsection.3.1.3) 261 0 R (subsection.3.2.1) 269 0 R (subsection.3.2.2) 273 0 R (subsection.3.5.1) 289 0 R (subsection.3.5.2) 293 0 R] -/Limits [(subsection.3.1.2) (subsection.3.5.2)] +/Names [(subsection.2.7.1) 225 0 R (subsection.2.7.2) 229 0 R (subsection.2.8.1) 237 0 R (subsection.2.8.2) 241 0 R (subsection.3.1.1) 253 0 R (subsection.3.1.2) 257 0 R] +/Limits [(subsection.2.7.1) (subsection.3.1.2)] >> endobj -1044 0 obj +1091 0 obj +<< +/Names [(subsection.3.1.3) 261 0 R (subsection.3.10.1) 349 0 R (subsection.3.2.1) 269 0 R (subsection.3.2.2) 273 0 R (subsection.3.5.1) 289 0 R (subsection.3.5.2) 293 0 R] +/Limits [(subsection.3.1.3) (subsection.3.5.2)] +>> +endobj +1092 0 obj << /Names [(subsection.3.6.1) 301 0 R (subsection.3.6.2) 305 0 R (subsection.3.6.3) 309 0 R (subsection.3.6.4) 313 0 R (subsection.3.7.1) 321 0 R (subsection.3.7.2) 325 0 R] /Limits [(subsection.3.6.1) (subsection.3.7.2)] >> endobj -1045 0 obj +1093 0 obj << -/Names [(subsection.3.7.3) 329 0 R (table.3.1) 787 0 R (theorem.1.1) 524 0 R (theorem.1.10) 536 0 R (theorem.1.11) 537 0 R (theorem.1.12) 545 0 R] -/Limits [(subsection.3.7.3) (theorem.1.12)] +/Names [(subsection.3.7.3) 329 0 R (subsection.3.9.1) 341 0 R (table.3.1) 814 0 R (theorem.1.1) 551 0 R (theorem.1.10) 563 0 R (theorem.1.11) 564 0 R] +/Limits [(subsection.3.7.3) (theorem.1.11)] >> endobj -1046 0 obj +1094 0 obj << -/Names [(theorem.1.13) 546 0 R (theorem.1.14) 550 0 R (theorem.1.15) 551 0 R (theorem.1.16) 560 0 R (theorem.1.17) 561 0 R (theorem.1.18) 564 0 R] -/Limits [(theorem.1.13) (theorem.1.18)] +/Names [(theorem.1.12) 572 0 R (theorem.1.13) 573 0 R (theorem.1.14) 577 0 R (theorem.1.15) 578 0 R (theorem.1.16) 587 0 R (theorem.1.17) 588 0 R] +/Limits [(theorem.1.12) (theorem.1.17)] >> endobj -1047 0 obj +1095 0 obj << -/Names [(theorem.1.19) 565 0 R (theorem.1.2) 525 0 R (theorem.1.20) 566 0 R (theorem.1.21) 567 0 R (theorem.1.22) 568 0 R (theorem.1.23) 569 0 R] -/Limits [(theorem.1.19) (theorem.1.23)] +/Names [(theorem.1.18) 591 0 R (theorem.1.19) 592 0 R (theorem.1.2) 552 0 R (theorem.1.20) 593 0 R (theorem.1.21) 594 0 R (theorem.1.22) 595 0 R] +/Limits [(theorem.1.18) (theorem.1.22)] >> endobj -1048 0 obj +1096 0 obj << -/Names [(theorem.1.24) 570 0 R (theorem.1.25) 571 0 R (theorem.1.26) 577 0 R (theorem.1.27) 578 0 R (theorem.1.28) 579 0 R (theorem.1.29) 580 0 R] -/Limits [(theorem.1.24) (theorem.1.29)] +/Names [(theorem.1.23) 596 0 R (theorem.1.24) 597 0 R (theorem.1.25) 598 0 R (theorem.1.26) 604 0 R (theorem.1.27) 605 0 R (theorem.1.28) 606 0 R] +/Limits [(theorem.1.23) (theorem.1.28)] >> endobj -1049 0 obj +1097 0 obj << -/Names [(theorem.1.3) 526 0 R (theorem.1.30) 581 0 R (theorem.1.31) 582 0 R (theorem.1.32) 583 0 R (theorem.1.33) 584 0 R (theorem.1.34) 585 0 R] -/Limits [(theorem.1.3) (theorem.1.34)] +/Names [(theorem.1.29) 607 0 R (theorem.1.3) 553 0 R (theorem.1.30) 608 0 R (theorem.1.31) 609 0 R (theorem.1.32) 610 0 R (theorem.1.33) 611 0 R] +/Limits [(theorem.1.29) (theorem.1.33)] >> endobj -1050 0 obj +1098 0 obj << -/Names [(theorem.1.35) 586 0 R (theorem.1.36) 587 0 R (theorem.1.37) 589 0 R (theorem.1.38) 590 0 R (theorem.1.39) 591 0 R (theorem.1.4) 530 0 R] -/Limits [(theorem.1.35) (theorem.1.4)] +/Names [(theorem.1.34) 612 0 R (theorem.1.35) 613 0 R (theorem.1.36) 614 0 R (theorem.1.37) 616 0 R (theorem.1.38) 617 0 R (theorem.1.39) 618 0 R] +/Limits [(theorem.1.34) (theorem.1.39)] >> endobj -1051 0 obj +1099 0 obj << -/Names [(theorem.1.40) 596 0 R (theorem.1.41) 597 0 R (theorem.1.42) 598 0 R (theorem.1.43) 599 0 R (theorem.1.44) 600 0 R (theorem.1.45) 601 0 R] -/Limits [(theorem.1.40) (theorem.1.45)] +/Names [(theorem.1.4) 557 0 R (theorem.1.40) 623 0 R (theorem.1.41) 624 0 R (theorem.1.42) 625 0 R (theorem.1.43) 626 0 R (theorem.1.44) 627 0 R] +/Limits [(theorem.1.4) (theorem.1.44)] >> endobj -1052 0 obj +1100 0 obj << -/Names [(theorem.1.46) 602 0 R (theorem.1.47) 603 0 R (theorem.1.5) 531 0 R (theorem.1.6) 532 0 R (theorem.1.7) 533 0 R (theorem.1.8) 534 0 R] -/Limits [(theorem.1.46) (theorem.1.8)] +/Names [(theorem.1.45) 628 0 R (theorem.1.46) 629 0 R (theorem.1.47) 630 0 R (theorem.1.5) 558 0 R (theorem.1.6) 559 0 R (theorem.1.7) 560 0 R] +/Limits [(theorem.1.45) (theorem.1.7)] >> endobj -1053 0 obj +1101 0 obj << -/Names [(theorem.1.9) 535 0 R (theorem.2.1) 609 0 R (theorem.2.10) 620 0 R (theorem.2.100) 768 0 R (theorem.2.101) 769 0 R (theorem.2.102) 778 0 R] -/Limits [(theorem.1.9) (theorem.2.102)] +/Names [(theorem.1.8) 561 0 R (theorem.1.9) 562 0 R (theorem.2.1) 636 0 R (theorem.2.10) 647 0 R (theorem.2.100) 795 0 R (theorem.2.101) 796 0 R] +/Limits [(theorem.1.8) (theorem.2.101)] >> endobj -1054 0 obj +1102 0 obj << -/Names [(theorem.2.103) 780 0 R (theorem.2.104) 781 0 R (theorem.2.105) 782 0 R (theorem.2.106) 783 0 R (theorem.2.107) 784 0 R (theorem.2.108) 785 0 R] -/Limits [(theorem.2.103) (theorem.2.108)] +/Names [(theorem.2.102) 805 0 R (theorem.2.103) 807 0 R (theorem.2.104) 808 0 R (theorem.2.105) 809 0 R (theorem.2.106) 810 0 R (theorem.2.107) 811 0 R] +/Limits [(theorem.2.102) (theorem.2.107)] >> endobj -1055 0 obj +1103 0 obj << -/Names [(theorem.2.11) 621 0 R (theorem.2.12) 622 0 R (theorem.2.13) 628 0 R (theorem.2.14) 629 0 R (theorem.2.15) 630 0 R (theorem.2.16) 631 0 R] -/Limits [(theorem.2.11) (theorem.2.16)] +/Names [(theorem.2.108) 812 0 R (theorem.2.11) 648 0 R (theorem.2.12) 649 0 R (theorem.2.13) 655 0 R (theorem.2.14) 656 0 R (theorem.2.15) 657 0 R] +/Limits [(theorem.2.108) (theorem.2.15)] >> endobj -1056 0 obj +1104 0 obj << -/Names [(theorem.2.17) 632 0 R (theorem.2.18) 633 0 R (theorem.2.19) 634 0 R (theorem.2.2) 610 0 R (theorem.2.20) 635 0 R (theorem.2.21) 636 0 R] -/Limits [(theorem.2.17) (theorem.2.21)] +/Names [(theorem.2.16) 658 0 R (theorem.2.17) 659 0 R (theorem.2.18) 660 0 R (theorem.2.19) 661 0 R (theorem.2.2) 637 0 R (theorem.2.20) 662 0 R] +/Limits [(theorem.2.16) (theorem.2.20)] >> endobj -1057 0 obj +1105 0 obj << -/Names [(theorem.2.22) 637 0 R (theorem.2.23) 638 0 R (theorem.2.24) 639 0 R (theorem.2.25) 640 0 R (theorem.2.26) 641 0 R (theorem.2.27) 648 0 R] -/Limits [(theorem.2.22) (theorem.2.27)] +/Names [(theorem.2.21) 663 0 R (theorem.2.22) 664 0 R (theorem.2.23) 665 0 R (theorem.2.24) 666 0 R (theorem.2.25) 667 0 R (theorem.2.26) 668 0 R] +/Limits [(theorem.2.21) (theorem.2.26)] >> endobj -1058 0 obj +1106 0 obj << -/Names [(theorem.2.28) 649 0 R (theorem.2.29) 650 0 R (theorem.2.3) 613 0 R (theorem.2.30) 651 0 R (theorem.2.31) 652 0 R (theorem.2.32) 653 0 R] -/Limits [(theorem.2.28) (theorem.2.32)] +/Names [(theorem.2.27) 675 0 R (theorem.2.28) 676 0 R (theorem.2.29) 677 0 R (theorem.2.3) 640 0 R (theorem.2.30) 678 0 R (theorem.2.31) 679 0 R] +/Limits [(theorem.2.27) (theorem.2.31)] >> endobj -1059 0 obj +1107 0 obj << -/Names [(theorem.2.33) 654 0 R (theorem.2.34) 655 0 R (theorem.2.35) 656 0 R (theorem.2.36) 657 0 R (theorem.2.37) 658 0 R (theorem.2.38) 660 0 R] -/Limits [(theorem.2.33) (theorem.2.38)] +/Names [(theorem.2.32) 680 0 R (theorem.2.33) 681 0 R (theorem.2.34) 682 0 R (theorem.2.35) 683 0 R (theorem.2.36) 684 0 R (theorem.2.37) 685 0 R] +/Limits [(theorem.2.32) (theorem.2.37)] >> endobj -1060 0 obj +1108 0 obj << -/Names [(theorem.2.39) 661 0 R (theorem.2.4) 614 0 R (theorem.2.40) 662 0 R (theorem.2.41) 663 0 R (theorem.2.42) 670 0 R (theorem.2.43) 671 0 R] -/Limits [(theorem.2.39) (theorem.2.43)] +/Names [(theorem.2.38) 687 0 R (theorem.2.39) 688 0 R (theorem.2.4) 641 0 R (theorem.2.40) 689 0 R (theorem.2.41) 690 0 R (theorem.2.42) 697 0 R] +/Limits [(theorem.2.38) (theorem.2.42)] >> endobj -1061 0 obj +1109 0 obj << -/Names [(theorem.2.44) 672 0 R (theorem.2.45) 673 0 R (theorem.2.46) 676 0 R (theorem.2.47) 677 0 R (theorem.2.48) 678 0 R (theorem.2.49) 679 0 R] -/Limits [(theorem.2.44) (theorem.2.49)] +/Names [(theorem.2.43) 698 0 R (theorem.2.44) 699 0 R (theorem.2.45) 700 0 R (theorem.2.46) 703 0 R (theorem.2.47) 704 0 R (theorem.2.48) 705 0 R] +/Limits [(theorem.2.43) (theorem.2.48)] >> endobj -1062 0 obj +1110 0 obj << -/Names [(theorem.2.5) 615 0 R (theorem.2.50) 680 0 R (theorem.2.51) 681 0 R (theorem.2.52) 682 0 R (theorem.2.53) 685 0 R (theorem.2.54) 693 0 R] -/Limits [(theorem.2.5) (theorem.2.54)] +/Names [(theorem.2.49) 706 0 R (theorem.2.5) 642 0 R (theorem.2.50) 707 0 R (theorem.2.51) 708 0 R (theorem.2.52) 709 0 R (theorem.2.53) 712 0 R] +/Limits [(theorem.2.49) (theorem.2.53)] >> endobj -1063 0 obj +1111 0 obj << -/Names [(theorem.2.55) 700 0 R (theorem.2.56) 701 0 R (theorem.2.57) 702 0 R (theorem.2.58) 703 0 R (theorem.2.59) 704 0 R (theorem.2.6) 616 0 R] -/Limits [(theorem.2.55) (theorem.2.6)] +/Names [(theorem.2.54) 720 0 R (theorem.2.55) 727 0 R (theorem.2.56) 728 0 R (theorem.2.57) 729 0 R (theorem.2.58) 730 0 R (theorem.2.59) 731 0 R] +/Limits [(theorem.2.54) (theorem.2.59)] >> endobj -1064 0 obj +1112 0 obj << -/Names [(theorem.2.60) 705 0 R (theorem.2.61) 706 0 R (theorem.2.62) 707 0 R (theorem.2.63) 708 0 R (theorem.2.64) 709 0 R (theorem.2.65) 710 0 R] -/Limits [(theorem.2.60) (theorem.2.65)] +/Names [(theorem.2.6) 643 0 R (theorem.2.60) 732 0 R (theorem.2.61) 733 0 R (theorem.2.62) 734 0 R (theorem.2.63) 735 0 R (theorem.2.64) 736 0 R] +/Limits [(theorem.2.6) (theorem.2.64)] >> endobj -1065 0 obj +1113 0 obj << -/Names [(theorem.2.66) 711 0 R (theorem.2.67) 719 0 R (theorem.2.68) 720 0 R (theorem.2.69) 721 0 R (theorem.2.7) 617 0 R (theorem.2.70) 722 0 R] -/Limits [(theorem.2.66) (theorem.2.70)] +/Names [(theorem.2.65) 737 0 R (theorem.2.66) 738 0 R (theorem.2.67) 746 0 R (theorem.2.68) 747 0 R (theorem.2.69) 748 0 R (theorem.2.7) 644 0 R] +/Limits [(theorem.2.65) (theorem.2.7)] >> endobj -1066 0 obj +1114 0 obj << -/Names [(theorem.2.71) 723 0 R (theorem.2.72) 724 0 R (theorem.2.73) 725 0 R (theorem.2.74) 726 0 R (theorem.2.75) 727 0 R (theorem.2.76) 728 0 R] -/Limits [(theorem.2.71) (theorem.2.76)] +/Names [(theorem.2.70) 749 0 R (theorem.2.71) 750 0 R (theorem.2.72) 751 0 R (theorem.2.73) 752 0 R (theorem.2.74) 753 0 R (theorem.2.75) 754 0 R] +/Limits [(theorem.2.70) (theorem.2.75)] >> endobj -1067 0 obj +1115 0 obj << -/Names [(theorem.2.77) 729 0 R (theorem.2.78) 730 0 R (theorem.2.79) 731 0 R (theorem.2.8) 618 0 R (theorem.2.80) 732 0 R (theorem.2.81) 733 0 R] -/Limits [(theorem.2.77) (theorem.2.81)] +/Names [(theorem.2.76) 755 0 R (theorem.2.77) 756 0 R (theorem.2.78) 757 0 R (theorem.2.79) 758 0 R (theorem.2.8) 645 0 R (theorem.2.80) 759 0 R] +/Limits [(theorem.2.76) (theorem.2.80)] >> endobj -1068 0 obj +1116 0 obj << -/Names [(theorem.2.82) 743 0 R (theorem.2.83) 744 0 R (theorem.2.84) 745 0 R (theorem.2.85) 746 0 R (theorem.2.86) 747 0 R (theorem.2.87) 748 0 R] -/Limits [(theorem.2.82) (theorem.2.87)] +/Names [(theorem.2.81) 760 0 R (theorem.2.82) 770 0 R (theorem.2.83) 771 0 R (theorem.2.84) 772 0 R (theorem.2.85) 773 0 R (theorem.2.86) 774 0 R] +/Limits [(theorem.2.81) (theorem.2.86)] >> endobj -1069 0 obj +1117 0 obj << -/Names [(theorem.2.88) 749 0 R (theorem.2.89) 750 0 R (theorem.2.9) 619 0 R (theorem.2.90) 751 0 R (theorem.2.91) 752 0 R (theorem.2.92) 755 0 R] -/Limits [(theorem.2.88) (theorem.2.92)] +/Names [(theorem.2.87) 775 0 R (theorem.2.88) 776 0 R (theorem.2.89) 777 0 R (theorem.2.9) 646 0 R (theorem.2.90) 778 0 R (theorem.2.91) 779 0 R] +/Limits [(theorem.2.87) (theorem.2.91)] >> endobj -1070 0 obj +1118 0 obj << -/Names [(theorem.2.93) 756 0 R (theorem.2.94) 757 0 R (theorem.2.95) 763 0 R (theorem.2.96) 764 0 R (theorem.2.97) 765 0 R (theorem.2.98) 766 0 R] -/Limits [(theorem.2.93) (theorem.2.98)] +/Names [(theorem.2.92) 782 0 R (theorem.2.93) 783 0 R (theorem.2.94) 784 0 R (theorem.2.95) 790 0 R (theorem.2.96) 791 0 R (theorem.2.97) 792 0 R] +/Limits [(theorem.2.92) (theorem.2.97)] >> endobj -1071 0 obj +1119 0 obj << -/Names [(theorem.2.99) 767 0 R (theorem.3.1) 797 0 R (theorem.3.10) 821 0 R (theorem.3.11) 828 0 R (theorem.3.12) 829 0 R (theorem.3.13) 832 0 R] -/Limits [(theorem.2.99) (theorem.3.13)] +/Names [(theorem.2.98) 793 0 R (theorem.2.99) 794 0 R (theorem.3.1) 824 0 R (theorem.3.10) 848 0 R (theorem.3.11) 855 0 R (theorem.3.12) 856 0 R] +/Limits [(theorem.2.98) (theorem.3.12)] >> endobj -1072 0 obj +1120 0 obj << -/Names [(theorem.3.14) 833 0 R (theorem.3.15) 834 0 R (theorem.3.16) 835 0 R (theorem.3.17) 836 0 R (theorem.3.18) 837 0 R (theorem.3.19) 838 0 R] -/Limits [(theorem.3.14) (theorem.3.19)] +/Names [(theorem.3.13) 859 0 R (theorem.3.14) 860 0 R (theorem.3.15) 861 0 R (theorem.3.16) 862 0 R (theorem.3.17) 863 0 R (theorem.3.18) 864 0 R] +/Limits [(theorem.3.13) (theorem.3.18)] >> endobj -1073 0 obj +1121 0 obj << -/Names [(theorem.3.2) 798 0 R (theorem.3.20) 839 0 R (theorem.3.21) 840 0 R (theorem.3.22) 853 0 R (theorem.3.23) 858 0 R (theorem.3.24) 859 0 R] -/Limits [(theorem.3.2) (theorem.3.24)] +/Names [(theorem.3.19) 865 0 R (theorem.3.2) 825 0 R (theorem.3.20) 866 0 R (theorem.3.21) 867 0 R (theorem.3.22) 880 0 R (theorem.3.23) 885 0 R] +/Limits [(theorem.3.19) (theorem.3.23)] >> endobj -1074 0 obj +1122 0 obj << -/Names [(theorem.3.25) 860 0 R (theorem.3.26) 861 0 R (theorem.3.27) 862 0 R (theorem.3.28) 863 0 R (theorem.3.29) 864 0 R (theorem.3.3) 802 0 R] -/Limits [(theorem.3.25) (theorem.3.3)] +/Names [(theorem.3.24) 886 0 R (theorem.3.25) 887 0 R (theorem.3.26) 888 0 R (theorem.3.27) 889 0 R (theorem.3.28) 890 0 R (theorem.3.29) 891 0 R] +/Limits [(theorem.3.24) (theorem.3.29)] >> endobj -1075 0 obj +1123 0 obj << -/Names [(theorem.3.30) 869 0 R (theorem.3.31) 870 0 R (theorem.3.32) 871 0 R (theorem.3.33) 872 0 R (theorem.3.34) 873 0 R (theorem.3.35) 874 0 R] -/Limits [(theorem.3.30) (theorem.3.35)] +/Names [(theorem.3.3) 829 0 R (theorem.3.30) 896 0 R (theorem.3.31) 897 0 R (theorem.3.32) 898 0 R (theorem.3.33) 899 0 R (theorem.3.34) 900 0 R] +/Limits [(theorem.3.3) (theorem.3.34)] >> endobj -1076 0 obj +1124 0 obj << -/Names [(theorem.3.36) 877 0 R (theorem.3.37) 880 0 R (theorem.3.38) 881 0 R (theorem.3.39) 882 0 R (theorem.3.4) 808 0 R (theorem.3.40) 883 0 R] -/Limits [(theorem.3.36) (theorem.3.40)] +/Names [(theorem.3.35) 901 0 R (theorem.3.36) 904 0 R (theorem.3.37) 907 0 R (theorem.3.38) 908 0 R (theorem.3.39) 909 0 R (theorem.3.4) 835 0 R] +/Limits [(theorem.3.35) (theorem.3.4)] >> endobj -1077 0 obj +1125 0 obj << -/Names [(theorem.3.41) 884 0 R (theorem.3.42) 885 0 R (theorem.3.43) 886 0 R (theorem.3.44) 887 0 R (theorem.3.45) 894 0 R (theorem.3.46) 895 0 R] -/Limits [(theorem.3.41) (theorem.3.46)] +/Names [(theorem.3.40) 910 0 R (theorem.3.41) 911 0 R 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\newcommand{\PP}{\mathcal{P}} \newcommand{\NN}{\mathbb{N}} +\newcommand{\ZZ}{\mathbb{Z}} \newcommand{\RR}{\mathbb{R}} \newcommand{\QQ}{\mathbb{Q}} diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex b/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex index 7033215..3cf156b 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/0-notazioni.tex @@ -66,9 +66,11 @@ modo $f = f^+ - f^-$. \item $\exp$ -- funzione esponenziale $e^x$. \item $\log \equiv \ln = \log_e$ -- logaritmo naturale, ossia logaritmo in base $e$. - \item $C^n$ -- classe di funzioni derivabili $n$ volte con $n$-esima derivata continua. Per $n = 0$, classe di funzioni continue. - \item $C^\infty$ -- classe di funzioni derivabili un numero illimitato di volte. - \item $\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \dt$ -- funzione gamma. È tale per cui $\Gamma(n+1) = n!$ per ogni $n \in \NN$. + \item $C^n$, $C^n(\RR)$ -- classe delle funzioni derivabili $n$ volte con $n$-esima derivata continua. Per $n = 0$, classe di funzioni continue. + \item $C^\infty$ -- classe delle funzioni derivabili un numero illimitato di volte. + \item $C_b$, $C_b(\RR)$ -- classe delle funzioni reali, continue e limitate. + \item $\Gamma(x) = \int_0^\infty t^{x-1} e^{-t} \dt$ -- funzione gamma. È tale per cui $\Gamma(n+1) = n!$ per ogni $n \in \NN$. + \item $f * g$ -- convoluzione di funzioni; tale che $(f * g)(z)$ sia pari a $\int_\RR f(x) g(z-x) \dx$. \end{itemize} \section*{Combinatoria} diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/2-probabilità-discreta.tex b/Secondo anno/Elementi di probabilità e statistica/sections/2-probabilità-discreta.tex index 26fe79c..0436f67 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/2-probabilità-discreta.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/2-probabilità-discreta.tex @@ -986,6 +986,8 @@ sono definiti altri due indici di centralità celebri. \subsection{Definizioni ed enunciato} +\label{sec:lgn} + \begin{definition}[Media campionaria $n$-esima] Data una famiglia di v.a.~reali $(X_i)_{i \in \NN}$ i.i.d.~dotate di momento secondo finito\footnote{ diff --git a/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex b/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex index 1d28645..a11addd 100644 --- a/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex +++ b/Secondo anno/Elementi di probabilità e statistica/sections/3-probabilità-reale.tex @@ -726,7 +726,14 @@ che $\mu$ assuma valori finiti per le funzioni semplici). \] \end{remark} -\section{Momenti e disuguaglianze, (co)varianza, dev.~standard, mediana e moda} +\begin{remark} + La formula presentata si estende in modo naturale al caso di più variabili: + \[ + \EE[\varphi(X_1, \ldots, X_n)] = \int_{\RR^n} \varphi(x_1, \ldots, x_n) f(x_1, \ldots, x_n) \dx_1 \cdots \dx_n. + \] +\end{remark} + +\section{Momenti e disuguaglianze, varianza, covarianza, dev.~standard, mediana e moda} Tutti le disuguaglianze sul valore atteso (e.g.~Markov) e tutti i risultati riguardanti i momenti (assoluti e non), la covarianza, la varianza, la @@ -748,4 +755,142 @@ $\EE[\cdot]$. Si rimanda dunque alla \hyperref[sec:momenti_assoluti]{\textit{Par di tale intervallo è mediana. \end{proposition} +\section{Trasformazioni di variabili aleatorie} + +\begin{proposition}[Formula del cambio di variabili] Sia $X : \Omega \to \RR^d$ una v.a. assolutamente + continua con densità $f_X$ tale per cui $f_X \equiv 0$ fuori da un aperto $O \subseteq \RR^d$. Sia + $\varphi : O \to O'$ un diffeomorfismo con $O' \subseteq \RR^d$ aperto (i.e.~$\varphi$ è $C^1$, invertibile e + $\varphi\inv$ è $C^1$). Allora la v.a.~$Y = \varphi(X)$ è assolutamente continua con densità: + \[ + f_Y(y) = f_X(\varphi\inv(y)) \, \abs{\det D\varphi\inv(y)} \, 1_O(y), \quad y \in \RR^d. + \] + Segue dall'usuale formula del cambio di variabili per l'integrale di Lebesgue. +\end{proposition} + +\begin{proposition}[Densità della somma] + Siano $X$, $Y : \Omega \groupto \RR$ v.a.~reali con $(X, Y)$ AC di densità $f_{(X, Y)}$. Allora + $X+Y$ è assolutamente continua con densità: + \[ + f_{X+Y}(z) = \int_\RR f_{(X, Y)}(x, z-x) \dx = \int_\RR f_{(X, Y)}(z-y, y) \dy. + \] +\end{proposition} + +\begin{corollary} + Siano $X$, $Y : \Omega \groupto \RR$ v.a.~reali, AC con densità $f_X$ e $f_Y$, e indipendenti. Allora + $X+Y$ è assolutamente continua con densità: + \[ + f_{X+Y}(z) = (f_X * f_Y)(z) = \int_\RR f_X(x) f_Y(z-x) \dx = \int_\RR f_X(z-y) f_Y(y) \dy. + \] +\end{corollary} + +\begin{remark} + Per il caso discreto vale una formula analoga. In particolare: + \[ + p_{X+Y}(k) = \sum_{j \in \ZZ} p_{(X,Y)}(j, k-j) = \sum_{j \in \ZZ} p_{(X,Y)}(k-j, j). + \] + Dunque, se $X$ e $Y$ sono indipendenti: + \[ + p_{X+Y}(k) = \sum_{j \in \ZZ} p_X(j) p_Y(k-j) = \sum_{j \in \ZZ} p_X(k-j) p_Y(j). + \] +\end{remark} + +\subsection{Standardizzazione e riproducibilità di v.a.~gaussiane} + +\begin{proposition} + Sia $X \sim N(m, \sigma^2)$ e siano $a$, $b \in \RR$ con $a \neq 0$. Allora + $aX + b \sim N(am+b, a^2 \sigma^2)$. +\end{proposition} + +A partire da questa proposizione si enuncia il corollario riguardante la +\textit{standardizzazione}, ovverosia il processo con cui si riconduce una +qualsiasi gaussiana alla gaussiana standard: + +\begin{corollary}[Standardizzazione] + $X \sim N(m, \sigma^2) \iff \frac{X-m}{\sigma} \sim N(0, 1)$. Equivalentemente + $Z \sim N(0, 1) \iff \sigma Z + m \sim N(m, \sigma^2)$. +\end{corollary} + +\begin{remark} + Pertanto, tramite il processo di standardizzazione, si ricava facilmente che + per $X \sim N(m, \sigma^2)$ vale che: + \[ + P(a \leq X \leq b) = \Phi\left(\frac{b-m}{\sigma}\right) - \Phi\left(\frac{a-m}{\sigma}\right). + \] +\end{remark} + +\begin{proposition}[Riproducibilità delle gaussiane] + Siano $X \sim N(m_1, \sigma_1^2)$ e $Y \sim N(m_2, \sigma_2^2)$ indipendenti. Allora + $X+Y \sim N(m_1 + m_2, \sigma_1^2 + \sigma_2^2)$. +\end{proposition} + +\section{Legge dei grandi numeri (LGN)} + +La legge (debole) dei grandi numeri segue lo stesso enunciato (e la stessa dimostrazione) del caso +discreto, e si rimanda dunque alla \hyperref[sec:lgn]{\textit{Parte 2}}. + +\subsection{Metodo di Monte-Carlo per il calcolo di integrali} + +Tramite l'uso delle variabili aleatorie è possibile approssimare integrali sfruttando +modelli di probabilità uniforme. Supponiamo di voler calcolare $\int_a^b \varphi(x) \dx$ con +$\varphi : (a, b) \to \RR$ boreliana e integrabile. Si assuma anche che +$\int_a^b \abs{\varphi(x)}^2 \dx < \infty$. \smallskip + +Si osserva che $\frac{1}{b-a} \int_a^b \varphi(x) \dx = \EE[\varphi(U)]$ con $U \sim U(a, b)$. +Infatti vale che: +\[ + \EE[\varphi(U)] = \int_a^b \varphi(x) \underbrace{\frac{1}{b-a}}_{=\,f_U(x)} \dx. +\] +Per la LGN\footnote{In questo momento si sfrutta l'ipotesi per cui $\varphi(U_i)$ ha momento secondo finito, +ossia che $\int_a^b \abs{\varphi(x)}^2 \dx < \infty$. Questa ipotesi è tuttavia rimovibile.}, +prese $U_i$ i.i.d.~come $U(a, b)$, allora le $\varphi(U_i)$ sono a loro volta +i.i.d.~e vale che $\frac{1}{n} \sum_{i=1}^n \varphi(U_i) \toprob \EE[\varphi(U)]$, ovverosia: +\[ + \frac{1}{n} \sum_{i=1}^n \varphi(U_i) \toprob \frac{1}{b-a} \int_a^b \varphi(x) \dx. +\] +Pertanto è sufficiente calcolare un ``grande numero'' di $\varphi(U_i)$ uniformi per approssimare +l'integrale scelto. \smallskip + +Lo stesso procedimento si può applicare per calcolare integrali della forma $\int_{(a, b)^d} \varphi(x) \dx$ con +$\varphi : (a, b)^d \to \RR$ boreliana, integrabile e tale per cui $\int_{(a, b)^d} \abs{\varphi(x)}^2 \dx$, +approssimandolo con v.a.~i.i.d.~come $U((a, b)^d)$. \smallskip + +\section{Teorema centrale del limite (TCL, o TLC)} + +\begin{definition}[Convergenza in legge] + Si dice che una successione $(X_i)_{i \in I}$ di v.a.~reali con $I \subseteq \NN$ converge in legge + a $X$ se, detta $F_n$ la f.d.r.~di $X_n$ e $F$ la f.d.r.~di $X$, allora: + \[ + \lim_{n \to \infty} F_n(x) = F(x), \quad \forall x \in \RR, + \] + ossia se $F_n$ converge puntualmente a $F$. +\end{definition} + +\begin{theorem}[Teorema centrale del limite] + Sia $(X_i)_{i \in \NN^+}$ una successione di v.a.~reali i.i.d~dotate di momento + secondo finito e q.c.~non costanti. Siano $m = \EE[X_1]$, $\sigma^2 = \Var(X_1)$. Si definisca + la successione $(Z_i)_{i \in \NN^+}$ in modo tale che: + \[ + Z_n = \frac{\left(\sum_{i=1}^n X_i\right)-nm}{\sqrt{n} \sigma} = \frac{\sqrt{n}}{\sigma} (\overline{X_n} - m). + \] + Allora la successione $(Z_i)_{i \in \NN^+}$ converge in legge a $N(0, 1)$, ovverosia: + \[ + \lim_{n \to \infty} F_n(x) = \Phi(x), \quad \forall x \in \RR, + \] + dove $F_n$ è la f.d.r.~di $Z_n$. +\end{theorem} + +\begin{remark} + Si può dimostrare che vale anche: + \[ + \EE\left[\varphi\!\left(\frac{\sqrt{n}}{\sigma} (\overline{X_n} - m)\right)\right] \to \EE[\varphi(Z)], + \] + per ogni $\varphi \in C_b(\RR)$, ovverosia per ogni $\varphi$ continua limitata. +\end{remark} + +\begin{remark} + Sfruttando il procedimento di standardizzazione si può dimostrare che + $\frac{1}{\sqrt{n}} \left(\sum_{i=1}^n X_i - nm\right) = \sqrt{n}(\overline{X_n} - m)$ converge + in legge a $Z_\sigma \sim N(0, \sigma^2)$. +\end{remark} + \end{multicols*} \ No newline at end of file