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���� JFIF � � �� Photoshop 3.0 8BIM �� 4Exif MM * ( �i & 222 0 obj << >> /Filter /DCTDecode /ColorSpace /DeviceRGB (Strong Markov Property) If ˝is a stopping time for a random walk fSngn 0, then the post-˝sequence fS˝+j gj 0 is also a random walk, with the same step distribution, started at S˝, and is independent of the random path fSj gj ˝. -fyFE ���4�����@��X0���$� �+&�h -�~�"�gE�+~��H `\�t)qE\ n ����� ޸@*�=�o>%�A������\M�*�J���}�>�� �d�ǝ��AK�C9,j���,s�z՝a$��4AOA(c���FXI����S�V}z0�G��=�']��˧kb����Xy��;JП��8S�`a�ya2}>�ȹ�0���'�P �rQA�0i[E�HX�h�X�Γ�a�x��r��N+a��Q����ͣ��%X��g�:h9������(� �o� ��q �)='�;eޠY��ê�7��|��z�����81i�LH[< ݭ-R�J_`Y$�-����a�`ٍ��Fj����u�5�.��aRQ�Ӆ��]�˄+���s��]��q>�ů���~r �!���~�&q�B"�?�7RV�. �1(�П�z h���wր-}�P�$z��747�?p� $�J.�lVI$���,�N�J�I�Y�/� At each time unit, a walker flips /First 811 In mathematics, a random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.. An elementary example of a random walk is the random walk on the integer number line, , which starts at 0 and at each step moves +1 or −1 with equal probability. We use this chapter to illustrate a number of useful concepts for one-dimensional random walk. [� In later chapters we will consider d-dimensional random walk as well. 1 Simple Random Walk We consider one of the basic models for random walk, simple random walk on the integer lattice Zd. /N 100 /Type /ObjStm J*)+IH*��D�H��1JI&$ʌ��G��O>��и���f�RIi�KJJ�FN*K(դrE*�,�A ��,�3�QV`��K��r�q�r��R� ��S�Ӥ��9')��"Ux� endobj /Width 136 �� %���� [��.c+q#������ 6����� ���i%� ����R�`i#J ��\�LN�ɜ+�I������ER��6Y�H0O�`��CY�;����K��I�e�ZM��`�AK�^���g��?���-=�WUMx!���׳gﭳ��n�2��s��~M_��{� �3u�޽c�[��p�M=,��E߶�-)ڵ�+Ѳ�h*4݆����u�ј�T��b�v��նm�Th��BM�u���>P\�te;)�~���{ȏ"�?����Y>�y�ӛ�;c�ov�2���u����]��I��oS��1�5�b��\�Ґ۬/L�$y��ڸ���f�WƇ)��]����͕����l�M�Ն����#��x�r٨�i�`�ϩon������� stream xڅS�n�0��+x$����|�GIڴN��n�qp-6V-+����%� ( >> stream }-��Bu�I�h��s�5�T|C�>�T;���ZګMu��A�v�J���}@.v���Yx� �?��aߜDe�y�{x�&�x�E"��}��'�_���ǹ��oo������q=�aܑ%j>Д�fKh���P�n o�8�Q�L��PS�e7��Q�k����͈Y�\�ƺ��~ݥa���;E��t�������#�� 1T�/u�X�\��ҵ����� ~7��ƳE�)V���tCW���&� n�U�}���]��9KKAk�b���s�w����{Ѣ�3�dߥ�*u���y@����4�0���܊"�[� �x��{ݫ�ZXMbAh�v�'v��UMӟ�:o9s9�J��B�����jBl���!��~H#.�9���s�Q� and, in fact, we do not say anything about the probability law of the sequence at all. I�P�S�ヴ�� 1.1 One dimension We start by studying simple random walk on the integers. /BitsPerComponent 8 /Filter /FlateDecode /Height 156