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- Title
Comparing backwards and forwards random walk maxima.
- Authors
Sigman, Karl
- Abstract
What makes this case easier (and elegant) to analyse is that one can construct the time reversal by using the time-reversed transition probabilities of the underlying Markov chain with stationary distribution HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mi> </mi></math> ht : HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><msubsup><mi>P</mi><mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msubsup><mo>=</mo><mfrac><msub><mi> </mi><mi>j</mi></msub><msub><mi> </mi><mi>i</mi></msub></mfrac><msub><mi>P</mi><mrow><mi>j</mi><mo>,</mo><mi>i</mi></mrow></msub><mo>.</mo></mrow></math> ht Other key tools used are the Perron-Frobenius theorem for finite matrices and importance sampling/change of measure. ( HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mover><mo>=</mo><mi mathvariant="normal">d</mi></mover></math> ht denotes "equal in distribution".) </mo></mrow></mtd></mtr></mtable></mrow></math> ht Graph I Does it always hold that i HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mi>M</mi><mi>k</mi></msup><mo stretchy="false">)</mo><mo> </mo><mi> </mi></mrow></math> ht I if and only if i HT <math xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mi>E</mi><mo stretchy="false">(</mo><msup><mrow><msup><mi>M</mi><mrow><mo stretchy="false">(</mo><mi>r</mi><mo stretchy="false">)</mo></mrow></msup></mrow><mi>k</mi></msup><mo stretchy="false">)</mo><mo> </mo><mi> </mi><mo>,</mo><mspace width="4pt" /><mi>k</mi><mo»=</mo><mn>1</mn></mrow></math> ht ?.
- Subjects
RANDOM walks; MARKOV processes; POINT processes; TIME reversal; CENTRAL limit theorem; RANDOM variables
- Publication
Queueing Systems, 2022, Vol 100, Issue 3/4, p349
- ISSN
0257-0130
- Publication type
Article
- DOI
10.1007/s11134-022-09815-1