The Properties of Probability of Normal Chain

Transcription

1 I. J. Coep. Mah. Sceces Vol o HIKARI Ld The Properes of Proaly of Noral Cha L Che School of Maheacs ad Sascs Zheghou Noral Uversy Zheghou Cy Hea Provce 4544 Cha cluu6697@sa.co Malg address: No.6 Ygca sree Norh Uversy Cy Zheghou Cy Hea Prov Cha 4544 Zhog-guag Fa School of Maheacs ad Sascs Zheghou Noral Uversy. Zheghou Cy Hea Provce 4544 Cha Copyrgh 23 L Che ad Zhog-guag Fa. Ths s a ope access arcle dsrued uder he Creave Coos Aruo Lcese whch pers uresrced use dsruo ad reproduco ay edu provded he orgal work s properly ced. Asrac I order o eer sudy he Markov cha of rack srucure ypcal laeral rh-deah process aure has regular cha of -deph sudy hs paper aly dscusses soe proalsc properes of oral cha. Keywords: Noral cha; Soppg e; Jupg rack Iroduco For he Markov cha of sochasc process odel laeral rh-deah process s oe of ypcal Markov cha. However how o fd ou all soluos of laeral rh-deah process aracs he eresg of ay epers ad eeds a lo of work. Acadeca Wag Zku e. al proposed he ehod of cog he fuco srucure ad proaly ehods ad hs ehod s cosdered o e ore reasoale ad araced a lo of aeo[]-[3]. Therefore s ecessary o

2 434 L Che ad Zhog-guag Fa sudy he proalsc properes of oral cha correspodg o Q ar. Ths paper wll dscuss proalsc properes of oral cha. 2 Prelary oes Le E { L 2 2 L } a ± ± 2 L s a group of posve Q q ( ) s laeral rh-deah ar of E where a q ( a + ) + else Le p () e a al rasfer fuco whch s defed y he laeral rh-deah ar Q ( q ). Ad p p () δ () p ( ) k f f f E E E where E E {} U E p () s hoes al raso fuco Q q q ( ) s desy of p (). For E p () p () hus q. Ovously s asorg sae ad E q. X ( Ω FF X θ P ) whch s oral cha correspodg o p (). Le T s he up es of { } assue T. For E f { X } X 2 L For coveece δ.

3 The properes of proaly of oral cha 435 E > ( ) F P { X T } () > hus: () F q () e δ ( ) F + P { X T } + P { X T T X E T } q qs ( ) e δ + e q F ( s) ds k k k E By he defo of al rasfer fuco { X T } { X δ > } l F ( ) p ( ). > so ha ( ) ( ) { δ > } { } P X l F () p () P X ad P { X E δ } P { X E} > ha s alos ceraly geg δ > fro X E. The ha s alos ceraly geg X fro δ. Therefore X s alos [ δ ). 3. Proalsc properes of oral cha <. Propery P { δ < δ} P { δ δ } Prove: For E + le P { δ < δ } + L. Ovously. Ad for < < { } P δ < δ { δ < δ } { δ < δ T T } P X + P X +

4 436 L Che ad Zhog-guag Fa a P a + a + { δ < δ } P { δ < δ } + + Thus a + + a + a + a a a La ( ) L ( ) + + L a a La L so ha a a La k + k + L k L [ ] + aa 2 La [ ] cos a + Because of so. Le ge P { δ < δ } P { δ < δ}. Propery2 { δ < δ } { δ < δ } L. P P

5 The properes of proaly of oral cha 437 Prove: Assue a > >. For le P { δ < δ } so ha.accordg o Propery ge a + Thus a + < a + a + + a aa La ( ) L + L aa L L a so ha a. If he ha s a. P P L. Therefore { δ δ } { δ δ } Noe : propery ad 2 descre he proaly of 2 L. Propery3 Assue R < le y E { e } λσ Ad y E sasfes equaos: so ha l y.

6 438 L Che ad Zhog-guag Fa y λ μ ( y )( ) + λ( ) μ ( y ) k k k k k k k k ( )( ) y ( ) λ μ + λ μ ( y ) k k k k k k k The process of provg ca e see referece [4]. Propery 4 Assue R < so ha E { σ } μ ( ) + ( ) μ k k k k k k ( ) ( ) μ + μ k k k k k Prove: Based o Propery 3 ge { σ } { } P < l E e λσ λ Whleλ e λσ λ σ hus E e E { σ } l λ λ λσ { } Cog forula y of propery 3 ad aove forula so ha E { σ } μ ( ) + ( ) μ k k k k k k ( ) ( ) μ + μ k k k k k We ca ge he followg propery fro propery 3 ad 4. Propery 5 If R P { σ }.

7 The properes of proaly of oral cha 439 Propery 6 For k k k E P { σ } or { } P σ < ad hese wo kds of proales are refleced y R & R <. Noe 2: Propery 5 ad 6 descre he proaly of caocal easure. 4 Cocluso Blaeral rh-deah process s oe of pora Markov cha. I s wdely used he acual odels of chesry physcs edce ec. Ad laeral rh-deah process has pora heorecal sgfcace. Therey o provde heorecal ass for fdg ou all soluos of he rregular ar Q ased o he research o proaly propery of oral cha correspodg o laeral rh-deah ar Q of se E. Refereces [] Wag Zku Markov process ad aheacs oday [M]. Chagsha: Hua scece ad Techology Press [2] Dea L. Isaecso Rchard W.Mdase.Makrov Chas Theory ad Applcaos [M]. Joh Wley&Sos Ic [3] Hou Zheg The oly crero Q process[j] scece Cha seres A 974 ( 2 ). [4] Wag Zku Yag ag-qu. Brh ad deah processes ad Markov chas ( Secod Edo ) [M]. Beg: Scece Press ; [5] Hou Zheg Zou Jehog Zhag Hau Q ar Markov process of Chagsha Hua scece ad Techology Press Receved: Feruary 23