This is from Durrett's probability:Let X n ∈ [ 0 , 1 ] be adpated to F n . Let α , β &gt; 0 with α + β = 1 and suppose P ( X n + 1 = α + β X n ∣ F n ) = X n , P ( X n + 1 = β X n ∣ F n ) = 1 − X n Show P ( lim n X n = 0 or 1 ) = 1.I know there is question Martingale convergence proof here but can't we prove this using levy's 0-1 law?What can we say about levy's 0-1 law regarding Martingale Sequence?

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This is from Durrett&#039;s probability:Let       X    n    ∈  [  0  ,  1  ] be adpated to             F        n  . Let   α  ,  β  &amp;gt;  0 with   α  +  β  =  1 and suppose      P    (      X          n      +      1        =  α  +  β      X    n    ∣            F        n    )  =      X    n    ,        P    (      X          n      +      1        =  β      X    n    ∣            F        n    )  =  1  −      X    n  Show       P    (      lim    n        X    n    =  0    or    1  )  =  1.I know there is question Martingale convergence proof here but can&#039;t we prove this using levy&#039;s 0-1 law?What can we say about levy&#039;s 0-1 law regarding Martingale Sequence?

Oliver Shepherd · Accepted Answer

X    n   is a martingale, and so converges (boundedly) to a random variable       X    ∞  . You&#039;ll be able to write a recursion for the second moments       E    [      X    n    2    ], leading, after a passage to the limit as   n  →  ∞, to the conclusion that       E    [      X    ∞    2    ]  =      E    [      X    ∞    ].

This is from Durrett's probability: Let X n </msub> ∈ [

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