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extra - ΘΕΩΡΙΑ ΠΙΘΑΝΟΤΗΤΩΝ...

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ΘΕΩΡΙΑ ΠΙΘΑΝΟΤΗΤΩΝ ΑΣΚΗΣΕΙΣ ΣΤΑ MARTINGALES Άσκηση : Έστω { } ακολουθία ανεξάρτητων τ .μ. μ ε > 0 P- σ . β . και Ε ( )=1 . Θέτου μ ε . ∆είξτε ότι η { } είναι F - martingale μ ε F n X ,n ` n ` n n X ` n X n 1 2 n Z X X ... X ,n = n 1 n (X ,...,X ). σ = n Z , n ` Λύση Επειδή ανεξάρτητες μ ε Ε ( < συ μ περαίνου μ ε ότι . Είναι ακό μ α προφανές ότι η τ .μ. Ζ είναι F - μ ετρήσι μ η . Τώρα F )= F )= F )= i X ,i 1,...,n = 1 n X ) E(X ) = i X ) ` Z n n 1 X / + 1 n E(X E(X ) n ⋅⋅⋅ ⋅⋅⋅ < ∞ ∀ n n 1 E(Z / + n E( n E(X n n Z n 1 E(X / + n n Z n 1 ) + Λόγω της ανεξαρτησίας της από την F . n 1 X + n Ό μ ως =1 και συνεπώς F n ) P- σ . β . n 1 E(X ) + n 1 E(Z / + n Z = Άσκηση : Έστω ακολουθία τ .μ. { }, ανεξάρτητων μ ε Ε ( )=0 και V(X i )= σ για κάθε i=1,2,… . ∆είξτε ότι η ακολουθία Y ( είναι F - martingale μ ε F n X ,n ` n i X 2 2 n ,n 2 n n k k 1 X ) σ = = ` n n 1 (X ,...,X ). σ = Λύση Εύκολα διαπιστώνεται ότι Ε (| Y )< και είναι
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