Metrik değerli dizilerin istatistiksel yakınsaklığı / Statistical convergence of metric valued sequences

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METRİK DEĞERLİ DİZİLERİN İSTATİSTİKSEL YAKINSAKLIĞI

YÜKSEK LİSANS TEZİ Muharrem KARATAŞ

(141121111) Anabilim Dalı : Matematik Program : Analiz ve Fonksiyonlar Teorisi

Tez Danışmanı: Prof. Dr. Mikail ET Ağustos-2018

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: : + _: : : : : lim : 1 : : ( ) :

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( ) = ( ) 0 = 0 = ¡1 0

= ( ) 0 1 [ ]

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0 1 [ ] : ( ) 0 0 + 1 [ ] 2 [ ( )] 0

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: ( ) ( ) 0 ( ) = 0 = ( ) = ( ) ( ) ( ) + ( ) ( ) = ( ) 0 0 ( ) 0 ( ) lim !1 = ( ) ( ) : ( ) = ( = 1 2 3 ) = ( ) ( = 1 2 3 ) = ( ) = ( ) + = ( ) + ( ) = ( )

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: ( ) = lim !1 1 : = ( ) 0 lim !1 1 : = 0 lim = = 0

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0 = ( ) = 1 = 2 _{= 1 2} 0 = 2 = ( ) : = 0 lim !1 1 : = 0 lim !1 = 0 = = 2 _{= 1 2} 0 = 2 = ( 1 1 1 ) 0 0

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= 0( ) lim !1 1 : 0 = 0 = lim = lim = lim ( + ) = + lim = = ( ) 0 = 0 1 = 1 = _¡1 = ( ) = ( ) = ( _¡1 ] = _¡1+1 = 2 = ¡1 = ( ) lim 1 =1 = 0 1 1 = sup 1 =1

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= ( ) lim 1 2 = 0 = sup 1 = (2 ) = 1 = ( ) 0 lim 1 : = 0 = ( ) lim = ( )

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0 1 " = : ( ) = lim !1 1 [ ] : = 1 = (2 ) 0 1 ( ) ( ) 0 1 ( ) ( ) [ ] [ ] 1 [ ] 1 [ ] lim !1 1 [ ] : lim!1 1 [ ] :

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( ) ( ) = ( ) (0 1] 0 lim !1 1 [ ] : = 0 = ( ) = ( ) lim = (0 1] = 0 0 = 1 = (2 ) 1 = 1 = 2 0 = 2 = 1 2 3 1 lim !1 1 [ ] : 1 lim!1 ¡1 2 [ ] = lim!12 [ ] = 0 lim !1 1 [ ] : 0 lim!1 ¡1 2 [ ] = lim!12 [ ] = 0 1 0 = ( ) = ( ) + = ( ) + ( ) = ( )

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= (2 ) = ( ) = (2 0 0 2 0 0 0 0 2 0 0 0 0 0 0 2 0 ) 1 : 0 1 2 + 1 (1₂ 1] lim = 0 (0 1] 0 1 (0 1] 0 1 (0 1] 0 1 0 1 [ ] : 1 [ ] : = (2 ) = ( ) = 3 = 3 0 = 3 1 3 1 lim = 0 0 1 3

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( ) = ( ) 0 1 lim !1 1 [ ] 2 = 0 = ( ) = ( ) ( ) lim = (0 1] ( ) = 0 0( ) = 1 = (2 ) = ( ) (0 1] 0_{( )} ₁ = sup 1 [ ] 0 2 1 0 1 1 = sup 1 [ ] 0 2 0_{( )} _{(4 2)} ( ) = ( ) = ( ₁ ₂ ) 0_{( )} = sup 1 [ ] 0 2 1 0 1 [ ] 0 2 1 0

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0 ( ) = ( 1 2 ₎ lim = 0 0 0 = 0( ) 0 1 [ ] 0 2 lim 1 [ ] 0 2 = 1 [ ] 0 2 = sup 1 [ ] 0 2 1 0 1 [ ] 0 2 2 1 [ ] 0 2 0 ₊ 1 [ ] 0 2 0 0_{( )} 0 1 0_{( )} _{(4 3)} (0 1] 0 1 = ( ) = = 1 2 3 0

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1 2 1 0 1 2 ( ) 0 1 0 1 + 1 2 1 2 = (2 ) (4 1) 1 =1 0 2 3 = 2 ¡1 3 0 (1 3 1] ( ) (0 1 3) 1 =1 0 2 3 ₁ ( )

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( ) 0 lim !1 1 : ( ) = 0 ( ) ( ) = : ( ) lim !1 1 : ( ) = 0 ( ) lim !1 1 2 [ ( )] = 0 = ( ) = ( ) = ( ) 2 [ ( )] 2 2 ( ) [ ( )] + 2 2 ( ) [ ( )] 2 2 ( ) [ ( )] : ( )

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1 2 [ ( )] : ( ) lim 1 2 [ ( )] lim : ( ) = 0 = = ( ) = = 1 2 3 0 0 lim 1 : 0 = lim = 0 = 1 1 2 0 = 1 2 = 1 = ( ) ( ) 0 ¡1 _{: (} ₎ 2 1 2 = : ( ) 2 1 ¡1 2 [ ( )] = ¡1 2 2 [ ( )] + ¡1 2 2 [ ( )] 2 + ¡1 2 = = ( )

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= ( ) = ( ) lim !1inf 1 = ( ) lim !1sup lim !1sup 0 = : ( ) 0 0 0 0 = max : 1 0 ¡1 1 : ( ) 1 ¡1 : ( ) = 1 ¡1 1 + 2+ + 0 + 0+1+ + ¡1 0 + 1 ¡1 sup 0+1+ + 0 ¡1 + 0 ¡1 +

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= ( ) = = ( ) = 0 = 1 0 (2 1)! (2 )! = 1 2 3 1 (2 )! (2 + 1)! = (2 )! 1 +1 +1 : = 0 1 (2 +1)!¡(2 )! 2( +1)!¡(2 )! 0 1 0 ( ) 0 = (2 + 1)! 1 +1 +1 : 1 = 0 1 (2 )!¡(2 ¡1)! (2 +1)!¡(2 ¡1)! 0 1 1 ( 0) 0 0 1 = 0 lim inf 1 ( ) ( ) = ( ) lim sup ( ) ( ) ( 1) ( 2) 1 2 ( 1) ( 2)

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( 1) 1 ( 2) 2 ( ) = ( ) 0 = 0 = _¡1 0 = ( ) 0 1 = ( _¡1 ] [ ] [ ] = ([ ] ) = ([ 1] [ 2] [ ] ) 0 lim !1 1 [ ] : ( ) = 0 lim !1 1 [ ] : ( ) = 0 1 = 0 1 1 = ( ) = = ( ) = 1 = 2 0 = 2 = 1 2 3 1 lim !1 1 [ ] : ( 1) lim!1 ¡1 2 [ ] = lim!12 [ ] = 0

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lim = 1 lim !1 1 [ ] : ( 0) lim!1 ¡1 2 [ ] = lim!12 [ ] = 0 lim = 0 = ( ) 1 0 = ( ) 0 + lim !1 1 [ ] 2 [ ( )] = 0 ( ) 0 1 + _{( )} ( ) 0 ( ) 2 [ ( )] 2 ( )¸ [ ( )] + 2 ( ) [ ( )] 2 ( )¸ [ ( )] : ( ) 1 [ ] 2 [ ( )] [ ] : ( ) [ ] : ( ) ( ) = = 1

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= = 1 2 3 0 = ( ) 0 1₂ 1 1 [ ] : ( 0) = [ ] 0 1 2 1 0 0 1 = 1 1 [ ] 2 [ ( )] = 1 [ ] 2 [ ( )] = [ ] = 1 1 [ ] 2 [ ( )] = [ ] 1 0 1 0 ( ) 1₂ 1 ( ) ( ) = ( ) = 0 1 lim inf 1 ( ) lim inf 1 1 + 0 1 + ¡1 1 + ¡1 1 1 + 1 ¡1 1 1 1 + 0 1 1 + 1 + 1 [ ] (1 + ) 1 [ ] 1 [ ] : ( ) (1 + ) 1 [ ] : ( )

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= 0 1 lim sup lim sup 0 = : ( ) 0 1 lim !1 1 [ ] : ( ) = 0 0 0 1 0 0 [ ] [ ] [ ] = max : 1 0

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¡1 1 : ( ) 1 ¡1 : ( ) = 1 ¡1 1 + ₂ + + ₀ + ₀₊₁ + + ¡1 0 + 1 ¡1 0+1 0+1 ₊ ₊ 0 ¡1 + 1 ¡1 sup 0+1+ + 0 ¡1 + 0 ¡1 0 ¡1 + 0 ¡1 + ( ) = 0 1 lim inf [ ] 0 = 0 lim inf 1 ( ) 0 = 1 [ ] 2 [ ( )] = 1 [ ] =1 [ ( )] 1 [ ] ¡1 =1 [ ( )] = [ ] [ ] 1 [ ] =1 [ ( )] [ ¡1] [ ] 1 [ _¡1] ¡1 =1 [ ( )] = _¡1 [ ] [ ] (1 + ) [ ¡1] [ ] 1 1 [ ] =1 [ ( )] 0

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1 [ _¡1] ¡1 =1 [ ( )] 0 0 ( ) ( ) 0 1 + _{( )} _{( )} = ( ) 0 1 0 1 [ ] ₂ [ ( )] 1 [ ] 2 [ ( )] ( ) ( ) ( ) = = 1 0 = 1 2 3 1 2 1 ( ) 0 1 2 ( ) 0 1 ( ) = = ( ) = = 1 2 3 0 1 2 1 0 1 2 lim sup [ _¡1] (0 1] ( )

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lim sup [ _¡1] 1 _[ ¡1] 0 0( ) 0 = 1 2 3 sup_¸ 0 0 ¡1 1 =1 [ ( 0)] 1 [ _¡1] =1 [ ( 0)] = 1 [ _¡1] 1 [ ( 0)] + 2 [ ( 0)] + + [ ( 0)] = 1 [ _¡1] 1+ 2 1 [ _¡1] 2+ + ¡1 [ _¡1] + +1 [ _¡1] +1 + + ¡1 [ ¡1] sup ¸1 [ ¡1] + sup ¸ [ ¡1] [ _¡1] + ¡1 1 =1 [ ( 0)] 0 0 (0 1] lim sup [ _¡1] ( ) = ( )

lim = lim = 0 = 0 lim sup _[

¡1] 0 ( ) 0 0 0( ) 0 0 = 1 1 =1 ( 0) 0 1 =1 ( 0) + 1 =1 ( ) 1 ( 0) 0

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= 0 0 1 = 0 = lim !1inf [ ] [ ] 0 0 lim !1inf _{[ ]} = 1 0 0 1 = 0 = (5 1) 0 ( ) ( ) (5 2) 1( ) ( ) 0 ( ) (5 2 1) 1 [ ] ₂ [ ( )] [ ] [ ] 1 [ ] 2 [ ( )] 0 ( ) ( ) = ( ) ₁( ) ( ) (5 2 2) = ( ) ₁( ) ( ) 0

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1 [ ] ₂ [ ( )] = 1 [ ] _{2 ¡} [ ( )] + 1 [ ] ₂ [ ( )] [ ] + 1 [ ] ₂ [ ( )] [ ] [ ] + 1 [ ] ₂ [ ( )] [ ] 1 + 1 [ ] 2 [ ( )] 1( ) ( ) 0 ( ) 0 1 = 0 = (5 1) 0 ( ) (5 2) 0 ( ) = ( ) 0 2 [ ( )] = 2 ( )¸ [ ( )] + 2 ( ) [ ( )] 2 ( )¸ [ ( )] 2 [ ( )] : ( ) 1 [ ] ₂ [ ( )] 1 [ ] : ( ) [ ] [ ] 1 [ ] : ( )

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(5 1) = ( ) 0 ( ) lim = = ( ) ₁( ) ₁( ) ( ) 0 0 1 [ ] ₂ [ ( )] = 1 [ ] _{2 ¡} [ ( )] + 1 [ ] ₂ [ ( )] [ ] + 1 [ ] ₂ [ ( )] [ ] [ ] + 1 [ ] ₂ [ ( )] [ ] 1 + 1 [ ] ₂ ( )¸ [ ( )] + 1 [ ] ₂ ( ) [ ( )] [ ] 1 +[ ] : ( ) +[ ] [ ] 1 +[ ] : ( ) + [ ] (5 2) = ( ) 0 ( )

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