Rachunek prawdopodobieństwa 1, 2
\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\ep }{\varepsilon }\)
\(\newcommand {\ve }{\varepsilon }\)
\(\newcommand {\s }{\sigma }\)
\(\newcommand {\o }{\omega }\)
\(\newcommand {\f }{\varphi }\)
\(\newcommand {\vr }{\varrho }\)
\(\newcommand {\a }{\mathcal {A}}\)
\(\newcommand {\h }{\mathcal {H}}\)
\(\newcommand {\b }{\mathcal {B}}\)
\(\newcommand {\G } {\mathbb {G}}\)
\(\newcommand {\r } {\mathbb {R}}\)
\(\newcommand {\rn } {\mathbb {R}^n}\)
\(\newcommand {\Z } {\mathbb {Z}}\)
\(\newcommand {\z } {\mathbb {Z}}\)
\(\newcommand {\N } {\mathbb {N}}\)
\(\newcommand {\C } {\mathbb {C}}\)
\(\newcommand {\K } {\mathbb {K}}\)
\(\newcommand {\Q } {\mathbb {Q}}\)
\(\newcommand {\bx }{\mathbf {x}}\)
\(\newcommand {\by }{\mathbf {y}}\)
\(\newcommand {\bY }{\mathbf {Y}}\)
\(\newcommand {\p }{\mathbf {p}}\)
\(\newcommand {\P }{\mathbf {P}}\)
\(\let \leq \leqslant \)
\(\let \le \leqslant \)
\(\newcommand {\O }{\emptyset }\)
\(\newcommand {\imp }{\Longrightarrow }\)
\(\newcommand {\str }{\longrightarrow }\)
\(\newcommand {\rwn }{\Longleftrightarrow }\)
\(\newcommand {\di }{\displaystyle }\)
15.4 Twierdzenie o zbieżności
Jednym z ważniejszych twierdzeń w teorii martyngałów jest:
-
Twierdzenie – 15.21 Niech \(\left (\{X_n\}, \{\a _n\} \right )\) będzie supmartyngałem, lub submartyngałen spełniającym warunek:
\(\seteqnumber{0}{15.}{0}\)
\begin{equation}
\sup _n E(|X_n|) < \infty . \label {wnzbnp}
\end{equation}
Wtedy istnieje taka zmienna losowa \(X\), że \(E(X) \in \r \) oraz:
\[X_n\stackrel {1}{\to } X. \]
Dowód. Pomijamy.
Łatwo sprawdzić (ćwiczenie)
Jako wniosek otrzymujemy:
-
Wniosek – 15.23 Niech \(\left (\{X_n\}, \{\a _n\} \right )\) będzie supmartyngałem, \(X_n \ge 0\) dla \(n =1, 2, 3, ...\).
Wtedy istnieje taka zmienna losowa \(X\), że \(E(X) \in \r \) oraz:
\[X_n\stackrel {1}{\to } X. \]