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| Title of object: |
Lindeberg's central limit theorem |
| Canonical Name: |
LindebergsCentralLimitTheorem |
| Type: |
Theorem |
| Created on: |
2002-12-10 10:55:37.65497-05 |
| Modified on: |
2002-12-10 13:15:59.881843-05 |
| Classification: |
msc:60F05 |
| Defines: |
normal convergence, liapunov's central limit theorem, liapunov condition |
| Synonyms: |
Lindeberg's central limit theorem=Lyapunov's central limit theorem
Lindeberg's central limit theorem=central limit theorem
Lindeberg's central limit theorem=lyapunov condition
Lindeberg's central limit theorem=lindeberg condition |
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Content:
\textbf{Theorem (Lindeberg's central limit theorem)}
Let $X_1, X_2,\dots$ be independent random variables with distribution functions $F_1,F_2,\dots$, respectively, such that $EX_n=\mu_n$ and $\operatorname{Var}X_n=\sigma_n^2<\infty$, with at least one $\sigma_n>0$.
Let \[S_n = X_1+\cdots+X_n\;\mbox{and}\; S_n=\sqrt{\operatorname{Var}(S_n)} =
\sqrt{\sigma_1^2+\cdots+\sigma_n^2}.\]
Then the normalized partial sums $\frac{S_n - ES_n}{s_n}$ converge
\PMlinkname{in distribution}{ConvergenceInDistribution} to a random variable with normal distribution $N(0,1)$, if the following \emph{Lindeberg condition} is satisfied:
\[\forall \varepsilon>0,\; \lim_{n\rightarrow\infty} \frac{1}{s_n^2}
\sum_{k=1}^n \int_{|x-\mu_k|>\varepsilon s_n} (x-\mu_k)^2 dF_k(x) = 0.\]
\textbf{Corollary 1 (Lyapunov's central limit theorem)}
If the Lyapunov condition
\[\frac{1}{s_n^{2+\delta}}\sum_{k=1}^n E|X_k-\mu_k|^{2+\delta}
\xrightarrow[n\rightarrow\infty]{} 0\]
is satisfied for some $\delta>0$, the normal convergence holds.
\textbf{Corollary 2}
If $X_1,X_2,\dots$ are identically distributed random variables, $EX_n=\mu$ and $\operatorname{Var}S_n = \sigma^2$, with $0<\sigma<\infty$, the normal convergence holds; i.e. $\frac{S_n-n\mu}{\sigma \sqrt{n}}$ converges \PMlinkname{in distribution}{ConvergenceInDistribution} to a random variable with distribution $N(0,1)$.
\textbf{Reciprocal (Feller)}
The reciprocal of Lindeberg's central limit theorem holds under the following additional assumption:
\[\max_{1\leq k\leq n} \left(\frac{\sigma_k^2}{s_n^2}\right)\xrightarrow[n\rightarrow\infty]{} 0.\]
\textbf{Historical remark}
\begin{itshape}
\PMlinkescapetext{The normal distribution was historically called the law of errors. It was used by Gauss to model errors in astronomical observations, which is why it is usually refered to as the Gaussian distribution. Gauss derived the normal distribution, not as a limit of sums of independent random variables, but from the consideration of certain ``natural'' hypotesis for the distribution of errors; e.g. considering the arithmetic mean of the observations to be the ``most probable'' value of the quantity being observed.}
\PMlinkescapetext{Nowadays, the central limit theorem supports the use of normal distribution as a distribution of errors, since in many real situations it is possible to consider the error of an observation as the result of many independent small at the two run parallel.
\item Given three strands $A$, $B$ and $C$ so that $A$ passes below $B$ and $C$, $B$ passes between $A$ and $C$, and $C$ passes above $A$ and $B$, the strand $A$ may be moved to either side of the crossing of $B$ and $C$.
\end{enumerate}
\end{defn}
Note that number 1. is the inverse of number 2. and number 3. is the inverse of number 4. Number 5 is its own inverse. In pictures:
\begin{center}
\includegraphics[width= 1in]{twist}
\hskip 0.3in \raisebox{0.5in}{$\longleftrightarrow$}
\includegraphics[width= 1in]{untwist}
\includegraphics[width= 1in]{parallel}
\hskip 0.3in \raisebox{0.5in}{$\longleftrightarrow$} \hskip 0.2in
\includegraphics[width= 1in]{passover}
\includegraphics[width= 1in]{r3}
\hskip 0.3in \raisebox{0.5in}{$\longleftrightarrow$} \hskip 0.2in
\includegraphics[width= 1in, angle=180, origin= c]{r3}
\end{center}
Finding such a sequence of Reidemeister moves is generally not easy, and proving that no such sequence exists can be very difficult, so other approaches must be taken.
Knot theorists have accumulated a large number of \emph{knot invariants}, values associated with a knot diagram which are unchanged when the diagram is modified by a Reidemeister move. Two diagrams with the same invariant may not represent the same knot, but two diagrams with different invariant never represent the same knot.
Knot theorists also study ways in which a complex knot may be described in terms of simple pieces --- for example every knot is the connected sum of non trivial prime knots and many knots can be described simply using Conway notation.
\subsection*{formal definitions of \emph{knot}}
\subsubsection*{polygonal knots}
This definition is used by Charles Livingston in his book \emph{Knot Theory}. It avoids the problem of wild knots by restricting knots to piece-wise linear (polygonal) curves. Every knot that is intuitively ``tame'' can be approximated by such knot. We also define the \emph{vertices}, \emph{elementary deformation}, and \emph{equivalence} of knots.
\begin{defn}
A \emph{knot} is a simple closed polygonal curve in $\mathbb{S}^3$.
\end{defn}
\begin{defn}
The vertices of a knot are the smallest ordered set of points such that the knot can be constructed by connecting them.
\end{defn}
\begin{defn}
A knot $J$ is an \emph{elementary deformation} of a knot $K$ if one is formed from the other by adding a single vertex $v_0$ not on the knot such that the triangle formed by $v_0$ together with its adjacent vertices $v_1$ and $v_2$ intersects the knot only along the segment $[v_1,v_2]$.
\end{defn}
\begin{defn}
A knot $K_0$ is \emph{equivalent} to a knot $K_n$ if there exists a sequence of knots $K_1,\hdots,K_{n-1}$ such that $K_i$ is an elementary deformation of $K_{i-1}$ for $1<i\leq n$.
\end{defn}
\subsubsection*{smooth submanifold}
This definition is used by Raymond Lickorish in \emph{An Introduction to Knot Theory}.
\begin{defn}
A \emph{link} is a smooth one dimensional submanifold of the 3-sphere $S^3$.
A \emph{knot} is a link consisting of one component.
\end{defn}
\begin{defn}
Links $L_1$ and $L_2$ are defined to be equivalent if there is an orientation-preserving homeomorphism $h:
S^3 \to
S^3$ such that $h(L_1)
h(L_2)$.
\end{defn} |
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