-
Recent Posts
Recent Comments
Archives
Categories
Meta
Tag Archives: probability theory
Notes on the t-distribution
Let $X_1, X_2, \ldots, X_n$ be IID normal variables with mean $\mu$ and variance $\sigma^2$. Let $\overline{X}$ be the mean of these variables, and define the deviations $D_i = \overline{X} – X_i$. Let $Y = [X_1 \ X_2 \ \ldots … Continue reading
Posted in Uncategorized
Tagged distributions, probability theory, statistics
Comments Off on Notes on the t-distribution
Portmanteau Theorem
Statement: $X_n \overset{D}{\to} X$ iff $E[g(X_n)] \to E[g(X)]$ for any bounded, continuous Lipschitz g. We prove this in two parts: Part 1 Here we show $X_n \overset{D}{\to} X$ implies $E[g(X_n)] \to E[g(X)]$ for any continuous bounded g. Let $F_n$ denote the cdf … Continue reading
Posted in Uncategorized
Tagged portmanteau theorem, probability theory
Comments Off on Portmanteau Theorem
Strong Law of Large Numbers
Truncation Lemma Statement: Let $X_n$ be a sequence of IID variables with finite mean, and let $Y_n$ = $X_n I_{|X_n| \leq n}$. Then $\sum \frac{Var(Y_n)}{n^2} < \infty$. Proof: Lemma: If $c \geq 0$, then $c^2 \sum_{n=1}^\infty \frac{I_{|c| \leq n}}{n^2} \leq 1 … Continue reading
Posted in Uncategorized
Tagged law of large numbers, probability theory
Comments Off on Strong Law of Large Numbers
Kolmogorov’s Inequality
Statement: Let $X_i$ be independent random variables with mean 0, finite variance and $S_n = X_1 + \ldots + X_n$. Then $P\left( \displaystyle \max_{1 \leq i \leq n} |S_i| \geq a \right) \leq \frac{1}{a^2} Var(S_n)$. Note that Chebyshev’s inequality already … Continue reading
Posted in Uncategorized
Tagged kolmogorov's inequality, probability theory
Comments Off on Kolmogorov’s Inequality