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Last updated on: Sun Jul 19 09:18:24 IST 2026

Miscellaneous tools

$\newcommand{\rightarrowD}{\stackrel{d}{\longrightarrow}}$ $\newcommand{\rightarrowP}{\stackrel{P}{\longrightarrow}}$ $\newcommand{\v}[1]{\boldsymbol{#1}}$ Miscellaneous tools

1. Slutsky's theorem

TheoremLet $(X_n), (Y_n), X$ be random variables defined on the same probability space. $c\in{\mathbb R}.$ Let $X_n\rightarrowD X$ and $Y_n\rightarrowP c.$

Then
  1. $X_n+Y_n\rightarrowD X+c.$
  2. $X_n-Y_n\rightarrowD X-c.$
  3. $X_nY_n\rightarrowD cX.$
  4. $X_n/Y_n\rightarrowD X/c$ if $c\neq 0.$

Proof:

[QED]

1.1. Problem set

EXERCISE 1: Let $X_n\rightarrowD N(0,1)$ and $Y_n\rightarrowP 5.$ Then what is the limiting distribution of $X_n+Y_n?$

EXERCISE 2: Let $X_n\rightarrowD X$ and $Y_n\rightarrowP Y.$ Show that $X_n+Y_n\rightarrowD X+Y$ need not hold.

EXERCISE 3: Let $X_n\rightarrowD N(0,1)$, $Y_n\rightarrowP 5$ and $Z_n\rightarrowP 4$ with $z_n > 0.$ Then what is the limiting distribution of $\frac{X_n+Y_n}{\sqrt {Z_n}}?$

EXERCISE 4: Suppose that $\sqrt n(X_n-\theta)\rightarrowD Z$ and $Y_n\rightarrowP a.$ Show that $\sqrt n(X_nY_n-a\theta)\rightarrowD aZ.$

EXERCISE 5: Let $X_n$ be asymptotically $N\left(\mu,\frac{\sigma^2}{n}\right).$ What is the asymptotic distribution of $\frac{X_n}{1+X_n}?$

EXERCISE 6: Let $T_n$ be a consistent estimator of $\theta,$ and let $S_n$ be a consistent estimator of $\sigma^2.$ Show that the Studentized statistic $\frac{T_n-\theta}{\sqrt{S_n}}$ has the the same asymptotic distribution as $\frac{T_n-\theta}{\sigma},$ whenever an asymptotic distribution exists.

EXERCISE 7: Let $X_n\rightarrowD X$ and $X_n+Y_n\rightarrowD X+1.$ Does this imply that $Y_n\rightarrowP 1?$

2. Delta method

Delta method Let $\frac{\sqrt{n}(X_n-\mu)}{\sigma}\rightarrowD$ some distribution. Let $f:{\mathbb R}\rightarrow{\mathbb R}$ be a differentiable function with $f'(\mu)\neq 0.$

Then $\frac{\sqrt{n}(f(X_n)-f(\mu))}{\sigma f'(\mu)}\rightarrowD$ the same distribution.

Proof:Nice proof using Skorohod in Resnick (p262).[QED]

2.1. Problem set

EXERCISE 8: Suppose that $\sqrt n(S_n^2-\sigma^2)\rightarrowD N(0,\theta^2).$ Show that $\sqrt n(S_n-\sigma)\rightarrowD N\left(0,\frac{\theta^2}{4 \sigma^2}\right).$

EXERCISE 9: If $\sqrt n(\bar X_n-\mu)\rightarrowD N(0,1),$ show that $\sqrt n(\log \bar X_n-\log \mu)\rightarrowD N(0,\frac{1}{\mu^2}).$

EXERCISE 10: If $\sqrt n(T_n-\theta)\rightarrowD N(0,\sigma^2)$ for some $\theta\neq 0,$ then show that $\sqrt n\left( \frac{1}{T_n}-\frac 1\theta\right) \rightarrowD N\left(0,\frac{\sigma^2}{\theta^2}\right).$

EXERCISE 11:  We toss a coin with unknown $P(H)=p\in (0,1).$ Let $X_n = $ proportion of heads. Find an asymptotic distribution for the odds ratio $\frac{X_n}{1-X_n}.$

EXERCISE 12: $T_n$ is an estimator for a parameter $\theta$ with asymptotic distribution $N\left(\theta, \frac{\sigma^2}{n}\right).$ Use deltan method to obtain an approximate 95% confidene interval for $e^\theta.$

3. Variance stabilising transform

Variance stabilising transform Let $\frac{\sqrt{n}(X_n-\mu)}{\sigma(\mu)}\rightarrowD$ some distribution. Let $f:{\mathbb R}\rightarrow{\mathbb R}$ be a differentiable function with $f'(\mu) = \frac{1}{\sigma(\mu)}.$

Then $\sqrt{n}(f(X_n)-f(\mu))\rightarrowD$ the same distribution.

Such an $f$ is called a variance stabilising transform.

Proof:Immediate from the last theorem.[QED]

The most common application of this is when we have an estimator $T_n$ for some parameter $\mu$ with $\frac{\sqrt n(T_n-\mu)}{\sigma}\rightarrow N(0,1).$ This allows us to obtain a (large sample) confidence interval for $\mu$ as $(T_n-a \sigma/\sqrt n,T_n+a \sigma/\sqrt n).$ But this requires $\sigma$ to be known. Often $\sigma$ is a function of the unknown $\mu.$ Then we can use a variance stabilising transform $f(\cdot)$ to get $\sqrt n(f(T_n)-f(\mu))\rightarrow N(0,1),$ from which we can obtain a confidence interval for $f(\mu).$ Since $f$ is one-one we can obtain a confidence et for $\mu$ from it.

3.1. Problem set

EXERCISE 13: We know that if $X\Poi(\lambda),$ then $V(X) = \lambda.$ Find a variance stabilising transform for $X,$ i.e., some function $f(\cdot)$ such that $f(X)$ has variance free of $\lambda$ (approximately). What is this constant variance?

EXERCISE 14: Let $X\sim Binom(n,p).$ Find a variance stabilising transform for $\frac Xn.$

EXERCISE 15: Let $X\sim$ Exponential distribution with rate $\lambda$. Find a variance stabilising transform for $X.$

EXERCISE 16: Let $X$ satisfy $V(X) = k E(X)^ \alpha.$, where $k$ and $\alpha$ are known constants. Find a variance stabilising transform for $X.$

4. Cramer-Wold device

EXAMPLE 1: Suppose that $X_n\rightarrowD X$ and $Y_n\rightarrowD Y.$ Does this imply $(X_n,Y_n)\rightarrow (X,Y)?$ ■

Cramer Wold theorem Let $(\v X_n), \v X $ be random vectors. Then $\v X_n\rightarrowD \v X$ if and only if $$\forall \v \ell\in{\mathbb R}^k~~\v\ell' \v X_n\rightarrowD \v X.$$

Proof: Use characteristic function (to be covered in the next page). [QED]

4.1. Multivariate CLT

EXAMPLE 2: Suppose that $X_n\rightarrowD X$ and $Y_n\rightarrowD Y.$ Does this imply $(X_n,Y_n)\rightarrow (X,Y)?$ ■

Multivariate CLT Let $(\v X_n)$ be iid $k$-dimensional random vectors with mean vector $\v \mu$ and variance matrix $\Sigma.$ Then $$\sqrt{n}(\v{\bar X}_n-\v\mu)\rightarrowD N_k(\v 0, \Sigma).$$

4.2. Problem set

EXERCISE 17: Prove multivariate CLT from univariate CLT using the Cramer-Wold device.

EXERCISE 18: Prove multivariate delta method from univariate delta method using the Cramer-Wold device.

EXERCISE 19: Let $(X_n,Y_n)$ be an iid sequence of random vectors with mean $(\mu, \nu)$ and some (finite) covariance matrix. Find the asymptotic distribution of $(\bar X_n,\bar Y_n).$

EXERCISE 20: A (pssibly biased) die is rolled $n$ times. Let $X_{i,n}$ be the proportion of face $i.$ Find the asymptotic joint distribution of $(X_{1,n},...,X_{6,n}).$

EXERCISE 21: Prove the following theorem using the Cramer-Wold device.

Multivariate delta method If $\sqrt(T_n-\theta)\rightarrowD N_d(\v 0, \Sigma),$ then $\sqrt n(g(T_n)-g(\theta))\rightarrowD N_d(\v 0, \nabla g(\theta)'\Sigma\nabla g(\theta))$

EXERCISE 22: Use the above theorem to obtain asymptotic variance of the sample CV.