Math Problems Vault

Problem 22.1

Let $\nu_n$ be the number of successes in $n$ Bernoulli trials with success probability $p$ . Prove that for every function $f(x)$ continuous on $[0,1]$ the following convergence holds:

\sup _{p \in [0,1]} \left|\mathbb {E}\left[f\left(\frac{\nu _n}{n}\right)\right] - f(p)\right| \to 0 \text{ as } n \to \infty .

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Remark.

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Let $\xi_1, \xi_2, \ldots$ be a sequence of independent identically distributed random variables with $\mathbb {E}\left[\left|\xi_1\right|\right] < \infty$ . Then the LLN holds: $S_n/n \xrightarrow {p} \mathbb {E}\left[\xi_1\right]$ as $n \to \infty$ .

Problem 22.2

Let $f: [0,1] \to \mathbb {R}$ be a continuous function. Prove that the Bernstein polynomials

B_n(x) = \sum _{i=0}^{n} f\left(\frac{i}{n}\right) C_n^i x^i (1-x)^{n-i}

converge to $f(x)$ as $n \to \infty$ uniformly in $x \in [0,1]$ :

\sup _{x \in [0,1]} \left|B_n(x) - x\right| \to 0 \text{ as } n \to \infty .

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Problem 22.3

Let $f: \mathbb {R}^{+} \to \mathbb {R}$ be a continuous bounded function, and let $K$ be a compact set in $[0,\infty )$ . Prove that the polynomials

B_n(x) = \sum _{i=0}^{\infty } f\left(\frac{i}{n}\right) \frac{(nx)^i}{i!} e^{-nx}

converge to $f(x)$ as $n \to \infty$ uniformly in $x \in K$ .

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Problem 22.4

Let $f: \mathbb {R}^{+} \to \mathbb {R}$ be a continuous bounded function, and let $K$ be a compact set in $[0,\infty )$ . Prove that the polynomials

B_n(x) = \sum _{i=0}^{\infty } f\left(\frac{i}{n+1}\right) \frac{C_{n+i}^i x^i}{(1+x)^{n+i+1}}

converge to $f(x)$ as $n \to \infty$ uniformly in $x \in K$ .

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Problem 22.5

(a)(b)(c)

Under the hypotheses of problem

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(a)

22.2;

(b)

22.3;

(c)

22.4

prove that if the function $f$ has a continuous derivative $f'$ , then the derivatives of the polynomials $B_n(x)$ converge uniformly to $f'(x)$ as $n \to \infty$ .

Problem 22.6

Let $f(x,y)$ be a function continuous on the triangle $x \geqslant 0$ , $y \geqslant 0$ , $x+y \leqslant 1$ . Prove that the Bernstein polynomials in $\mathbb {R}^{2}$

B_n(x,y) = \sum _{i,j=0}^{n} f\left(\frac{i}{n}, \frac{j}{n}\right) \frac{n!}{i! \, j! \, (n-i-j)!} x^i y^j (1-x-y)^{n-i-j}

converge to $f(x,y)$ as $n \to \infty$ uniformly in $x, y$ .

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Problem 22.7

Let $\xi_1, \xi_2, \ldots$ be a sequence of independent random variables with the standard Cauchy distribution. Does the LLN hold for the sequence $\{ \xi_n\}$ ?

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Problem 22.8

Let $\xi_1, \xi_2, \ldots$ be a sequence of independent identically distributed random variables having the standard normal distribution. Set

\chi _n^2 = \sum _{i=1}^{n} \xi _i^2 \quad \text{and} \quad \tau _n = \frac{\xi _{n+1}}{\sqrt{\chi _n^2/n}}.

Find the limiting distribution of $\tau_n$ as $n \to \infty$ .

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Problem 22.9

Let $\xi_1, \xi_2, \ldots$ be a sequence of independent identically distributed random variables with $\mathbb {E}\left[\xi_1\right] = a$ , $\operatorname {Var}\left[\xi_1\right] = \sigma^2$ . Prove that the sequence of random variables $\eta_n$ , defined by

\eta _n = \frac{\xi _1 + \ldots + \xi _n}{\xi _1^2 + \ldots + \xi _n^2},

converges in probability. Find the limit.

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Problem 22.10

(a)(b)(c)

Let the random variable $\eta_n$ have:

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(a)

the Poisson distribution with parameter $n$ ;

(b)

the binomial distribution with parameters $n$ and $p$ ;

(c)

the normal distribution with parameters $0$ and $n$ .

Determine whether the ratio $\eta_n/n$ has a limit in probability as $n \to \infty$ .