Lesson_09 – Statistics Course

Homework_22A

Refine your statistical application in the following way:

To the contingency table, add or make sure it has the following features: 2) the option to display the frequencies either in absolute or relative form, with totals 2) the option to display the histograms “around” the table, in a compact form.

Code VB.NET

https://drive.google.com/file/d/1WYN5gVGdhYOwm1wAyA7ByCm40G4ZC0PU/view?usp=sharing

Homework_21A

Consider the general scheme we have used so far to simulate some stochastic processes (such as the relative frequency of success in a sequence of trials, the sample mean, the random walk, the Poisson point process) and now add this new process to our simulator.
Same scheme as previous simulations programs, except changing the way to compute the values of the paths at each time. Starting from value 0 at time 0, for each of m paths, at each new time compute P(t) = P(t-1) + Random step(t), for t = 1, …, n, where Random step(t) is now: σ * sqrt(1/n) * Z(t), where Z(t) is a N(0,1) random variable (the deviation σ is a user parameter, to scale the process dispersion).
At time n (last time) and one (or more) other chosen inner time 1<j<n (j is a program parameter) create and represent with histogram the distribution of P(t). Observe the behavior of the process for large n.

Code VB.NET

https://drive.google.com/file/d/18VrAuvDyjSh2ntiNrCYFsxfhpS_3jtxi/view?usp=sharing

Homework_22R

An “analog” of the CLT for stochastic process: the Brownian motion as limit of random walk and the functional CLT (Donsker theorem). Explain the intuitive meaning of this result.

A (μ, σ) Brownian motion is the limiting case of random walk.

A particle moves ∆x to the left with probability 1−p.
It moves to the right with probability p after ∆t time.
Define:
X_i = +1, if the ith move is to the right
X_i = -1, if the ith move is to the left
X_i are independent with
P[X_i = 1] = p = 1 – P[X_i = -1]
Assume n = t/∆t is an integer
Its position at time t is
Y(t) = ∆x(X₁ + X₂ + … + X_n)
Recall
E[X_i] = 2p – 1
Var[X_i] = 1 – (2p – 1)²
And
E[Y(t)] = n(∆x)(2p – 1)
Var[Y(t)] = n(∆x)(1 – (2p – 1)²)
with ∆x = σ√∆t and p = [ 1 + (μ/σ) √∆t]/2
Thus, {Y(t), t≥0} converges to a (μ, σ) Brownian motion by the central limit theorem.
Brownian motion with zero drift is the limiting case of symmetric random walk by choosing μ = 0

Functional extension of the central limit theorem, the Donsker’s theorem.

Let X₁ , X₂ , X₃ , … be a sequence of independent and identically distributed (i.i.d.) random variables with mean 0 and variance 1.

Let $S_n:=\sum_{i=1}^n X_i$ . The stochastic process S := ( S_n ) n ∈ N is known as a random walk.

Define the diffusively rescaled random walk (partial-sum process) by

t ∈ [0,1]

The central limit theorem asserts that W⁽ⁿ⁾(1) converges in distribution to a standard Gaussian random variable W(1) as n -> ∞ .

In its modern form, Donsker’s invariance principle states that:
As random variables taking values in the Skorokhod space D [0,1] (The set of all càdlàg functions from E to M is often denoted by D(E,M) and is called Skorokhod space), the random function W⁽ⁿ⁾ converges in distribution to a standard Brownian motion W := (W(t))_{t ∈ [0,1]} as n -> ∞

https://www.csie.ntu.edu.tw/~lyuu/finance1/2014/20140423.pdf
https://en.wikipedia.org/wiki/Donsker%27s_theorem
https://en.wikipedia.org/wiki/C%C3%A0dl%C3%A0g#Skorokhod_space

Homework_21R

What is a Brownian diffusion process. History, importance, definition and applications.

Brownian motion

A stochastic process, defined on a common probability space
(Ω, Σ, P)
with the following properties:

Increments must be independent and stationary (Cause is a Lévy process)
Increments has the N(0,Δt) distribution
x₀ = 0
Path must be continuous with probability 1 (Almost sure):
P{ω ∈ Ω : X( ^. , ω) is continuous} = 1

Diffusion Processes

A continuous time stochastic process with(almost surely) continuous sample paths which has the Markov property (the conditional probability distribution of future states of the process depends only upon the present state, not on the sequence of events that preceded it.) is called a diffusion.

The simplest and most fundamental diffusion process is Brownian motion(The independent increments imply the Markov property).

Also called Wiener process, It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics, quantitative finance, evolutionary biology, and physics.

Brownian motion was discovered by the biologist Robert Brown in 1827, who observed through a microscope the random swarming motion of pollen grains in water.
The theory of Brownian motion was developed by Bachelier in his 1900 PhD Thesis and independently by Einstein in his 1905 paper which used Brownian motion to estimate Avogadro’s number and the size of molecules.
Wiener in 1923 proved that there exists a version of BM with continuous paths.

https://en.wikipedia.org/wiki/Markov_property
https://arxiv.org/pdf/1802.09679.pdf
http://dept.stat.lsa.umich.edu/~ionides/620/notes/diffusions.pdf
https://dlib.bc.edu/islandora/object/bc-ir%3A102098/datastream/PDF/view