Coin tossing

Svi su dobro upoznati s eksperimentom bacanja simetričnog novčića. U ovom radu se obrađuje malo drugačija tema: u prvom koraku bacamo simetričan novčić, a zatim u \(n \) -tom koraku okrećemo novčić na drugu stranu s nekom vjerojatnošću \(p_n\), \(n\geq 2\).Preciznije, uz proizvoljan vjerojatnosti niz \( (p_n)_{n\in\mathbb{N}}\) promatramo slučajne varijable \( (X_n)_{n \in\mathbb{N}}\) za koje je \(X_1\sim B\left(\frac12\right) \) i za \(n \geq 2\) \[ X_n:=\begin{cases} X_{n-1},& \text{s vjerojatnošću } p_n\\ 1-X_{n-1},& \text{s vjerojatnošću } 1-p_n \end{cases}.\] Cilj rada je odgovorit na pitanje kada granična distribucija od \[\frac{1}{N}\sum_{k=1}^NX_k\] konvergira, i ako konvergira, ka kojoj distribuciji konvergira. Vidjet ćemo da uz određene uvjete na niz vjerojatnosti \(p_n\), dobivamo različite vrste konvergencije. Na primjer, ako niz "jako brzo" konvergira k nuli, tada se dobiva konvergencija po distribuciji k Bernoullijevoj slučajnoj varijabli. Nadalje, pokazat ćemo i da naša granična distribucija može slijediti normalnu i beta razdiobu uz određene uvjete. Odgovorit ćemo i na pitanje, uz koje uvjete vrijede zakoni velikih brojeva za našu graničnu distribuciju. Za kraj ćemo odgovoriti na pitanje hoće li pretpostavka o simetričnosti novčića utjecati na sve dokazane rezultate o konvergenciji.The coin tossing experiment is one of the most famous experiments in mathematics. In this thesis we are going to analyze a different kind of experiment involving a coin: in the first step we toss a standard coin, after that, in every \(n \)-th step we turn the coin on the other side with probability - \(p_n\), \(n\geq 2\). Therefore, with a given sequence of probabilities \( (p_n)_{n\in\mathbb{N}}\) we deffine a sequence of random variables \( (X_n)_{n \in\mathbb{N}}\) such that \(X_1\sim B\left(\frac12\right) \) i za \(n \geq 2\) \[ X_n:=\begin{cases} X_{n-1},& \text{with probability } p_n\\ 1-X_{n-1},& \text{with probability } 1-p_n \end{cases}.\] The main goal of this thesis is to determine when the limiting distribution of \[\frac{1}{N}\sum_{k=1}^NX_k\] converges, and to which distribution it converges. We are going to see that under different conditions on the sequence (pn), we obtain different types of convergence, and different types of distributions. To give a more clear example, let's say that the sequence of probabilities \(p_n\) is going to zero "very quickly", then we obtain the convergence in distribution, where the limiting distribution is a Bernoulli random variable. Among other distributions, we will obtain normal and beta distributions. We will also obtain that under some conditions, the laws of large numbers hold. At the end, we will answer the question whether or not the symmetry of the coin affects the conclusions of the theorems we are going to prove