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\fancyhead[HCO]{9th Int. Conf. on Mathematics and Informatics,
	September 7-8, 2023, T\^argu Mure\c s, Romania}
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\begin{center}{\Large\bf
Limit theorems for runs
containing two types of contaminations
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\begin{center}{\large\bf\noindent
{Istv\'an Fazekas,
	Borb\'ala Fazekas,
	Michael Suja }
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Department of Mathematics\\
University of Debrecen
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\\[1mm]\texttt{
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fazekas.istvan@inf.unideb.hu \\
borbala.fazekas@science.unideb.hu\\
michael.suja@science.unideb.hu
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\vspace*{7mm}

The problem of the length of the longest head run for $n$ Bernoulli random variables was studied 
in the classical paper by 
Erd{\H{o}}s and R{\'e}nyi \cite{erdos1970law}.
In this paper, we define and study the limiting distribution of the first hitting time and the 
accompanying distribution for the length of the longest at most 1 + 1 contaminated sequence of runs with trinary trials.

 Let $X_1,X_2,\ldots,X_N$ be a sequence of independent random variables with three possible outcomes; 
 $0$, $+1$ and $-1$ labelled as success, failure of type I and failure of type II, respectively with the distribution
 $$
 P(X_i = 0) = p,  \quad P(X_i= +1) = q_1 \ \  {\text {and}} \ \ P(X_i= -1) = q_2,
 $$ 
 where $p+ q_1+q_2 = 1$ and $p>0$,\quad$q_1>0$,\quad$q_2 > 0$.
 
An $m$ length section of the above sequence is called a pure run if it contains only 0 values. 
It is called a one-type contaminated run if it contains precisely one non-zero element either a +1 or a -1. 
On the other hand, it is called a two-type contaminated run if contains precisely one +1, and one -1 
while the rest of the elements are 0's. 
A run is called at most 1 + 1  contaminated if it is either pure, or one-type contaminated, or two-type contaminated.
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So for an arbitrary fixed $m$, let $A_n = A_{n,m}$ denote the occurrence of the event at the $n^{th}$ step, that is, there is an at most 1 + 1  contaminated run in the sequence $X_n, X_{n+1}, \ldots X_{n+m-1}$ and $\bar{A_n}$.
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\begin{thm}\label{T1}
Let $\tau_m$ be the first hitting time of the at most 1 + 1 contaminated run of heads having length $m$. 
Then, for $x>0$,
	\begin{equation}
		P(\tau_m \alpha P(A_1)> x) \sim e^{-x}
	\end{equation}
	as $m\to \infty$. 
Here $P(A_1) = p^m + m(1-p)p^{m-1} + m(m-1)p^{m-2} q_1 q_2$ and $\alpha$ is defined by the parameters of the process.
\end{thm}
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Let $\mu(N)$ be the length of the longest at most two-type contaminated run in $X_1,X_2,\ldots,X_N$.
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\begin{thm}\label{T2}
For integer $k>0$,
	\begin{equation}
		P(\mu(N)-[m(N)]<k)=\exp \left(-p^{-\left(\log \left(C_0 p^{-2} q_1 q_2\right)+H(k-\{m(N)\})\right)}\right)\left(1+\mathrm{O}\left(\frac{1}{(\log N)^3}\right)\right),
	\end{equation}  
	where $m(N)$, $C_0$ and $H(x)$ are defined by the parameters of the process, 
	$[m(N)]$ denotes the integer part of $m(N)$ and $\{m(N)\}$ denotes the fractional part of $m(N)$. 
\end{thm}
The proofs of the above theorems depends on the fulfilment of some conditions given in the main Lemma 
of Cs\'aki et al. \cite{CFK}.
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\bibitem{erdos1970law} 
Erd\H{o}s, P.; R\'enyi, A.
On a new law of large numbers, {\em J. Analyse Math.} {\bf 23} (1970), 103-111.

\bibitem{CFK}
Cs\'aki, E.; F\"oldes, A.; Koml\'os, J.
Limit theorems for Erd\H{o}s-R\'enyi type problems.
{\em Studia Sci. Math. Hungar.} {\bf 22} (1987), 321--332.

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