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\title{Rate of convergence in the strong law of large numbers for records}
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\author{O. I. Klesov, O. V. Kolesnik}
\address{National Technical University of Ukraine ``Igor Sikorsky Kyiv Politechnic Institute,
Kyiv, Ukraine}
\email{voselk@gmail.com}
\thanks{The first author is supported in part by DFG}
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Consider a sequence of independent identically distributed random variables
$\{X_k\}$ and the corresponding sequence of \textit{record times}:
$$
T_1=1, \qquad T_k=\min\{j:\ X_j>X_{T_{k-1}}\}, \quad k>1.
$$
The random variables $X_{T_k}$, $k\ge1$, are called \textit{records}.
The 
$$
\mu_k=\text{number of records up to moment $k$}
$$
is of interest in this talk.
The strong law of large numbers for $\{\mu_k\}$ tells us that
$$
\lim_{n\to\infty} \frac{\mu_n}{\ln(n)}=1 \qquad \text{almost surely}.
$$
An improvement of this statement could be called {the Marcinkiewicz--Zygmund strong law of large numbers}:
$$
\lim_{n\to\infty} \frac{\mu_n-\ln(n)}{\left(\ln(n)\right)^{1/r}}=0 \qquad \text{almost surely}
$$
for any $1\le r<2$.
We study a similar result in a more general setting called $F^\alpha$-scheme.

Let $X$ be a random variable and $F_X$ be its distribution function being nondegenerate in the sense that
$$
P(X=0)\ne 0.
$$
Let $\{\alpha_k\}$ be a sequence of nonnegative numbers.
A sequence of independent random variables $\{X_k\}$ is called the $F^\alpha$-scheme
associated with $X$ and $\{\alpha_k\}$ if
$$
F_{X_k}=F_X^{\alpha_k}.
$$
Define $T_k$ and $\mu_k$, $k\ge1$, in the same way as above.
Then one can prove for the $F^\alpha$-scheme that
$$
\lim_{n\to\infty} \left[\frac{\mu_n}{\ln(A_n)}-\frac{1}{A_n}\sum_{k=1}^n\frac{\alpha_k}{A_k}\right]=0
\qquad\text{almost surely,}
$$
where $A_n=\alpha_1+\dots+\alpha_n$ (see~\cite{DKS}).
We propose a method for finding the sequences $\{f(n)\}$ and $\{g(n)\}$ such that
$$
\lim_{n\to\infty} \left[\frac{\mu_n}{g(n)}-f(n)\right]=0
\qquad\text{almost surely.}
$$

Based on this result, we are able to solve the same problem for record times $\{T_k\}$.
In doing so we use the idea of pseudo-regularly functions, since $T_n$ is an
generalized renewal process constructed from $\mu_n$ (see~\cite{BIKS}).

\begin{thebibliography}{9}
  \bibitem{BIKS}
    V. V. Buldygin, K.-H. Indlekofer, O. I. Klesov, J. G. Steinebach (2018).
    {\em Pseudo-Regularly Varying Functions and Generalized Renewal Processes}.
    Springer, Cham, 482+XXIIpp. 
  \bibitem{DKS}
    P. Doukhan, O. I. Klesov, J. G. Steinebach (2015)
    {\em Strong laws of large numbers in an $F^\alpha$-scheme}
    in book {\em Festschrift dedicated to 65$^{\text{th}}$ birthday
    of Prof. Paul Deheuvels}, Springer, Cham, 287--303
   
\end{thebibliography}

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