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Miklós Csörgő

Publications and source records attributed to Miklós Csörgő.

11 recordsLinked to original sources
arXiv2018

Two-dimensional anisotropic random walks: fixed versus random column configurations for transport phenomena

We consider random walks on the square lattice of the plane along the lines of Heyde (1982, 1993) and den Hollander (1994), whose studies have in part been inspired by the so-called transport phenomena of statistical physics. Two-dimensional anisotropic random walks with anisotropic density conditions a' la Heyde (1982, 1993) yield fixed column configurations and nearest-neighbour random walks in a random environment on the square lattice of the plane as in den Hollander (1994) result in random column configurations. In both cases we conclude simultaneous weak Donsker and strong Strassen type invariance principles in terms of appropriately constructed anisotropic Brownian motions on the plane, with self-contained proofs in both cases. The style of presentation throughout will be that of a semi-expository survey of related results in a historical context.

math.PR↗
arXiv2018

Randomized Empirical Processes and Confidence Bands via Virtual Resampling

Let $X,X_1,X_2,\cdots$ be independent real valued random variables with a common distribution function $F$, and consider $\{X_1,\cdots,X_N \}$, possibly a big concrete data set, or an imaginary random sample of size $N\geq 1$ on $X$. In the latter case, or when a concrete data set in hand is too big to be entirely processed, then the sample distribution function $F_N$ and the the population distribution function $F$ are both to be estimated. This, in this paper, is achieved via viewing $\{X_1,\cdots,X_N \}$ as above, as a finite population of real valued random variables with $N$ labeled units, and sampling its indices $\{1,\cdots,N \}$ with replacement $m_N:= \sum_{i=1}^N w_{i}^{(N)}$ times so that for each $1\leq i \leq N$, $w_{i}^{(N)}$ is the count of number of times the index $i$ of $X_i$ is chosen in this virtual resampling process. This exposition extends the Doob-Donsker classical theory of weak convergence of empirical processes to that of the thus created randomly weighted empirical processes when $N, m_N \rightarrow \infty$ so that $m_N=o(N^2)$.

stat.ME↗
arXiv2015

Some limit theorems for heights of random walks on spider

A simple symmetric random walk is considered on a spider that is a collection of half lines (we call them legs) joined at the origin. We establish a strong approximation of this random walk by the so-called Brownian spider. Transition probabilities are studied, and for a fixed number of legs we investigate how high the walker can go on the legs in $n$ steps. The heights on the legs are also investigated when the number of legs goes to infinity.

math.PR↗
arXiv2013

Weak convergence of self-normalized partial sums processes

Let $\{X, X_n, n\geq 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \sum^n_{i=1} X_i$ and $V_n^2=\sum^n_{i=1} X_i^2, n\ge 1.$ A weak convergence theorem is established for the self-normalized partial sums processes $\{S_{[nt]}/V_n, 0\le t\le 1\}$ when $X$ belongs to the domain of attraction of a stable law with index $α\in (0,2]$. The respective limiting distributions of the random variables ${\max_{1\le i\le n}|X_i|}/{S_n}$ and ${\max_{1\le i\le n}|X_i|}/{V_n}$ are also obtained under the same condition.

math.PR↗
arXiv2008

Reduction principles for quantile and Bahadur-Kiefer processes of long-range dependent linear sequences

In this paper we consider quantile and Bahadur-Kiefer processes for long range dependent linear sequences. These processes, unlike in previous studies, are considered on the whole interval $(0,1)$. As it is well-known, quantile processes can have very erratic behavior on the tails. We overcome this problem by considering these processes with appropriate weight functions. In this way we conclude strong approximations that yield some remarkable phenomena that are not shared with i.i.d. sequences, including weak convergence of the Bahadur-Kiefer processes, a different pointwise behavior of the general and uniform Bahadur-Kiefer processes, and a somewhat "strange" behavior of the general quantile process.

math.ST↗
arXiv2007

On weighted approximations in $D[0, 1]$ with applications to self-normalized partial sum processes

Let $X, X_1, X_2,...$ be a sequence of non-degenerate i.i.d. random variables with mean zero. The best possible weighted approximations are investigated in $D[0, 1]$ for the partial sum processes $\{S_{[nt]}, 0\le t\le 1\}$, where $S_n=\sum_{j=1}^nX_j$, under the assumption that $X$ belongs to the domain of attraction of the normal law. The conclusions then are used to establish similar results for the sequence of self-normalized partial sum processes $\{S_{[nt]}/V_n, 0\le t\le 1\}$, where $V_n^2=\sum_{j=1}^nX_j^2$. $L_p$ approximations of self-normalized partial sum processes are also discussed.

math.PR↗
arXiv2007

Asymptotics of Studentized U-type processes for changepoint problems

This paper investigates weighted approximations for studentized $U$-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition $E|h(X_1,X_2)|^2 < \infty$ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence.

math.PR↗
arXiv2006

Strong invariance principles for sequential Bahadur--Kiefer and Vervaat error processes of long-range dependent sequences

In this paper we study strong approximations (invariance principles) of the sequential uniform and general Bahadur--Kiefer processes of long-range dependent sequences. We also investigate the strong and weak asymptotic behavior of the sequential Vervaat process, that is, the integrated sequential Bahadur--Kiefer process, properly normalized, as well as that of its deviation from its limiting process, the so-called Vervaat error process. It is well known that the Bahadur--Kiefer and the Vervaat error processes cannot converge weakly in the i.i.d. case. In contrast to this, we conclude that the Bahadur--Kiefer and Vervaat error processes, as well as their sequential versions, do converge weakly to a Dehling--Taqqu type limit process for certain long-range dependent sequences.

math.ST↗
arXiv2004

On Vervaat and Vervaat-error type processes for partial sums and renewals

We study the asymptotic behaviour of stochastic processes that are generated by sums of partial sums of i.i.d. random variables and their renewals. We conclude that these processes cannot converge weakly to any nondegenerate random element of the space $D[0,1]$. On the other hand we show that their properly normalized integrals as Vervaat-type stochastic processes converge weakly to a squared Wiener process. Moreover, we also deal with the asymptotic behaviour of the deviations of these processes, the so-called Vervaat-error type processes.

math.PR↗