Limit theorems for some sequential occupancy problems

1983 ◽  
Vol 20 (03) ◽  
pp. 545-553 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Asymptotic Results ◽  
Occupancy Problems ◽  
Number Of Cells

Consider n cells into which balls are thrown at random until all but m cells contain at least l + 1 balls each. Asymptotic results when n →∞, m and l held fixed, are given for the number of cells containing exactly k balls and for related random variables.

Limit theorems for some sequential occupancy problems

1983 ◽  
Vol 20 (3) ◽  
pp. 545-553 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Limit Theorems ◽  
Random Variables ◽  
Asymptotic Results ◽  
Occupancy Problems ◽  
Number Of Cells

Consider n cells into which balls are thrown at random until all but m cells contain at least l + 1 balls each. Asymptotic results when n →∞, m and l held fixed, are given for the number of cells containing exactly k balls and for related random variables.

On sequential occupancy problems

1981 ◽  
Vol 18 (02) ◽  
pp. 435-442 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Waiting Time ◽  
Random Variables ◽  
Characteristic Functions ◽  
Occupancy Problems ◽  
Number Of Cells

Consider n cells into which balls are thrown at random until k cells contain at least l + 1 balls each. Let Y l, · ··, Yn be the number of balls in the cells when stopping. In this paper two representations are given for the characteristic functions of random variables of the form The usefulness of these representations are illustrated by two examples. In the first the number of cells with exactly one ball when each cell contains at least one ball is considered. In the second the waiting time until the ball-throwing process stops is discussed.

scholarly journals LIMIT THEOREMS FOR RADIAL RANDOM WALKS ON EUCLIDEAN SPACES OF HIGH DIMENSIONS

2014 ◽  
Vol 97 (2) ◽  
pp. 212-236 ◽  
Author(s):  
WALDEMAR GRUNDMANN
Keyword(s):  
Random Walks ◽  
Method Of Moments ◽  
Limit Theorems ◽  
Random Variables ◽  
High Dimensions ◽  
Asymptotic Results ◽  
Explicit Formulas ◽  
Normal Limits ◽  
Fixed Dimension ◽  
Radial Random Walks

AbstractLet $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\nu \in M^1([0,\infty [)$ be a fixed probability measure. For each dimension $p\in \mathbb{N}$, let $(X_n^{p})_{n\geq 1}$ be independent and identically distributed $\mathbb{R}^p$-valued random variables with radially symmetric distributions and radial distribution $\nu $. We investigate the distribution of the Euclidean length of $S_n^{p}:=X_1^{p}+\cdots + X_n^{p}$ for large parameters $n$ and $p$. Depending on the growth of the dimension $p=p_n$ we derive by the method of moments two complementary central limit theorems (CLTs) for the functional $\| S_n^{p}\| _2$ with normal limits, namely for $n/p_n \to \infty $ and $n/p_n \to 0$. Moreover, we present a CLT for the case $n/p_n \to c\in \, (0,\infty )$. Thereby we derive explicit formulas and asymptotic results for moments of radial distributed random variables on $\mathbb{R}^p$. All limit theorems are also considered for orthogonal invariant random walks on the space $\mathbb{M}_{p,q}(\mathbb{R})$ of $p\times q$ matrices instead of $\mathbb{R}^p$ for $p\to \infty $ and some fixed dimension $q$.

Central limit theorems for logarithmic averages

2001 ◽  
Vol 38 (1-4) ◽  
pp. 79-96
Author(s):  
I. Berkes ◽  
L. Horváth ◽  
X. Chen
Keyword(s):  
Wiener Process ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Variables ◽  
Central Limit Theorems ◽  
Partial Sums ◽  
Asymptotic Results

We prove central limit theorems and related asymptotic results for where W is a Wiener process and Sk are partial sums of i.i.d. random variables with mean 0 and variance 1. The integrability and smoothness conditions made on f are optimal in a number of important cases.

On sequential occupancy problems

1981 ◽  
Vol 18 (2) ◽  
pp. 435-442 ◽  
Author(s):  
Lars Holst
Keyword(s):  
Waiting Time ◽  
Random Variables ◽  
Characteristic Functions ◽  
Occupancy Problems ◽  
Number Of Cells

Consider n cells into which balls are thrown at random until k cells contain at least l + 1 balls each. Let Yl, · ··, Yn be the number of balls in the cells when stopping. In this paper two representations are given for the characteristic functions of random variables of the form The usefulness of these representations are illustrated by two examples. In the first the number of cells with exactly one ball when each cell contains at least one ball is considered. In the second the waiting time until the ball-throwing process stops is discussed.

Moment convergence in conditional limit theorems

2001 ◽  
Vol 38 (2) ◽  
pp. 421-437 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Normal Distribution ◽  
Random Forests ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Variables ◽  
Triangular Array ◽  
Moment Convergence ◽  
Conditional Limit Theorems ◽  
Occupancy Problems ◽  
Conditional Limit

Consider a sum ∑1NYi of random variables conditioned on a given value of the sum ∑1NXi of some other variables, where Xi and Yi are dependent but the pairs (Xi,Yi) form an i.i.d. sequence. We consider here the case when each Xi is discrete. We prove, for a triangular array ((Xni,Yni)) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

Moment convergence in conditional limit theorems

2001 ◽  
Vol 38 (02) ◽  
pp. 421-437 ◽  
Author(s):  
Svante Janson
Keyword(s):  
Normal Distribution ◽  
Random Forests ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Variables ◽  
Triangular Array ◽  
Moment Convergence ◽  
Conditional Limit Theorems ◽  
Occupancy Problems ◽  
Conditional Limit

Consider a sum ∑1 N Y i of random variables conditioned on a given value of the sum ∑1 N X i of some other variables, where X i and Y i are dependent but the pairs (X i ,Y i ) form an i.i.d. sequence. We consider here the case when each X i is discrete. We prove, for a triangular array ((X ni ,Y ni )) of such pairs satisfying certain conditions, both convergence of the distribution of the conditioned sum (after suitable normalization) to a normal distribution, and convergence of its moments. The results are motivated by an application to hashing with linear probing; we give also some other applications to occupancy problems, random forests, and branching processes.

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