## Moment Convergence In Conditional Limit Theorems (2000)

Citations: | 5 - 4 self |

### BibTeX

@MISC{Janson00momentconvergence,

author = {Svante Janson},

title = {Moment Convergence In Conditional Limit Theorems},

year = {2000}

}

### OpenURL

### Abstract

. Consider a sum P N 1 Y i of random variables conditioned on a given value of the sum P N 1 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 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. 1. Introduction Many random variables arising in different areas of probability theory, combinatorics and statistics turn out to have the same distribution as a sum of independent random variables conditioned on a specific value of another such sum. More precisely, we are concerned with variables with the distribution of P N 1 Y i conditioned on P N 1 X...

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Citation Context ...rm a sequence of i.i.d. random vectors; in general, the variables Xi can be discrete or continuous, or even vector-valued. Some examples are given in Section 3 below, and many others in, for example, =-=[5, 6, 7, 11, 12]-=-. General limit theorems yielding the asymptotic behaviour of the conditioned sum under suitable assumptions are given by, among others, Steck [21], Holst [5, 7] and Kudlaev [13]. (See also results on... |

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Citation Context ... observe that if x1, . . . , xn � are any non-negative integers such that n 1 (xi − 1) = −1, then there is exactly one cyclical shift x ′ i := xi+l (mod n) such that �k 1 (x′ i − 1) ≥ 0 for 1 ≤ k < n =-=[24, 18]-=- (l is the smallest number for which �l 1 xi = −1). Consequently, if we ignore (or randomize) the ordersMOMENT CONVERGENCE IN CONDITIONAL LIMIT THEOREMS 9 of X1, . . . , Xn, they have the same distrib... |

30 | G.: Phase transition for parking blocks, Brownian excursion and coalescence
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Citation Context ... with the Borel distribution (3.1) for some λ [16, 11, 17]. Thus, the distribution of the tree sizes is the same as the distribution of the block lengths in Example 3.2, cf. [10, Exercise 6.4-31] and =-=[1]-=-. Let N = Nn and m = mn = n+Nn and consider the number Wnk of trees of size k. Pavlov [16, 17] has shown, using similar methods, asymptotic normality of Wnk for every fixed k ≥ 1, provided n, Nn → ∞ a... |

23 |
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Citation Context ...α ∈ (0, 1), so mn/Nn → 1/(1 − α) and λn → λ ∈ (0, e −1 ), Corollary 2.1 immediately shows that that the total displacement is asymptotically normal, as previously shown by Flajolet, Poblete and Viola =-=[3]-=-; we further obtain moment convergence. Moreover, straightforward estimates, see [9] for details, show that Theorem 2.1 applies also when n/mn → 0 or n/mn → 1 (with γ = 2 and γ = 1/2, respectively), p... |

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Citation Context ...following such a block as belonging to the block, and we regard an empty urn following another empty urn as a block with zero occupied urns. Thus there are N := m − n blocks, and it can be shown, see =-=[9]-=- for details, that the lengths of the blocks (counting the empty urn) and the sums of displacements inside each block are distributed as (X1, Y1), . . . , (XN, YN) conditioned on �N i=1 Xi = m, where ... |

9 |
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Citation Context ... observe that if x1, . . . , xn � are any non-negative integers such that n 1 (xi − 1) = −1, then there is exactly one cyclical shift x ′ i := xi+l (mod n) such that �k 1 (x′ i − 1) ≥ 0 for 1 ≤ k < n =-=[24, 18]-=- (l is the smallest number for which �l 1 xi = −1). Consequently, if we ignore (or randomize) the ordersMOMENT CONVERGENCE IN CONDITIONAL LIMIT THEOREMS 9 of X1, . . . , Xn, they have the same distrib... |

7 |
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Citation Context ...rm a sequence of i.i.d. random vectors; in general, the variables Xi can be discrete or continuous, or even vector-valued. Some examples are given in Section 3 below, and many others in, for example, =-=[5, 6, 7, 11, 12]-=-. General limit theorems yielding the asymptotic behaviour of the conditioned sum under suitable assumptions are given by, among others, Steck [21], Holst [5, 7] and Kudlaev [13]. (See also results on... |

7 |
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(Show Context)
Citation Context ...rm a sequence of i.i.d. random vectors; in general, the variables Xi can be discrete or continuous, or even vector-valued. Some examples are given in Section 3 below, and many others in, for example, =-=[5, 6, 7, 11, 12]-=-. General limit theorems yielding the asymptotic behaviour of the conditioned sum under suitable assumptions are given by, among others, Steck [21], Holst [5, 7] and Kudlaev [13]. (See also results on... |

7 |
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Citation Context ...nt convergence. His estimates of the moments uses a complicated combinatorial analysis, however, and the proof here seems simpler. (For other proofs of asymptotic normality, not treating moments, see =-=[20, 12, 5]-=-; the last reference uses the same method as this paper.) Asymptotic normality in the other cases was shown by Rényi [20, 12]; we do not know any earlier proof of moment convergence in these cases. (I... |

7 |
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Citation Context ...case overlapping with ours has earlier been studied by Swensen [22] in a somewhat different formulation. The mean, in the equivalent form E(Y1 | � N 1 Xi), has been studied by Portnoy [19] and Zabell =-=[25, 26]-=-.) Date: May 8, 2000; revised May 2, 2001. 1991 Mathematics Subject Classification. 60F05. Key words and phrases. Conditional distribution, limit theorem, moment convergence, occupancy, hashing, rando... |

6 |
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Citation Context ...− N be the number of non-roots. The sizes of the N trees in the forest are distributed as X1, . . . , XN conditioned on �N i=1 Xi = m, where Xi are i.i.d. with the Borel distribution (3.1) for some λ =-=[16, 11, 17]-=-. Thus, the distribution of the tree sizes is the same as the distribution of the block lengths in Example 3.2, cf. [10, Exercise 6.4-31] and [1]. Let N = Nn and m = mn = n+Nn and consider the number ... |

6 |
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(Show Context)
Citation Context ...below, and many others in, for example, [5, 6, 7, 11, 12]. General limit theorems yielding the asymptotic behaviour of the conditioned sum under suitable assumptions are given by, among others, Steck =-=[21]-=-, Holst [5, 7] and Kudlaev [13]. (See also results on asymptotic expansions in [2, 14, 4, 8] and on rates of convergence in [15].) The purpose of this paper is to give a set of sufficient conditions f... |

6 |
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(Show Context)
Citation Context ...eorem 2.1 thus proves asymptotic normality of Wn in all cases where it holds, together with moment convergence. Actually, asymptotic normality in the case mn/Nn → a ∈ (0, ∞) was first proved by Weiss =-=[23]-=- using the method of moments, i.e. by first showing moment convergence. His estimates of the moments uses a complicated combinatorial analysis, however, and the proof here seems simpler. (For other pr... |

5 |
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(Show Context)
Citation Context ...− N be the number of non-roots. The sizes of the N trees in the forest are distributed as X1, . . . , XN conditioned on �N i=1 Xi = m, where Xi are i.i.d. with the Borel distribution (3.1) for some λ =-=[16, 11, 17]-=-. Thus, the distribution of the tree sizes is the same as the distribution of the block lengths in Example 3.2, cf. [10, Exercise 6.4-31] and [1]. Let N = Nn and m = mn = n+Nn and consider the number ... |

5 | Random Forests - Pavlov - 2000 |

4 |
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(Show Context)
Citation Context |

3 |
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(Show Context)
Citation Context ...ielding the asymptotic behaviour of the conditioned sum under suitable assumptions are given by, among others, Steck [21], Holst [5, 7] and Kudlaev [13]. (See also results on asymptotic expansions in =-=[2, 14, 4, 8]-=- and on rates of convergence in [15].) The purpose of this paper is to give a set of sufficient conditions for convergence of not only the distribution of the conditioned sum but also its moments. We ... |

2 |
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(Show Context)
Citation Context ...ielding the asymptotic behaviour of the conditioned sum under suitable assumptions are given by, among others, Steck [21], Holst [5, 7] and Kudlaev [13]. (See also results on asymptotic expansions in =-=[2, 14, 4, 8]-=- and on rates of convergence in [15].) The purpose of this paper is to give a set of sufficient conditions for convergence of not only the distribution of the conditioned sum but also its moments. We ... |

2 |
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(Show Context)
Citation Context ...d variance in a case overlapping with ours has earlier been studied by Swensen [22] in a somewhat different formulation. The mean, in the equivalent form E(Y1 | � N 1 Xi), has been studied by Portnoy =-=[19]-=- and Zabell [25, 26].) Date: May 8, 2000; revised May 2, 2001. 1991 Mathematics Subject Classification. 60F05. Key words and phrases. Conditional distribution, limit theorem, moment convergence, occup... |

2 |
A limit theorem for expectations conditional on a sum
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(Show Context)
Citation Context ...case overlapping with ours has earlier been studied by Swensen [22] in a somewhat different formulation. The mean, in the equivalent form E(Y1 | � N 1 Xi), has been studied by Portnoy [19] and Zabell =-=[25, 26]-=-.) Date: May 8, 2000; revised May 2, 2001. 1991 Mathematics Subject Classification. 60F05. Key words and phrases. Conditional distribution, limit theorem, moment convergence, occupancy, hashing, rando... |

1 |
Asymptotic expansions for conditional distributions
- Ismatullaev
(Show Context)
Citation Context ...ielding the asymptotic behaviour of the conditioned sum under suitable assumptions are given by, among others, Steck [21], Holst [5, 7] and Kudlaev [13]. (See also results on asymptotic expansions in =-=[2, 14, 4, 8]-=- and on rates of convergence in [15].) The purpose of this paper is to give a set of sufficient conditions for convergence of not only the distribution of the conditioned sum but also its moments. We ... |

1 | i Primenen. 29 - Veroyatnost - 1984 |

1 |
Asymptotic expansions for conditional distributions
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(Show Context)
Citation Context |

1 |
Limit theorems for conditional distributions
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(Show Context)
Citation Context ...tioned sum under suitable assumptions are given by, among others, Steck [21], Holst [5, 7] and Kudlaev [13]. (See also results on asymptotic expansions in [2, 14, 4, 8] and on rates of convergence in =-=[15]-=-.) The purpose of this paper is to give a set of sufficient conditions for convergence of not only the distribution of the conditioned sum but also its moments. We consider only the case when the vari... |

1 |
A note on convergence of distributions of conditional moments
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(Show Context)
Citation Context ...vial. For simplicity we further only consider the case when the limit distribution is normal. (Asymptotics of the mean and variance in a case overlapping with ours has earlier been studied by Swensen =-=[22]-=- in a somewhat different formulation. The mean, in the equivalent form E(Y1 | � N 1 Xi), has been studied by Portnoy [19] and Zabell [25, 26].) Date: May 8, 2000; revised May 2, 2001. 1991 Mathematics... |