scholarly journals Convergence in Total Variation of Random Sums

Mathematics ◽  
2021 ◽  
Vol 9 (2) ◽  
pp. 194
Author(s):  
Luca Pratelli ◽  
Pietro Rigo
Keyword(s):  
Asymptotic Behavior ◽  
Total Variation ◽  
Random Variables ◽  
Total Variation Distance ◽  
Random Sums ◽  
Variation Distance ◽  
Random Indices

Let (Xn) be a sequence of real random variables, (Tn) a sequence of random indices, and (τn) a sequence of constants such that τn→∞. The asymptotic behavior of Ln=(1/τn)∑i=1TnXi, as n→∞, is investigated when (Xn) is exchangeable and independent of (Tn). We give conditions for Mn=τn(Ln−L)⟶M in distribution, where L and M are suitable random variables. Moreover, when (Xn) is i.i.d., we find constants an and bn such that supA∈B(R)|P(Ln∈A)−P(L∈A)|≤an and supA∈B(R)|P(Mn∈A)−P(M∈A)|≤bn for every n. In particular, Ln→L or Mn→M in total variation distance provided an→0 or bn→0, as it happens in some situations.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (03) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variablesX1,X2, …,Xnare said to be totally negatively dependent (TND) if and only if the random variablesXiand ∑j≠iXjare negatively quadrant dependent for alli. Our main result provides, for TND 0-1 indicatorsX1,x2, …,Xnwith P[Xi= 1] =pi= 1 - P[Xi= 0], an upper bound for the total variation distance between ∑ni=1Xiand a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

Poisson approximation for a sum of dependent indicators: an alternative approach

2002 ◽  
Vol 34 (3) ◽  
pp. 609-625 ◽  
Author(s):  
N. Papadatos ◽  
V. Papathanasiou
Keyword(s):  
Total Variation ◽  
Upper Bound ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Birthday Problem ◽  
Alternative Approach ◽  
Variation Distance

The random variables X1, X2, …, Xn are said to be totally negatively dependent (TND) if and only if the random variables Xi and ∑j≠iXj are negatively quadrant dependent for all i. Our main result provides, for TND 0-1 indicators X1, x2, …, Xn with P[Xi = 1] = pi = 1 - P[Xi = 0], an upper bound for the total variation distance between ∑ni=1Xi and a Poisson random variable with mean λ ≥ ∑ni=1pi. An application to a generalized birthday problem is considered and, moreover, some related results concerning the existence of monotone couplings are discussed.

Poisson approximation for some statistics based on exchangeable trials

1983 ◽  
Vol 15 (3) ◽  
pp. 585-600 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Total Variation ◽  
Limit Theorems ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Variation Distance ◽  
Image Position ◽  
Poisson Approximations

Stein's (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables.Let be exchangeable random elements of a space and, for I a k-subset of , let XI be a 0–1 function. The statistics studied here are of the form where N is some collection of k -subsets of .An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.

On the distance between the distributions of random sums

2003 ◽  
Vol 40 (01) ◽  
pp. 87-106 ◽  
Author(s):  
Bero Roos ◽  
Dietmar Pfeifer
Keyword(s):  
Total Variation ◽  
Approximation Problem ◽  
Upper Bounds ◽  
Total Variation Distance ◽  
Random Sum ◽  
Random Sums ◽  
Random Summation ◽  
Variation Distance ◽  
Stop Loss

In this paper, we consider the total variation distance between the distributions of two random sums S M and S N with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum S N if M and N have the same first moments.

Poisson approximation for some statistics based on exchangeable trials

1983 ◽  
Vol 15 (03) ◽  
pp. 585-600 ◽  
Author(s):  
A. D. Barbour ◽  
G. K. Eagleson
Keyword(s):  
Total Variation ◽  
Limit Theorems ◽  
Random Variables ◽  
Random Variable ◽  
Poisson Approximation ◽  
Total Variation Distance ◽  
Poisson Random Variable ◽  
Variation Distance ◽  
Image Position ◽  
Poisson Approximations

Stein's (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables. Let be exchangeable random elements of a space and, for I a k-subset of , let XI be a 0–1 function. The statistics studied here are of the form where N is some collection of k -subsets of . An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.

On the distance between the distributions of random sums

2003 ◽  
Vol 40 (1) ◽  
pp. 87-106 ◽  
Author(s):  
Bero Roos ◽  
Dietmar Pfeifer
Keyword(s):  
Total Variation ◽  
Approximation Problem ◽  
Upper Bounds ◽  
Total Variation Distance ◽  
Random Sum ◽  
Random Sums ◽  
Random Summation ◽  
Variation Distance ◽  
Stop Loss

In this paper, we consider the total variation distance between the distributions of two random sums SM and SN with different random summation indices M and N. We derive upper bounds, some of which are sharp. Further, bounds with so-called magic factors are possible. Better results are possible when M and N are stochastically or stop-loss ordered. It turns out that the solution of this approximation problem strongly depends on how many of the first moments of M and N coincide. As approximations, we therefore choose suitable finite signed measures, which coincide with the distribution of the approximating random sum SN if M and N have the same first moments.

scholarly journals Distribution of the combinatorial multisets component vectors

2012 ◽  
Vol 53 ◽  
Author(s):  
Eugenijus Manstavičius ◽  
Robertas Petuchovas
Keyword(s):  
Total Variation ◽  
Negative Binomial ◽  
Random Variables ◽  
Total Variation Distance ◽  
Equal Probability ◽  
Combinatorial Structures ◽  
Variation Distance ◽  
Dependent Coordinates ◽  
Binomial Random Variables

We explore a class of random combinatorial structures called weighted multisets. Their components are taken from an initial set satisfying general boundedness conditions posed on the number of elements with a given weight. The component vector of a multiset of weight n taken with equal probability has dependent coordinates, nevertheless, up to r = o(n) of them as n→∞, we approximate by an appropriate vector comprised from independent negative binomial random variables. The main result is an estimate of the total variation distance.

scholarly journals An inequality between total variation and L2 distances for polynomials in log-concave random vectors

2019 ◽  
Vol 488 (2) ◽  
pp. 123-125
Author(s):  
E. D. Kosov
Keyword(s):  
Total Variation ◽  
Random Variables ◽  
Total Variation Distance ◽  
Random Vectors ◽  
Variation Distance

In the paper we discuss a new bound of the total variation distance in terms of L2 distance for random variables that are polynominals in log-concave random vectors.

Poisson and compound Poisson approximations for random sums of random variables

1996 ◽  
Vol 33 (01) ◽  
pp. 127-137 ◽  
Author(s):  
P. Vellaisamy ◽  
B. Chaudhuri
Keyword(s):  
Lower Bound ◽  
Total Variation ◽  
Random Variables ◽  
Upper Bounds ◽  
Random Sums ◽  
Compound Poisson ◽  
Sums Of Random Variables ◽  
Variation Distance ◽  
Poisson Approximations ◽  
Better Than

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

Poisson and compound Poisson approximations for random sums of random variables

1996 ◽  
Vol 33 (1) ◽  
pp. 127-137 ◽  
Author(s):  
P. Vellaisamy ◽  
B. Chaudhuri
Keyword(s):  
Lower Bound ◽  
Total Variation ◽  
Random Variables ◽  
Upper Bounds ◽  
Random Sums ◽  
Compound Poisson ◽  
Sums Of Random Variables ◽  
Variation Distance ◽  
Poisson Approximations ◽  
Better Than

We derive upper bounds for the total variation distance, d, between the distributions of two random sums of non-negative integer-valued random variables. The main results are then applied to some important random sums, including cluster binomial and cluster multinomial distributions, to obtain bounds on approximating them to suitable Poisson or compound Poisson distributions. These bounds are generally better than the known results on Poisson and compound Poisson approximations. We also obtain a lower bound for d and illustrate it with an example.

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