Multivariate CLT follows from strong Rayleigh property

Ghosh, Subhroshekhar; Liggett, Thomas M.; Pemantle, Robin

doi:10.1137/1.9781611974775.14

Cited by 10 publications

(13 citation statements)

References 16 publications

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“…We note in passing that Theorem 1 also implies an improvement on the work of Ghosh, Liggett and Pemantle [15] who considered a similar situation for vector-valued random variables. Let {Y n } be a sequence of random variables taking values in {0, .…”

Section: Resultsmentioning

confidence: 77%

“…From the choice of b n at (14) and equation (13), we see that b n ≥ |C k0 | 1/k0 = (1 + o(1))( x n · |B(k 0 )|) 1/k0 (16) and so from lines (15) and (16) we have…”

Section: Proof Of Claimmentioning

confidence: 99%

“…To do this we use the well known theorem of Cramér and Wold (see Corollary 6.3 in [15]), which says a sequence of random variables X n = (X (1) n , . .…”

Section: Proof Of Claimmentioning

confidence: 99%

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Central limit theorems from the roots of probability generating functions

Michelen

Sahasrabudhe

2019

Advances in Mathematics

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For each n, let Xn ∈ {0, . . . , n} be a random variable with mean µn, standard deviation σn, and let Pn(z) = n k=0 P(Xn = k)z k , be its probability generating function. We show that if none of the complex zeros of the polynomials {Pn(z)} is contained in a neighbourhood of 1 ∈ C and σn > n ε for some ε > 0, then X * n = (Xn − µn)σ −1 n is asymptotically normal as n → ∞: that is tends in distribution to a random variable Z ∼ N (0, 1).Moreover, we show this result is sharp in the sense that there exist sequences of random variables {Xn} with σn > C log n for which Pn(z) has no roots near 1 and X * n is not asymptotically normal. These results disprove a conjecture of Pemantle and improve upon various results in the literature. We go on to prove several other results connecting the location of the zeros of Pn(z) and the distribution of the random variable Xn. 1 arXiv:1804.07696v2 [math.PR] 12 Jun 2018 P X (z) are non-negative. Hence, we may write P X (z) = n i=1 (q i z + 1 − q i ), for some q 1 , . . . q n ∈ [0, 1]. A little further thought reveals that this special expression for the probability generating function corresponds to an expression of X as a sum of independent random variables X = X 1 + · · · + X n , where X i is the {0, 1}-random variable, taking 1 with probability q i . Thus, with an appropriate central limit theorem at hand, we see that X must be approximately normal, provided the variance of X is sufficiently large. In other words, from this one piece of information (albeit a strong piece of information) about the zeros of P X (z), one can quickly deduce quite a bit of information about the distribution of X. The aim of this paper is to show that this assumption of real rootedness of P X (z) can be related quite considerably while yielding similar results. In particular, we give three different results each of which says that "if X has large variance and the roots of P X (z) avoid a region in the complex plane, then X is approximately normal." HistoryBefore turning to our contributions, we take a brief moment to situate our results in an old and well-studied field centred around the following question: What does information about the coefficients of P (z) tell us about the distribution of the complex roots of P (and vice versa)? This question has a long and rich history, reaching back to the seminal work of Littlewood, Szegő, Pólya, and perhaps even Cauchy, due to his 1829 proof [8] of the fundamental theorem of algebra, which gives explicit bounds on the magnitude of the complex roots (see [7, Theorem 1.2.1] for a modern treatment of this proof). One line of research, initiated by the 1938 -1943 work of Littlewood and Offord [21, 22, 23], concerns the typical distribution of roots of random polynomials. For example Kac [18] gave an exact integral formulafor the number of real roots of random polynomial, with coefficients sampled independently from a normal distribution. Later, Erdős and Offord [11] showed that as n → ∞ almost all polynomials of the form n i=0 ε i x i , where ε 1 , . . . , ε n ∈ {0...

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Section: Resultsmentioning

confidence: 77%

“…From the choice of b n at (14) and equation (13), we see that b n ≥ |C k0 | 1/k0 = (1 + o(1))( x n · |B(k 0 )|) 1/k0 (16) and so from lines (15) and (16) we have…”

Section: Proof Of Claimmentioning

confidence: 99%

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Central limit theorems from the roots of probability generating functions

Michelen

Sahasrabudhe

2019

Advances in Mathematics

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“…In this paper, one of our main motivations is to finish a program set in motion by Ghosh, Liggett and Pemantle [27] to show that if X n ∈ {0, . .…”

Section: Introductionmentioning

confidence: 99%

Central limit theorems and the geometry of polynomials

Michelen,

Sahasrabudhe

2019

Preprint

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Let X ∈ {0, . . . , n} be a random variable, with mean µ and standard deviation σ and letbe its probability generating function. Pemantle conjectured that if σ is large and f X has no roots close to 1 ∈ C then X must be approximately normal. We completely resolve this conjecture in the following strong quantitative form, obtaining sharp bounds. If δ = min ζ |ζ − 1| over the complex roots ζ of f X , andwhere Z ∼ N (0, 1) is a standard normal. This gives the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. We also show that if f X has no roots with small argument, then X must be approximately normal, again in a sharp quantitative form:Using this result, we answer a question of Ghosh, Liggett and Pemantle by proving a sharp multivariate central limit theorem for random variables with real-stable probability generating functions.

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“…Our original motivation derives from a conjecture of Pemantle [24] and a related conjecture of Ghosh, Liggett and Pemantle [13] on random variables with real stable probability generating functions. Pemantle conjectured that random variables X ∈ {0, .…”

Section: Introductionmentioning

confidence: 99%

Anti-concentration of random variables from zero-free regions

Michelen¹,

Sahasrabudhe²

2021

Preprint

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This paper provides a connection between the concentration of a random variable and the distribution of the roots of its probability generating function. Let X be a random variable taking values in {0, . . . , n} with P(X = 0)P(X = n) > 0 and with probability generating function f X . We show that if all of the zeroswhere c > 0 is a absolute constant. We show that this result is sharp, up to the factor 2 in the exponent of R. As a consequence, we are able to deduce a Littlewood-Offord type theorem for random variables that are not necessarily sums of i.i.d. random variables.

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Multivariate CLT follows from strong Rayleigh property

Cited by 10 publications

References 16 publications

Central limit theorems from the roots of probability generating functions

Central limit theorems from the roots of probability generating functions

Central limit theorems and the geometry of polynomials

Anti-concentration of random variables from zero-free regions

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