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Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.

2 votes
0 answers
82 views

Asymptotic symmetry of distributions

Suppose we have a triangular array of positive random variables $\{\{X_{1i},\ldots,X_{ii}\},i=1,\ldots,n\}$ such that the random variables in the array are independent, all random variables have mean …
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  • 907
2 votes
0 answers
221 views

Local limit theorem for the exponential of chi-squared random variables

Let's denote a chi-squared distribution with $k$ degrees of freedom as $\chi^2_k$, and a random variable $Z_k=\frac{X_k-k}{\sqrt{2k}}$, where $X_k\sim\chi^2_k$. Thus, $Z_k$ is a chi-squared random va …
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  • 907
8 votes
1 answer
866 views

CLT for the squares of unbounded non-identically independently distributed random variables

I have a sequence of independent but not identically-distributed random variables $X_1, X_2, \ldots, X_n$ where $X_i\sim A_i$, with each $A_i$ having a support over $\mathbb{R}$ and subject to the fol …
Bullmoose's user avatar
  • 907
3 votes
3 answers
5k views

Sharp lower bound for the tail of Chi-squared distribution

Let $X_n$ be a chi-squared random variable with $n$ degrees of freedom. What are the sharpest known lower bounds on the tails of its distribution? Specifically, I am looking for the lower bounds in …
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  • 907
2 votes
0 answers
1k views

Converse for Levy's continuity theorem

Levy's continuity theorem states that, for a sequence of random variables $\{X_n\}$ with characteristic functions $\{\varphi_n(t)\}$ and a random variable $X$ with a characteristic function $\varphi(t …
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  • 907
1 vote
0 answers
447 views

How far away is the maximum of $n$ i.i.d. chi-squared random variables from the rest of the ...

Suppose that I have a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom $X_1, X_2, \ldots, X_n$, and denote $X_{\max}=\max(X_1, X_2, \ldots, X_n)$. Let $k$ be increasing …
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  • 907
0 votes
0 answers
212 views

Behavior of the sum of the exponents of chi-squared random variables normalized by their max...

Let $X_1,X_2,\ldots,X_n$ be a sequence of $n$ i.i.d. chi-squared random variables with $k$ degrees of freedom, and denote by $X_\max$ the maximum of this sequence. Furthermore, let $k=\omega(1)$ incr …
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  • 907
4 votes
2 answers
450 views

Bounding the tail of an average using the the tail of individual members

Let $X_1,X_2,\ldots,X_n$ be an i.i.d. sequence of $n$ positive random variables with mean $E[X_1]=\mu_X<\infty$ and the second moment $E[X_1^2]=\infty$. I am interested in upper-bounding $P\left(|\ …
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  • 907
3 votes
1 answer
450 views

What is known about the distribution of the errors in empirical approximation of a CDF?

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows: $$\hat{F}_ …
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  • 907
2 votes
1 answer
635 views

Estimating the variance of error in empirical approximation to a distribution

Let $X_1,X_2,\ldots,X_n$ be i.i.d. random variables in $\mathbb{R}$ with common cumulative distribution function (CDF) $F(x)$. The empirical approximation to $F(x)$ is defined as follows: $$\hat{F}_ …
Bullmoose's user avatar
  • 907
2 votes
1 answer
848 views

Maximum of a sequence of $n$ positive random variables where variance is an increasing funct...

Suppose I have a sequence of $n$ i.i.d. random variables $X_1,X_2,\ldots,X_n$. Each $X_i$ is positive and has variance $\sigma(n)$ that is an increasing function of the number of variables in the sequ …
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  • 907
3 votes
0 answers
552 views

Berry-Esseen result for triangular arrays

Let $\{\{X_{n1},\ldots,X_{nn}\},n=1,2,\ldots\}$ be a triangular array of independent random variables where each row contains identically distributed random variables. Let $E[X_{n1}]=\mu_n<\infty$ an …
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  • 907
4 votes
1 answer
5k views

Asymptotic behavior of max of chi-squared distribution

Suppose $X_{\max}$ is the maximum in a sequence $X_1,X_2,\ldots,X_n$ where each $X_i\sim\chi^2_k$ is an i.i.d. chi-squared random variable with $k$ degrees of freedom. Since chi squared distribution …
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  • 907
4 votes
1 answer
815 views

Does bounding moments make distributions close in total variation distance?

Let $W\sim\mathcal{N}(0,\sigma^2)$ be a "reference" Gaussian random variable. Suppose I have a set of distributions, $\mathcal{W}$, where $W_a\in\mathcal{W}$ if it satisfies the following criteria: …
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  • 907