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Register a new userOn a functional equation appearing in characterization of distributions by the optimality of an estimate
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Let $X$ be a second countable locally compact Abelian group containing no subgroup topologically isomorphic to the circle group $mathbb{T}$. Let $mu$ be a probability distribution on $X$ such that its characteristic function $hatmu(y)$ does not vanish and $hatmu(y)$ for some $n geq 3$ satisfies the equation $$ prod_{j=1}^{n} hatmu(y_j + y) = prod_{j=1}^{n}hatmu(y_j - y), quad sum_{j=1}^{n} y_j = 0, quad y_1,dots,y_n,y in Y. $$ Then $mu$ is a convolution of a Gaussian distribution and a distribution supported in the subgroup of $X$ generated by elements of order 2.
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We give a necessary and sufficient condition for symmetric infinitely divisible distribution to have Gaussian component. The result can be applied to approximation the distribution of finite sums of random variables. Particularly, it shows that for a large class of distributions with finite variance stable approximation appears to be better than Gaussian. keywords: infinitely divisible distributions; Gaussian component; approximations of sums of random variables.
Given a domain G, a reflection vector field d(.) on the boundary of G, and drift and dispersion coefficients b(.) and sigma(.), let L be the usual second-order elliptic operator associated with b(.) and sigma(.). Under suitable assumptions that, in particular, ensure that the associated submartingale problem is well posed, it is shown that a probability measure $pi$ on bar{G} is a stationary distribution for the corresponding reflected diffusion if and only if $pi (partial G) = 0$ and $int_{bar{G}} L f (x) pi (dx) leq 0$ for every f in a certain class of test functions. Moreover, the assumptions are shown to be satisfied by a large class of reflected diffusions in piecewise smooth multi-dimensional domains with possibly oblique reflection.
Consider finite sequences $X_{[1,n]}=X_1dots X_n$ and $Y_{[1,n]}=Y_1dots Y_n$ of length $n$, consisting of i.i.d. samples of random letters from a finite alphabet, and let $S$ and $T$ be chosen i.i.d. randomly from the unit ball in the space of symmetric scoring functions over this alphabet augmented by a gap symbol. We prove a probabilistic upper bound of linear order in $n^{0.75}$ for the deviation of the score relative to $T$ of optimal alignments with gaps of $X_{[1,n]}$ and $Y_{[1,n]}$ relative to $S$. It remains an open problem to prove a lower bound. Our result contributes to the understanding of the microstructure of optimal alignments relative to one given scoring function, extending a theory begun by the first two authors.
Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability $p$, then the probability of a correct verdict tends to one as the jury size tends to infinity (Condorcet, 1785). Recently, the first two authors developed a model where jurors sequentially receive signals from an interval according to a distribution, which depends on the state of Nature and on the jurors ability, and vote sequentially. This paper shows that to mimic Condorcets binary signal, such a distribution must satisfy a functional equation related to tail-balance, that is, to the ratio $alpha(t)$ of the probability that a mean-zero random variable satisfies $X >t$ given that $|X|>t$. In particular, we show that under natural symmetry assumptions the tail-balances $alpha(t)$ uniquely determine the distribution.
The stochastic dynamics of biochemical networks are usually modelled with the chemical master equation (CME). The stationary distributions of CMEs are seldom solvable analytically, and numerical methods typically produce estimates with uncontrolled errors. Here, we introduce mathematical programming approaches that yield approximations of these distributions with computable error bounds which enable the verification of their accuracy. First, we use semidefinite programming to compute increasingly tighter upper and lower bounds on the moments of the stationary distributions for networks with rational propensities. Second, we use these moment bounds to formulate linear programs that yield convergent upper and lower bounds on the stationary distributions themselves, their marginals and stationary averages. The bounds obtained also provide a computational test for the uniqueness of the distribution. In the unique case, the bounds form an approximation of the stationary distribution with a computable bound on its error. In the non-unique case, our approach yields converging approximations of the ergodic distributions. We illustrate our methodology through several biochemical examples taken from the literature: Schlogls model for a chemical bifurcation, a two-dimensional toggle switch, a model for bursty gene expression, and a dimerisation model with multiple stationary distributions.
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