We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition $xi$ is allowed to take the value +$infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., $xi$ is of the form $xi$ = g($Xi$ S) where $Xi$ is a continuous Markovian diffusion process and S is a hitting time of $Xi$ and g is a deterministic function 2) terminal conditions of the form $xi$ = $infty$ $times$ 1 {$tau$ $le$S} and 3) $xi$ 2 = $infty$ $times$ 1 {$tau$ >S} where $tau$ is another stopping time. For general $xi$ we prove the minimal supersolution is continuous at time S provided that F is left continuous at time S. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value $infty$ at terminal time S. The concept of solvability plays a key role in many of the arguments. Finally, we discuss implications of our results on the Markovian terminal conditions to solution of nonlinear elliptic PDE with singular boundary conditions.
{"title":"Continuity problem for singular BSDE with random terminal time","authors":"A. Popier, S. Samuel, A. Sezer","doi":"10.30757/alea.v19-49","DOIUrl":"https://doi.org/10.30757/alea.v19-49","url":null,"abstract":"We study a class of nonlinear BSDEs with a superlinear driver process f adapted to a filtration F and over a random time interval [[0, S]] where S is a stopping time of F. The terminal condition $xi$ is allowed to take the value +$infty$, i.e., singular. Our goal is to show existence of solutions to the BSDE in this setting. We will do so by proving that the minimal supersolution to the BSDE is a solution, i.e., attains the terminal values with probability 1. We consider three types of terminal values: 1) Markovian: i.e., $xi$ is of the form $xi$ = g($Xi$ S) where $Xi$ is a continuous Markovian diffusion process and S is a hitting time of $Xi$ and g is a deterministic function 2) terminal conditions of the form $xi$ = $infty$ $times$ 1 {$tau$ $le$S} and 3) $xi$ 2 = $infty$ $times$ 1 {$tau$ >S} where $tau$ is another stopping time. For general $xi$ we prove the minimal supersolution is continuous at time S provided that F is left continuous at time S. We call a stopping time S solvable with respect to a given BSDE and filtration if the BSDE has a minimal supersolution with terminal value $infty$ at terminal time S. The concept of solvability plays a key role in many of the arguments. Finally, we discuss implications of our results on the Markovian terminal conditions to solution of nonlinear elliptic PDE with singular boundary conditions.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49544036","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove the existence of non-trivial phase transitions for the intersection of two independent random interlacements and the complement of the intersection. Some asymptotic results about the phase curves are also obtained. Moreover, we show that at least one of these two sets percolates in high dimensions.
{"title":"On the percolative properties of the intersection\u0000of two independent interlacements","authors":"Zijie Zhuang","doi":"10.30757/alea.v18-40","DOIUrl":"https://doi.org/10.30757/alea.v18-40","url":null,"abstract":"We prove the existence of non-trivial phase transitions for the intersection of two independent random interlacements and the complement of the intersection. Some asymptotic results about the phase curves are also obtained. Moreover, we show that at least one of these two sets percolates in high dimensions.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47129940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the simple random walk in i.i.d. nonnegative potentials on the $d$-dimensional cubic lattice $mathbb{Z}^d$ ($d geq 1$). In this model, the so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential. The Lyapunov exponent depends on the distribution function of the potential, and the aim of this article is to prove that the Lyapunov exponent is strictly monotone in the distribution function of the potential with the order according to strict dominance. In particular, for the one-dimensional annealed situation, we observe that the Lyapunov exponents can coincide even under the strict dominance. Furthermore, the comparison for the Lyapunov exponent also provides that for the rate function of this model.
{"title":"Strict comparison for the Lyapunov exponents of the simple random walk in random potentials","authors":"Naoki Kubota","doi":"10.30757/alea.v20-36","DOIUrl":"https://doi.org/10.30757/alea.v20-36","url":null,"abstract":"We consider the simple random walk in i.i.d. nonnegative potentials on the $d$-dimensional cubic lattice $mathbb{Z}^d$ ($d geq 1$). In this model, the so-called Lyapunov exponent describes the cost of traveling for the simple random walk in the potential. The Lyapunov exponent depends on the distribution function of the potential, and the aim of this article is to prove that the Lyapunov exponent is strictly monotone in the distribution function of the potential with the order according to strict dominance. In particular, for the one-dimensional annealed situation, we observe that the Lyapunov exponents can coincide even under the strict dominance. Furthermore, the comparison for the Lyapunov exponent also provides that for the rate function of this model.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44171842","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We derive a lower bound for the probability that a random walk with i.i.d. increments and small negative drift $mu$ exceeds the value $x>0$ by time $N$. When the moment generating functions are bounded in an interval around the origin, this probability can be bounded below by $1-O(x|mu| log N)$. The approach is elementary and does not use strong approximation theorems.
{"title":"Bound on the running maximum of a random walk with small drift","authors":"Ofer Busani, T. Seppalainen","doi":"10.30757/alea.v19-03","DOIUrl":"https://doi.org/10.30757/alea.v19-03","url":null,"abstract":"We derive a lower bound for the probability that a random walk with i.i.d. increments and small negative drift $mu$ exceeds the value $x>0$ by time $N$. When the moment generating functions are bounded in an interval around the origin, this probability can be bounded below by $1-O(x|mu| log N)$. The approach is elementary and does not use strong approximation theorems.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45867613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuous-time Markov chain with an unknown stationary distribution. Specifically, we consider $N$ individuals such that each one of them is of one type among $K$ possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. As consequences of this result, we study the convergence in total variation of the process to stationarity and show a lower bound for the mixing time of the Moran process. Furthermore, we study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which is the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation, we also prove the existence of a cutoff phenomenon in the chi-square and the total variation distances when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the total variation distance to stationarity of the parent independent mutation process when initially all the individuals are of the same type has a Gaussian profile.
{"title":"On the spectrum and ergodicity of a neutral multi-allelic Moran model","authors":"J. Corujo","doi":"10.30757/ALEA.v20-18","DOIUrl":"https://doi.org/10.30757/ALEA.v20-18","url":null,"abstract":"The purpose of this paper is to provide a complete description of the eigenvalues of the generator of a neutral multi-type Moran model, and the applications to the study of the speed of convergence to stationarity. The Moran model we consider is a non-reversible in general, continuous-time Markov chain with an unknown stationary distribution. Specifically, we consider $N$ individuals such that each one of them is of one type among $K$ possible allelic types. The individuals interact in two ways: by an independent irreducible mutation process and by a reproduction process, where a pair of individuals is randomly chosen, one of them dies and the other reproduces. Our main result provides explicit expressions for the eigenvalues of the infinitesimal generator matrix of the Moran process, in terms of the eigenvalues of the jump rate matrix. As consequences of this result, we study the convergence in total variation of the process to stationarity and show a lower bound for the mixing time of the Moran process. Furthermore, we study in detail the spectral decomposition of the neutral multi-allelic Moran model with parent independent mutation scheme, which is the unique mutation scheme that makes the neutral Moran process reversible. Under the parent independent mutation, we also prove the existence of a cutoff phenomenon in the chi-square and the total variation distances when initially all the individuals are of the same type and the number of individuals tends to infinity. Additionally, in the absence of reproduction, we prove that the total variation distance to stationarity of the parent independent mutation process when initially all the individuals are of the same type has a Gaussian profile.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45652117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper analyzes the question of where one should stand when playing darts. If one stands at distance $d>0$ and aims at $ain mathbb{R}^n$, then the dart (modelled by a random vector $X$ in $mathbb{R}^n$) hits a random point given by $a+dX$. Next, given a payoff function $f$, one considers $$ sup_a Ef(a+dX) $$ and asks if this is decreasing in $d$; i.e., whether it is better to stand closer rather than farther from the target. Perhaps surprisingly, this is not always the case and understanding when this does or does not occur is the purpose of this paper. We show that if $X$ has a so-called selfdecomposable distribution, then it is always better to stand closer for any payoff function. This class includes all stable distributions as well as many more. On the other hand, if the payoff function is $cos(x)$, then it is always better to stand closer if and only if the characteristic function $|phi_X(t)|$ is decreasing on $[0,infty)$. We will then show that if there are at least two point masses, then it is not always better to stand closer using $cos(x)$. If there is a single point mass, one can find a different payoff function to obtain this phenomenon. Another large class of darts $X$ for which there are bounded continuous payoff functions for which it is not always better to stand closer are distributions with compact support. This will be obtained by using the fact that the Fourier transform of such distributions has a zero in the complex plane. This argument will work whenever there is a complex zero of the Fourier transform. Finally, we analyze if the property of it being better to stand closer is closed under convolution and/or limits.
{"title":"Where to stand when playing darts?","authors":"Björn Franzén, J. Steif, Johan Wastlund","doi":"10.30757/alea.v18-57","DOIUrl":"https://doi.org/10.30757/alea.v18-57","url":null,"abstract":"This paper analyzes the question of where one should stand when playing darts. If one stands at distance $d>0$ and aims at $ain mathbb{R}^n$, then the dart (modelled by a random vector $X$ in $mathbb{R}^n$) hits a random point given by $a+dX$. Next, given a payoff function $f$, one considers $$ sup_a Ef(a+dX) $$ and asks if this is decreasing in $d$; i.e., whether it is better to stand closer rather than farther from the target. Perhaps surprisingly, this is not always the case and understanding when this does or does not occur is the purpose of this paper. \u0000We show that if $X$ has a so-called selfdecomposable distribution, then it is always better to stand closer for any payoff function. This class includes all stable distributions as well as many more. \u0000On the other hand, if the payoff function is $cos(x)$, then it is always better to stand closer if and only if the characteristic function $|phi_X(t)|$ is decreasing on $[0,infty)$. We will then show that if there are at least two point masses, then it is not always better to stand closer using $cos(x)$. If there is a single point mass, one can find a different payoff function to obtain this phenomenon. \u0000Another large class of darts $X$ for which there are bounded continuous payoff functions for which it is not always better to stand closer are distributions with compact support. This will be obtained by using the fact that the Fourier transform of such distributions has a zero in the complex plane. This argument will work whenever there is a complex zero of the Fourier transform. \u0000Finally, we analyze if the property of it being better to stand closer is closed under convolution and/or limits.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46412255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider some problems related to the truncation question in long-range percolation. It is given probabilities that certain long-range oriented bonds are open; assuming that this probabilities are not summable, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. This question is still open if the set of vertices is $Z^2$. We give some conditions in which the answer is affirmative. One of these results generalize the previous result in [Alves, Hilario, de Lima, Valesin, Journ. Stat. Phys. {bf 122}, 972 (2017)].
{"title":"Truncation of long-range percolation models with square non-summable interactions","authors":"Alberto M. Campos, B. D. Lima","doi":"10.30757/alea.v19-41","DOIUrl":"https://doi.org/10.30757/alea.v19-41","url":null,"abstract":"We consider some problems related to the truncation question in long-range percolation. It is given probabilities that certain long-range oriented bonds are open; assuming that this probabilities are not summable, we ask if the probability of percolation is positive when we truncate the graph, disallowing bonds of range above a possibly large but finite threshold. This question is still open if the set of vertices is $Z^2$. We give some conditions in which the answer is affirmative. One of these results generalize the previous result in [Alves, Hilario, de Lima, Valesin, Journ. Stat. Phys. {bf 122}, 972 (2017)].","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42735318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric $U$-statistics, we demonstrate in which respect this approach yields fundamentally better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic.
{"title":"Normal approximation via non-linear exchangeable pairs","authors":"C. Dobler","doi":"10.30757/alea.v20-08","DOIUrl":"https://doi.org/10.30757/alea.v20-08","url":null,"abstract":"We propose a new functional analytic approach to Stein's method of exchangeable pairs that does not require the pair at hand to satisfy any approximate linear regression property. We make use of this theory in order to derive abstract bounds on the normal and Gamma approximation of certain functionals in the Wasserstein distance. Moreover, we illustrate the relevance of this approach by means of three instances of situations to which it can be applied: Functionals of independent random variables, finite population statistics and functionals on finite groups. In the independent case, and in particular for symmetric $U$-statistics, we demonstrate in which respect this approach yields fundamentally better bounds than those in the existing literature. Finally, we apply our results to provide Wasserstein bounds in a CLT for subgraph counts in geometric random graphs based on $n$ i.i.d. points in Euclidean space as well as to the normal approximation of Pearson's statistic.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42139579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P, we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation principles in an insurance market without arbitrage opportunities are discussed in [26] and [27], respectively.
{"title":"A characterization of progressively equivalent probability measures preserving the structure of a compound mixed renewal process","authors":"Spyridon M. Tzaninis, N. D. Macheras","doi":"10.30757/alea.v20-09","DOIUrl":"https://doi.org/10.30757/alea.v20-09","url":null,"abstract":"Generalizing earlier works of Delbaen & Haezendonck [5] as well as of [18] and [16] for given compound mixed renewal process S under a probability measure P, we characterize all those probability measures Q on the domain of P such that Q and P are progressively equivalent and S remains a compound mixed renewal process under Q with improved properties. As a consequence, we prove that any compound mixed renewal process can be converted into a compound mixed Poisson process through a change of measures. Applications related to the ruin problem and to the computation of premium calculation principles in an insurance market without arbitrage opportunities are discussed in [26] and [27], respectively.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47760171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Bermolen, Valeria Goicoechea, M. Jonckheere, E. Mordecki
We prove a large deviation principle (LDP) for a greedy exploration process on an Erdos-Renyi (ER) graph when the number of nodes goes to this http URL prove our main result we use the general strategy for the study of large deviations of processes proposed by Feng and Kurtz (2006), which is based on the convergence of non-linear semigroups. The rate function can be expressed in a closed form formula and associated optimization problems can be solved explicitly providing the trajectory of the large deviation. In addition we derive a LDP for the size of the maximum independent set discovered by such algorithm and analyze the probability that it exceeds known bounds for the maximal independent set. We also analyze the link between these results and the landscape complexity of the independent set and the exploration dynamic.
{"title":"Large Deviation Principle for the Greedy Exploration Algorithm over Erdös-Rényi Graphs","authors":"P. Bermolen, Valeria Goicoechea, M. Jonckheere, E. Mordecki","doi":"10.30757/alea.v19-16","DOIUrl":"https://doi.org/10.30757/alea.v19-16","url":null,"abstract":"We prove a large deviation principle (LDP) for a greedy exploration process on an Erdos-Renyi (ER) graph when the number of nodes goes to this http URL prove our main result we use the general strategy for the study of large deviations of processes proposed by Feng and Kurtz (2006), which is based on the convergence of non-linear semigroups. The rate function can be expressed in a closed form formula and associated optimization problems can be solved explicitly providing the trajectory of the large deviation. In addition we derive a LDP for the size of the maximum independent set discovered by such algorithm and analyze the probability that it exceeds known bounds for the maximal independent set. We also analyze the link between these results and the landscape complexity of the independent set and the exploration dynamic.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2020-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43018973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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