Questions tagged [pr.probability]
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
9,023 questions
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Probability of getting all pattern combinations in moving window over a vector of characters [closed]
So, I am in need of an indication of literature or where to start.
I am having a problem consisting of reading a vector of characters (for example, there are n=4 possible characters {A,B,C,D}) using a ...
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97
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Uniqueness of the solution to some SDE of state-dependent coefficient
This is a continuation of my question posted in Uniqueness of the solution to some SDE
Consider
$$X_t=X_0 + t + \int_0^t \frac{\sigma(s,X_s)}{1+m(s)}dW_s,\quad \forall t\ge 0,\quad\quad\quad (\ast)$$
...
- 1,334
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118
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Weak derivative of projection onto probabilist's simplex
Let $\Delta_n:=\{x\in [0,1]^n:\boldsymbol{1}^{\top}x=1\}$ denote the probabilist's $n$-simplex and let $P:\mathbb{R}^n\rightarrow\Delta_n$ denote the (Euclidean) metric projection onto this simplex ...
- 5,405
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114
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Why is Branching Brownian Motion log-correlated?
I need some references(or helps) on understanding why BBM is log-correlated. As I understand it, a random field on some metric space $V$ with distance $d$ is log-correlated if $$\mathbb{E}[X_u X_v]\...
- 715
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79
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Visualization PDF of distribution defined by quantiles
How can I visualise PDF of distribution defined by quantiles, that I predict with my neural network? Now I'm passing quantiles to the histogram, but I don't think it is the correct way for visualising....
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91
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Spectrally-weighted Stieltjes transform of random matrix $Z=XX^\top$ in terms of Stieltjes transform of $Z$ and the weighting function
Let $n$ and $d$ positive integers going to infinity such that $d/n \to \gamma \in (0,\infty)$. Let $X$ be a random $n \times d$ iid rows from $N(0,\Sigma)$, where $\Sigma = diag(\lambda_1,\ldots,\...
- 6,853
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84
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Determining the tails of a convolution from its behavior on a compact set
Let $p$ be a smooth (say, $C^\infty$, but this is not crucial) density on the interval $I=[0,1]$ and $g_\sigma$ be the density of $N(0,\sigma^2)$. Define $f=p\ast g_\sigma$. To what extent does the ...
- 19
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133
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Spectral CLT for random matrices with iid entries
Let $\lambda_1(A_n),...,\lambda_n(A_n)$ be the random eigenvalues of a random $(n \times n)$ matrix $A_n$. We can define the empirical spectral measure $\mu_n^{A_n}$ on $(\mathbb{C},\mathcal{B}(\...
- 1,295
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326
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Precise decay of density through Fourier transform
Suppose $f(x)$ is a probability density on $\mathbb{R}$. Let $\varphi(t)=\int e^{itx}f(x)dx$ denote the Fourier transform (characteristic function). It is well-known that if $\int |x|^p f(x)dx<\...
- 87
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Unexpected autocorrelations in sequence of primes modulo 4
It is well known that there is a little bias in the distribution of prime residues modulo 4. But the bias eventually vanishes. I looked at the first million primes, and the counts are as follows:
...
- 3,098
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162
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Iterated integrations by parts using the fractional Laplacian
Let $u \in C^\infty_c(\mathbb{\Omega})$ and $\varphi$ be an eigenfunction of the fractional Laplacian $(-\Delta)^s$ in $\Omega$ with eigenvalue $\lambda$. In what sense, if any, is it true that
$$\...
- 839
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255
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Sufficient conditions for decomposition of a bounded random variable into several small pieces
Given a random variable $X$ with $\mathsf{supp}\, X \subseteq [0,1]$ and $n$ positive numbers $h_1,\cdots,h_n$ with $\sum_{i=1}^n h_i=1$, I want to know some sufficient conditions for decomposing $X$ ...
- 550
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84
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Does fixed allocation increase the proportion of positively drifted Brownian motions surviving forever?
This is a continuation of Number of drifted Brownian motions that never hit zero under allocation
For each $n\ge 1$, consider $X^i_t=1+\beta t + W^i_t$ for $i=1,\ldots n$ and $t\ge 0$, where $\beta>...
- 1,334
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140
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Space derivative of Brownian local time
Let $\{L_x(t):(t,x)\in [0,T]\times\mathbb R\}$ be the local time of a Brownian motion $(B_t)_{t\in [0,T]}$, I know that the map $x\mapsto L_x(t)$ is $\alpha$-Hölder for $\alpha<1/2$ (uniformly in ...
- 515
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100
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Lognormal PDF in terms of the Meijer-G function
Is it possible to write this lognormal PDF in terms of the Meijer-G function?
$$f_{Y}(y)=\frac{10}{y\ln(10)\sqrt{2\pi}\sigma}\exp\left(-\frac{(-10\log_{10}(y) - \mu)^2}{2 \sigma^2}\right)$$
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81
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Relating sequence with or without replacement
I derived a relationship between sequences drawn with and without replacement for an application in genetics. The proof is easy enough, but I would rather find a source than provide a derivation of a ...
- 31
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86
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Expected diameter of a random point set
General problem: For a point set $S\subset X$ in a metric space $(X,d)$, let $\text{diam}(S)=\max_{x,y\in S}d(x,y)$. Given a distribution $P$ on $X$ and $m$ i.i.d. points $x_1,\ldots,x_m\sim P$, what ...
- 19
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LLN of random nearest neighbor function
There are two samples of iid random variates: $X=\{X_1,X_2,...,X_n\}$ and $Y=\{Y_1,Y_2,...,Y_n\}$. Further, $\forall i,j: X_i$ is independent of $Y_j$. The probability distributions $P,Q$ are unknown ...
- 89
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94
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"Cut norm" of conditional expectation has supremum on products of sets in sub-$\sigma$-algebra, or not?
I am reading Lovasz's book "Large networks and graph limits", and encountered the exercise that the stepping operator for graphons is contractive under the cut norm:
$$||W_P||_\square\leq||W|...
- 715
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249
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Variant on Janson-type inequality
Let us suppose that we are in the setting of Janson's inequality for Poisson-type deviations of increasing events. Specifically, we have independent Bernoulli variables $X_1, \dots, X_n$, and events $...
- 3,475
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117
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Estimate of cumulative probability of geometric Brownian motion
Let $B_\tau$ be the standard BM, $t$ be the initial time, $s$ be the time variable, $r$ and $\theta$ are positive constants. We also assume that $x$ is the initial position of the below geometric ...
- 54
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95
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Empirical estimation of Brenier map from data
Let $f:\mathbb R^d \to \mathbb R$ be a "nice" (say, continuous) function define $A = A_f := \{x \in \mathbb R^d \mid f(x) \ge 0\}$ and $B =B_f:= \{x \in \mathbb R^d \mid f(x) \le 0\}$, and ...
- 6,853
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60
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Reference request: Counting integer sequences in homogeneous linear recurrences
Are there references in the literature that deal with the probability of finding an integer sequence in a linear homogeneous recurrence with constant coefficients $ \in \mathbb{Z}$? (or provides a way ...
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101
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Realizations of alternative configurations
Consider a discrete distribution $P(\mathbf{X},Y)$ with $\mathbf X = \{ X_1, \dotsc, X_N \}$. I use the shorthand notation $p(\mathbf{x}, y)$ for $P(\mathbf{X}=\mathbf{x}, Y=y)$. Consider $P_\text{ind}...
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Independence of limits of asymptotically independent processes
Suppose $X, Y$ are $L^1$ random variables, and $X_t$ and $Y_t$ are real valued stochastic processes with $X_t, Y_t \in L^1$ for all $t$ such that the following convergences hold:
i) $X_t \to X$, $Y_t \...
- 6,213
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173
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Lemma 3.10 of paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain
I am reading a paper 'Periodic nonlinear Schrodinger Equation and Invariant measure' by J.Bourgain.
And I have a questions in the proof of lemma 3.10.
Please click the paper title for the link.
The ...
- 239
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99
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A martingale extension/interpolation problem
Let $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\geq 0},\mathbb{P})$ be a stochastic basis and let $N\in\mathbb{Z}^+$, $T>0$, $\{t_n\}_{n=1}^{N}$ be a partition of $[0,T]$ with $t_0=0,t_n<t_{n+1},t_N=...
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$|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)|=O_P(\frac{1}{\sqrt{n}})$ under $E(|X_1|)<\infty$?
For i.i.d. random variables $X_1,\dots, X_n$ with $E(|X_1|)<\infty$. Does the following equation hold?
$$
\left|\frac{1}{n}\sum_{i=1}^n X_i-E(X_1)\right|=O_P\left(\frac{1}{\sqrt{n}}\right)
$$
I ...
- 193
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Convergence of the probabilities that drifted Brownian motion with jump never hits zero (continuation)
This question can be seen as a continuation of my question at Convergence of the probabilities that drifted Brownian motion with jump never hits zero
Let $(W_t)_{t\ge 0}$ be a standard Brownian motion ...
user128095
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154
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Convergence of the probabilities that drifted Brownian motion with jump never hits zero
Let $X_t=2+t+W_t$ for $t\ge 0$, where $(W_t)_{t\ge 0}$ is a standard Brownian motion. For every $n\ge 1$, set $X^n_t:=X_t-{\bf 1}_{t\ge n}$. Denote respectively
$$\tau:=\inf\{t\ge 0:~ X_t\le 0\}\quad \...
user128095
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170
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Relationship between $L_1$ and $L_2$ distances of two Gaussian Mixture models
Given two Gaussian mixture models with
\begin{equation}
\begin{aligned}
f(x) &=\sum_{k=1}^{K} \pi_{k} \mathcal{N}\left(x \mid \mu_{k}, \sigma_{k}\right), \\
g(x) &=\sum_{i=1}^{N} \lambda_{i} \...
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330
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Lower-bound smallest eigenvalue of covariance matrix of $y = f(Ax)$, for $x$ uniform on unit-sphere
Let $A=(a_1,\ldots,a_)$ be a fixed $k \times d$ matrix (with $d$ large), and $x$ be a random vector uniformly distributed on the unit-sphere in $\mathbb R^d$. Let $f:\mathbb R \to \mathbb R$ be a ...
- 6,853
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70
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Where does the "mixing" occur in convex combination of Girsanov measures?
In this post, Ofer says that taking the convex combination of two Girsanov measures yields a drift $BF_1+(1-B)F_2$ where $B$ is a Bernoulli random variable with parameter $\lambda$, independent of the ...
user207651
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89
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Poisson spacings?
Assume that for every $n\geq 1 $ we are given a real random variable $X_n$ such that $(X_n-n)/\sqrt n$ follows the standard normal distribution. Furthermore, assume that the $X_n$ are independent. Fix ...
- 3,062
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185
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Probability that a $d$-dimensional Brownian bridge is greater than a given parameter
Let $(W_t)_{t\in[0,T]}$ be a Brownian bridge such that $W_0=a$ and $W_T=b$, the probability that $\forall t\in[0,T],W_t\geqslant x$ given the parameter $x\leqslant\min(a,b)$ is well known :
$$ \mathbb{...
- 186
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241
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k-secretary problem: not knowing the length of the queue
The secretary problem is a famous and old problem. You can find the basic definition of this problem here: https://en.wikipedia.org/wiki/Secretary_problem
Now I'm concerned with the k-secretary ...
- 109
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189
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Moment generating function of a stopped process from Wald's identity
In an exercise I am asked to prove the following Wald's identities: let $S_n$ be a simple random walk and $T$ a stopping time. Then for all $\lambda \in \mathbb R,$
$$
\mathbb E(e^{\lambda S_1}) = 1 \...
- 1,755
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133
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Convoluted Cantor-like measure which has a continuous component [duplicate]
Let $\mu$ be a finite measure on $\mathbb R$ which has no atoms, and no component continuous with respect to Lebesgue measure. An example is the law of the random variable
$$
\sum_{k\ge 1}3^{-k}X_k
$$...
- 1,352
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93
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Brownian Motion excursion distribution
Consider a Brownian Motion $(W_t)_t$ and to some $x\in\mathbb T_2$, where $\mathbb T_2$ is a two-dimensional torus, the circles $\partial B_{r_i}(x)$ around $x$ with radii $r_i= R(\frac \varepsilon R)^...
- 121
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47
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Exit probability on a finite interval
I have a question about the estimate of the exit probability on a finite interval. Given a $q$ function bounded and continuous, given the following SDE
\begin{cases}
dX_s=(\beta-q(s))X_sds+\frac{1}{2}...
- 1
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46
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Independence of variables predicted by the generator
Let $X$ be en compact metric set, and denote by $\mathcal{C}(X)$ the set of real continuous functions defined on $X$, endowed with the supremum norm $\|\cdot\|_{\infty}$. Let $\Omega$ be the generator ...
- 449
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148
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Classifying non atomic singular measures up to topological conjugacy
Write $\mathcal S$ for the set of probability measures on $[0, 1]$ that are non atomic and singular with respect to Lebesgue measure.
Two measures $\mu$ and $\nu$ in $\mathcal S$ are said to be ...
- 6,213
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769
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sub-exponential type upper bound on the Poisson probability
I posted this question on Math Stack Exchange, though I'm not satisfied with the answer I received.
Question:
For a Poisson random variable $Z$ with the parameter $\lambda,\,$ what would be a good ...
- 11
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0
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96
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Limit of a linear discrete-time stochastic process with uniform noise
I have posted this in the math and stats sites, but I am not sure where the proper forum for this question is. If it is not here, please go on and delete it.
Suppose we have a stochastic linear ...
- 1
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0
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136
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Diffusion process, hitting times, and harmonic functions
Let $E$ be a locally compact metric space. We consider a diffusion process $X=(\{X_t\}_{t \ge0 },\{P_x\}_{x \in E})$ on $E$ whose lifetime $\zeta$ may be finite: $P_x(\zeta<\infty)>0$ for some $...
- 721
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171
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A basic property of maximal correlation
Let $𝑋$ and $𝑌$ be random variables. Then the maximal correlation $\rho_{m}(X;Y)$ is defined as:
$$\rho_{m}(X;Y):=\max_{f,g}\mathbb{E}[f(X)g(Y)],$$
where the maximization is taken over real-valued ...
- 11
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133
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is there an example in planar graph that using probabilistic methods
The probabilistic method is a technique for proving the
existence of an object with certain properties by showing that
a random object chosen from an appropriate probability
distribution has the ...
- 1,851
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0
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72
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Integration of fractional function over Rice distribution
Let $a>2$ be a real variable. My objective is to find an approximation of the integral defined as
\begin{equation}
\int_0^{\infty } {\frac{1}{{1 + {x^a}}}} f\left( {x|y} \right)\, dx
\end{equation}...
- 377
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0
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340
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Lower-bound on expected value of norm of transformation of random vector with iid Rademacher coordinates
Let $n$ be a large positive integer. Let $A$ be a positive-definite matrix such with eigenvalues $\lambda_1 \ge \lambda_2 \ge \ldots \ge \lambda_n$ such that $\lambda_n = o(1) \to 0$ and $\lambda_i=\...
- 6,853
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86
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A non trivial example of a Gaussian semi-Markov process?
Let $(\Omega, \mathcal A, \mathbb P)$ be a probability space and $X=(X_t)$ a real Gaussian stochastic process.
Let $\mathcal F=(\mathcal F_t)$ be the filtration generated by $(X_t)$.
$X$ is Markov ...
- 143