Stochastic Processes Solutions
52 stochastic process problems covering Poisson processes, Brownian motion, and core topics.
52 problems
Problem 1: find the probability that the nearest-neighbor simple random walk on T is recurrent
Let T be a Galton-Watson tree with offspring distribution Poisson(2). Conditioned on non-extinction of the branching process, find the probability that the near
Problem 2: Prove that with probability 1 there exist sequences with , such that and
Let be standard Brownian motion and let . Prove that with probability 1 there exist sequences with , such that and .
Problem 3: Prove
Under probability measure , let be a Markov chain on state space with transition kernel and invariant distribution . Define a subset-valued Markov chain by: giv
Problem 4: Prove that almost surely and in , and that
Let be i.i.d. nonnegative integer-valued random variables with Define the Galton--Watson process by and set Prove that almost surely and in , and that .
Problem 5: Prove that and have the same distribution
Let . Prove that and have the same distribution.
Problem 6: Prove that the sequence of random variables converges in the mean-square sense
Let be a martingale with , where is a constant. Prove that the sequence of random variables converges in the mean-square sense.
Problem 7: Prove that can be expressed as the difference of two nonnegative martingales
Let be a martingale with , where is a constant. Prove that can be expressed as the difference of two nonnegative martingales.
Problem 8: Prove that
Let , and denote , , so that is a renewal process. (1) Prove that . (2) If the distribution function of is , find the exact analytical expression for . (3) Usin
Problem 9: Let the constant
Let the constant . Let be a jump process on whose embedded chain is a simple random walk, and whose transition rate at vertex is , where is the Euclidean distan
Problem 10: Prove that there exists a constant such that: where
Let be a martingale or a nonnegative submartingale. Prove that there exists a constant such that: where .
Problem 11: Prove that:
Constants satisfy , , and for all , , , . Let the discrete-time Markov chain have state space with transition probabilities: Let , and denote , . Prove that: (2
Problem 12: Prove that for any ,
Suppose is a standard Brownian motion, and suppose is a continuous function with . Prove that for any ,
Problem 13: Prove that exists and
Let be a one-dimensional standard Brownian motion. Define Prove that exists and . (If one assumes the limit exists and proves , half credit may be awarded.)
Problem 14: Prove that the density function of is where ,
Let be independent continuous random variables with common density function . Denote by the -th smallest among . (a) Prove that the density function of is where
Problem 15: Connect all adjacent points (distance 1) in by edges, then delete the set and all edges…
Connect all adjacent points (distance 1) in by edges, then delete the set and all edges incident to them. Is the nearest-neighbor random walk on this graph recu
Problem 16: Find the stationary distribution of this Markov chain
Let . Consider the following Markov chain with state space : at each step, a coordinate is chosen uniformly at random; if all of its neighbors (if or , there is
Problem 17: prove that
Let be a one-dimensional standard Brownian motion with . Given , prove that .
Problem 18: Let nonneg
Let nonneg. random variables be independent and identically distributed. Define . (1) What is the necessary and sufficient condition for to be a martingale? In
Problem 19: Prove that the density function of is where ,
Let be independent continuous random variables with common density function . Denote by the -th smallest among . (a) Prove that the density function of is where
Problem 20: Prove that as , in the sense, and that
Let be i.i.d. integer-valued random variables. Define the Galton--Watson branching process by Let and . Prove that as , in the sense, and that . (Hint: Consider
Problem 21: Prove your conclusions
Remove all horizontal edges from the two-dimensional integer lattice except those on the -axis, obtaining a "comb-shaped" graph. Is the simple random walk on th
Problem 22: Prove:
Constants , , satisfy , , and for all , , , . Let be a discrete-time Markov chain with state space and transition probabilities: Let , and denote , . Prove: (2)
Problem 23: Prove that is stationary and ergodic
Suppose is a random variable with the following density function: Given , the random variable is uniformly distributed on . Prove that is stationary and ergodic
Problem 24: Prove that there is positive probability that the system never returns to the all-healthy state
Consider the contact process on the infinite 100-regular tree, described as follows. Each vertex has two states: healthy or infected. Each vertex is equipped wi
Problem 25: Find
Consider the following random walk starting from the root of an infinite binary tree: if the current state is the root, jump to a uniformly chosen child; otherw
Problem 26: Prove that is also a Gaussian process, and find for
Let be a Gaussian process. Define for . Prove that is also a Gaussian process, and find for .
Problem 27: Let be nonnegative i.i.d
Let be nonnegative i.i.d. random variables. Define . (2) What is the necessary and sufficient condition for to be a uniformly integrable martingale?
Problem 28: prove that
Let be a one-dimensional (standard) Brownian motion with . Given , prove that .
Problem 29: Prove that
Let with . Define with . Let . Prove that .
Problem 30: Prove the following results: (a) If , then is a martingale
Consider the asymmetric random walk, i.e., , , and . Prove the following results: (a) If , then is a martingale. (b) With initial position , let . Then for , Fo
Problem 31: prove that
Let be a sequence of i.i.d. random variables with , , . Let be the order statistics of uniform random variables on . Set , . Choose an appropriate -field , , so
Problem 32: Prove: (a) The series defining converges almost surely; (b) holds in both the almost sure and…
Let be a sequence of i.i.d. random variables with and . For , define , where are constants satisfying . Prove: (a) The series defining converges almost surely;
Problem 33: Prove:
Constants , , satisfy , , and for all , , , . Let be a discrete-time Markov chain with state space and transition probabilities: Let , and denote , . Prove: (2)
Problem 34: Find the expected time for a simple random walk starting from the origin to first reach
Consider the grid. Remove all horizontal edges except those on the -axis (resulting in a comb-shaped graph). Find the expected time for a simple random walk sta
Problem 35: Prove: (1) is a Poisson process with rate
Let be independent Poisson processes with rates respectively. Prove: (1) is a Poisson process with rate . (2) For , the probability that the first event comes f
Problem 36: Prove that
Consider a jump process with state space . For , let be the transition probability matrix of the jump process at time . Prove that .
Problem 37: Prove that this Markov chain is positive recurrent
Starting from the number 0, a fair coin is flipped each time. If the coin lands heads, the number increases by 1; if it lands tails, the number is divided by 2
Problem 38: Prove:
Consider the simple symmetric random walk , , i.i.d., with . (Note .) Let , and . Prove:
Problem 39: Prove that a.s., and also in the sense
Let be a branching process with and . Define if . Set , and let . Prove that a.s., and also in the sense. Finally, prove that .
Problem 40: Prove that form a Markov chain
Let and . Let be random variables taking values in such that The above holds for any sequence taking values ; here is a normalizing constant ensuring has total
Problem 41: Prove that, with probability one, the set of local extrema of a standard Brownian path is…
Prove that, with probability one, the set of local extrema of a standard Brownian path is dense. (Note: A point t is a local extremum if and only if there exist
Problem 42: Prove that
Let , and define , , so that is a renewal process. (1) Prove that . (2) If has distribution function , find the exact analytic expression for . (3) Using the ke
Problem 43: Prove that
Let be a one-dimensional (standard) Brownian motion with . Let and be a measurable set. Prove that
Problem 44: Find the expectation and variance of
Suppose that in a game, killing a certain monster has a fixed "drop rate" for equipment 1 and 2. Assume that after killing the monster, equipment 1 drops with p
Problem 45: Prove that is both strictly stationary and wide-sense stationary
Let be a continuous-time stochastic process. Give reasonable definitions for to be strictly stationary and wide-sense stationary. Let be a Poisson process with
Problem 46: Prove: (1) is a Poisson process with rate
Let be independent Poisson processes with rates respectively. Prove: (1) is a Poisson process with rate . (2) For , the probability that the first event comes f
Problem 47: Prove that is a Markov chain and find its invariant distribution
Let be a jump process on state space with transition rate matrix , assumed to be positive recurrent with invariant distribution . Let be an independent Poisson
Problem 48: Prove that if is a three-dimensional standard Brownian motion, then almost surely
Prove that if is a three-dimensional standard Brownian motion, then almost surely .
Problem 49: Prove that is also a Gaussian process, and find ,
Given that is a Gaussian process, define , where . Prove that is also a Gaussian process, and find , .
Problem 50: Prove:
Consider the simple symmetric random walk , , where are i.i.d. with . (Note .) Let , with . Prove:
Problem 51: prove that as , in the convergence sense, and that
Let be i.i.d. integer-valued random variables. Define the Galton--Watson branching process satisfying: If and , prove that as , in the convergence sense, and th
Problem 52: Determine whether the location of the maximum of on is unique
Let be a one-dimensional standard Brownian motion. Determine whether the location of the maximum of on is unique.