With 80 million grandparents in the United States, you are one of many seniors searching for help in being a Godly grandparent. Pass the Legacy: 7 Keys for Grandparents Making a Difference is a book to encourage, inform and equip you in your important role while living in a culture that tells you your purpose is minimal. Many seniors desire to be Godly grandparents of their families. They simply do not know how. As you read this book, you will be empowered to live into the utmost calling God has on your life: to pass a legacy of faith in Jesus Christ to your grandchildren. What an awesome calling for someone living in a generation that struggles to find purpose in life. As you grab “The Seven Keys”, prepare to run an amazing race with the Lord. You will be living into His calling for you. Are you worried about your lovely granddaughter as she navigates this tumultuous world? Maybe you are concerned about your grown son. He seems to be moving farther and farther from the Lord. Then check out these “Seven Keys”. They are basic steps that can propel you towards leaving the most precious legacy of all: a legacy of faith in the Lord. FIRST KEY: Surrender Your Heart to Jesus Christ In order to pass a legacy of faith you must BE a person of faith! People are in one of three categories: those living in Egypt, those wandering around the desert or those abiding in the Promised Land. Discern which category describes you and learn the next step towards a life surrendered to the Lord. SECOND KEY: Read the Bible Daily Since we live in a noisy world, we must choose to set aside Quiet Time. By daily resting in God’s Word, we strengthen our spirits. Learn effective ways to delve into the Bible so that you can implement these Godly truths into your life. Then you will be powerfully prepared to encourage faith in your loved ones. THIRD KEY: Pray Fervently “Babushka Time”! Be like the Russian grandmothers in World War II who fell to their knees praying fervently for their loved ones. Discover powerful ways to prayerfully protect and guide the children and grandchildren the Lord has placed in your life and under your care. FOURTH KEY: Pursue Healthy Relationships with Loved Ones In a culture crowded with smart phones, computers and heavy schedules; relationships are diminished. This key encourages and equips grandparents and parents with grown children to run hard after healthy relationships with their children so that they can pass faith in Christ to the next generation. FIFTH KEY: Heal Broken Relationships We live in a tumultuous world. Many of us have strained, or broken, relationships with our children and grandchildren. Learn three steps towards resolving issues and mending broken relationships with loved ones. SIXTH KEY: Leave a Well-Written Legacy of Love With the world becoming heavily infiltrated by technology, the written word is decreasing. Study ways to write letters, create journals and scribe blessings to family members such that you communicate God’s love to your children and grandchildren. SEVENTH KEY: Pass Your Faith Every person has a God-story to tell. Maybe your story is how you accepted the Lord when you were eight years old, or forty eight years old. Maybe your God-story is that you have always struggled with faith in Christ. Regardless, you have a story that is worthy to be told to your children and grandchildren. God’s highest vision for grandparents is to pass a legacy of faith in His Son, Jesus Christ, to your loved ones. In this world where most seniors retire, God is calling YOU to become a life-changer in the lives of your children and grandchildren. You can make a difference by impacting their hearts for the Lord. Grab these keys and run the race marked divinely for you by God. Never has there been a more important job to do. Never has the need been more urgent.

Providing an elementary introduction to branching random walks, the main focus of these lecture notes is on the asymptotic properties of one-dimensional discrete-time supercritical branching random walks, and in particular, on extreme positions in each generation, as well as the evolution of these positions over time. Starting with the simple case of Galton-Watson trees, the text primarily concentrates on exploiting, in various contexts, the spinal structure of branching random walks. The notes end with some applications to biased random walks on trees.

The Bible teaches us to love God with all our hearts, minds, and strength, and our neighbors as ourselves. The Bible tells us that while we were yet sinners, Christ died for us. God loved us before we loved Him. The Bible tells us to walk in the love of Christ! It is my hope that as you read and study this book with the Holy Bible right there with you, you, too, will learn to walk in love!